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°and hEX. We want to solve U -
'\Au = h.
°
This is easy to do when 'P ~ c > on [0, 1] x JR. The boundary condition (Au(j) == 0) implies u(j) = h(j) for j = 0,1. Let f(x) = ax+{3 be the linear function such that k(x) = h(x)-f(x) vanishes at both endpoints x = 0,1. Let v(x) = u(x) - f(x). Then U- '\Au = h is equivalent to v - Bv = k, where
(Bv)(x) = '\{'P(x,v'(x) +a)v"(x) +1jJ(x,v(x) +f(x),v'(x) +a)} for v E :.D(B)
= {w
E C 2 (O, 1)
n X : Bw
-v
"
E X, w(O)
= w(l) = O}. This can be rewritten as
h - v - '\1jJ(x, v + f, v' + a) =--------'\'P(x, v'
+ a)
with (homogeneous) Dirichlet boundary conditions. Using the Green's function for the Dirichlet Laplacian on [0,1] this becomes the integral equation
1 1
v(x)
=
G(x, y)cp(y, v'(y)
+ a)-1[A- 1(h(y) -
v(y))
+ 'IjJ(y, v(y) + £(y), v'(y) + a)]dy
which can be solved by a fixed point argument (cf. [12]' [15]). For
0,
sup{IMi(x, TJ, ~)I : x E [0,1], ITJI ::; R, I~I ::; R, i == 0, I} < 00, aM sup{1 ax (x, TJ, ~)I: x E [0,1], ITJI ::; R, I~I::; R, i == 0, I} < 00, and Mo(x,TJ,~) ==
°
for all (TJ,~) E JR2 and x E {O, I}.
The integrability hypothesis is as follows. (B4) There exists a /-L E L1 (0,1) such that sup
VEC 1[O,1]
{exp[-
j,x1/2
0 on (anb n ). Then u(a n )
== u(b n ) == 0 and u'(a n ) >
bn
In
j [a(u'+ku+F(Ul)-F(U2))]'dx a(u' + ku + F(Ul) == °+ a(bn)u' (bn) - a( an)u' (an) :S °
=
an
F(U2))]~~
==
by the above and since Ul == U2 at an, bn . The other possible cases are similar, except that the boundary conditions must be used if an == or if bn == 00.
°
Condition (C3) can be dispensed with by replacing A by it closure. Condition (Cl) is the stumbling block. We want to solve u - AAu == h (given hEX and A > 0). Let
v(x) =
l
x
u(s)iJ(s)ds,
x E [0,00].
Then v E C[O, 00], v(O) == 0, v(oo) == Jooou{Jds E IR (since u E X). Now replace x E [0,00] by y == ~ tan-l(x) E [0,1]. Then A induces an operator B on C[O, 1] of the form
Bu == ')I(y)u + 'l/J(y, u, u')
Degenerate Nonlinear Parabolic Problems
109
with Wentzell boundary conditiflns. For the classical Kompaneets equation (2), "((y) ~ const. y2 near y == and "((y) ---* 00 as y ---* 1. In particular the previous work of Goldstein and Lin on (1) should be extended to cover the case when cp(x, ~), 1/J(x, 'TJ,~) are allowed to be discontinuous at x == 0,1. This has been done by Wang [20]. But the resulting theory (and the extension of the theorenl of Section 2 to this context) seems insufficient to produce a well-posedness result for (2).
°
Wang [19], [20] has extended the linear theory of (7) (cf. (4)) to the context of the Clement-Timrnermans theorem. The nonlinear extensions are still under investigation, so we shall not report on them here. We make two hypotheses. (D1) Let
a, a', k, k'
E C(O,
(0), with
0:,
(3
Define
> 1
°
on (0,00).
Au = j3[a(u'
+ ku)]'
for u E 1'(A) == {v E X == L 1 ((0, (0); (3(x)dx) : v E C 2 (0, (0), Au E X, a(v' as x ---* 0, 00 }.
°
(D2)
fooo f3(x) exp{ -
+ ku)
---*
fIX k(s)ds }dx < 00.
Theorem 2. (Wang [19]) Let (Dl), (D2) hold. Then the closure of A is densely defined and m-dissipative (i.e. (Cl) holds for A). Thus by the Hille- Yosida theorem (cf. [lO}), A generates a strongly continuous contraction (linear) semigroup on X.
Theorem 3. (Wang [19]) Let (Dl), (D2) hold. Let
y
= £2((0,00)); {3(x) exp{JX k(s)ds }dx).
Let A 2 be the operator A but with its domain modified in the obvious way so that it acts on Y rather than X. Then A 2 is a non-positive essentially selfadjoint operator.
Condition (D2) is the analogue of the Clement-Timmermans condition W E L 2 (0, 1) (cf. (Aa), (AI))' For the semigroup T == {T(t) : t 2: O} generated by A (or (A 2 ) on both X and Y, T is a positive semigroup in the sense that if f 2: (for f in X or Y), T(t)f 2: O. Because of Theorem 3, T is positive in two senses, namely T(t) is a positive operator (as above) in the lattice sense and T(t) is also a positive selfadjoint operator.
°
Let a(x) == x 4 , (3(x) == x 2 , k(x) == k == constant. When k > 0, then (D2) holds and both Theorem 2 and 3 apply. When k == 0, .A is m-dissipative and .A 2 is non-positive selfadjoint, but (D2) fails. (This corresponds to (Aa), (AI) holding but W 1-: L 1 (0,1).) When k < 0, the closure of A is not nl-dissipative; the conclusions of Theorems 2,3 fail in this case.
110
Goldstein et al.
REFERENCES 1. V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Space, Noordhoff, Leyden, 1976.
2. Ph. Benilan, Equations d'Evolution dans un Espace de Banach Quelconque et Applications, Thesis, Universite de Paris XI, Orsay, 1972. 3. Ph. Clement and C. A. Tilnmermans, On Co-selnigroups generated by differential operators satisfying Ventcel's boundary conditions, Indag. Math. 89 (1986), 379-386. 4. M.G. Crandall and T. M. Liggett, Generation of semigroups of nonlinear transformations on General Banach spaces, Amer. J. Math. 93 (1971), 265-298. 5. J. R. Dorroh and G. R. Goldstein, Quasilinear diffusions, in Evolution Equations, Control Theory and Biomathematics (ed. by Ph. Clement and G. Lumer), Dekker, New York (1994), 155-166. 6. J. R. Dorroh and G. R. Goldstein, A singular quasilinear parabolic problem in n dimensions, in preparation. 7. J. R. Dorroh and G. R. Rieder, A singular quasilinear parabolic problem in one space dimension, J. Diff. Equations 91 (1991), 1-23. 8. G. R. Goldstein, Nonlinear singular diffusion with nonlinear, boundary conditions, Math Meth. Appl. Sci. 16 (1993), 279-298. 9. G. R. Goldstein, J. A. Goldstein, and S. Oharu, The Favard class for a nonlinear parabolic problem, in Recent Development in Evolution Equations (ed. by A. C. McBride and G. F. Roach), Longman, Pitman Notes, Harlow (1995), 134-147. 10. J. A. Goldstein, Semigroups of Linear Operators and Applications, Oxford University Press, New York and Oxford, 1985. 11. J. A. Goldstein, The Kompaneets equation, in Differential Equations in Abstract Spaces (ed. by G. Dore, A. Favini, E. Obrecht, and A. Venni), Dekker, New York (1993), 115-123. 12. J.. A. Goldstein and C.- Y. Lin, Singular nonlinear parabolic boundary value problems in one space dimension, J. Diff. Equations 68 (1987), 429-443. 13. J. A. Goldstein and C.-Y. Lin, Highly degenerate parabolic boundary value problems, Diff. 1nl. Eqns. 2 (1989), 215-227. 14. J. A. Goldstein and C.-Y.Lin, An LP-semigroup approach to degenerate parabolic boundary value problems, Ann. Mat. Pura. Appl. 159 (1991), 211-227. 15. J. A. Goldstein and C.-Y. Lin, Parabolic problems with strong degeneracy at the spatial boundary, in Semigroup Theory and Evolution Equations (ed. by Ph. Clement, E. Mitidieri, and B. de Pagter), Dekker (1991), 181-191. 16. J.A. Goldstein and C.-Y. Lin, in preparation.
Degenerate Nonlinear Parabolic Problems
III
17. C.-Y. Lin, Degenerate nonlinear parabolic boundary value problems, Nonlinear Anal. TMA 13 (1989), 1303-1315. 18. G. Lumer, R. Redheffer, and W. WaIter, Estimates for solutions of degenerate second order differential equations and inequalities with application to diffusion, N onlinear Anal. TMA 12 (1988), 1105-1121. 19. K. Wang, The linear Kompaneets equation, J. Math. Anal. Appl., to appear. 20. K. Wang, The Generalized K ompaneets Equation, Ph.D. Thesis, Louisiana State University, 1995.
An Application of Measure Theory to Perfect Competition NEIL E. GRETSKY Department of rvlathematics, University of California, Riverside, CA 92521 (email: [email protected])
JOSEPH M. ()STROY Department of Economics, University of California, Los Angeles, CA 90024 (email: [email protected]) WILLIAM R,. ZAME Department of Econornics, University of California, Los Angeles, CA 90024 (email: [email protected])
Dedicated to Professor M.M. Rao on the occasion of his 65th birthday.
1
Introduction
This talk was given by the first author as part of the Festschrift held in November, 1994 at the University of California, Rjverside in honor of M.NI. R,ao's 65th birthday. The work presented here is a complement to a much larger project; this paper constitutes an alternative approach to a problerll discussed in (Gretsky, Ostroy, and Zame a). Other related material and background can be found in (Gretsky, Ostroy, and Zanle 1992; Gretsky, Ostroy, and Zame b). Since one of Professor Rao's influences was to instill in his students a love of vector measures, the purpose of this talk is to show how a measure theoretic approach allows an appropriate description of non-manipulation in very large econornies and to give a sample result. 113
Gretsky et al.
114
2
The Model
We start by describing a very large! assignment economy. There are many "stories" for which the assignment economy is an appropriate model; we choose one in which the commodities being traded are houses. As is the usual practice, an economy will be specified by listing the endowments and preferences of the agents who participate in the economy. Our story may be summarized as follows. There are two kinds of agents: buyers and sellers. Each buyer wishes to buy one house and is assumed to be initially endowed with enough money to buy any house; his preferences consist of a schedule describing what he is willing to pay for each house. Each seller is initially endowed with one house which she will be willing to sell if the price were right. The equilibria of the resulting exchange economy are the subject of study in the present paper 2 . We give a mathematical model for this story. Let the set of houses be a compact metric space H. For convenience, we introduce a fictional house 0 to H to indicate no trade. Denote H U 0 as Ho. The set of buyers is given by B == {b : H ---+ [0,1] I b is continuous} and the set of sellers is given by S == H x [0,1]. Denote the set of agents as I == BUS. We have assumed continuity for buyers partially for sirnplicity; a less restrictive condition on buyers leads to a much more complicated set of results as discussed in (Gretsky, Ostroy, and Zame
b). We interpret the information for buyers as meaning that b(h) is the reservation value that a buyer labeled as being of type b places on house h, i.e. b( h) is the maximum amount of money that a buyer of type b is willing to pay for house h. For sellers the interpretation is that a seller labeled as being of type s == (h, r) owns house h and places a reservation value r on this house, i.e. a seller of type (h, r) will not sell her house for an amount of money less than r. The reservation values are norrnalized to lie in the interval [0,1]. We can describe the preferences of an agent of type i by a utility junction defined on houses and money as:
vi(h)
vi(h) + m b(h) ifi == b,h E H if i == s_ == (h, r), h { -r o if h == 0
E H
Finally, the initial data of the economy consists of a population measure It E M+ (1) which is compactly supported. Such a measure specifies what types of, and in what relative quantities, agents are present. The condition of compact support is no restriction on the collection of sellers since the space S is a priori compact. However, this condition ensures that the collection of continuous functions for the buyers who are actually present in the economy form an equicontinuous family. The appropriate solution concept for such an exchange economy is Walrasian equilibrium, i.e. a price system on houses and a collection of trades among individuals such that individuals maximize utility, taking the price system as given, and such that markets clear. 1 Such economies are called non-atomic assignment economies since, in the attempt to describe large economies, it is common to assume the measures involved are non-atornic. The results given here do not actually depend on such an assumption. 2The goal of the assignment problem is to pair up buyers and sellers so as to maximize the total profit. The equilibria of the exchange economy provide an alternative, but equivalent, formulation to the optimization problem. See (Gretsky, Ostroy, and Zame 1992) for this and other equivalences.
Application of Measure Theory to Perfect Competition
115
A price system in our context should be a bounded Borel measurable function on the space of houses. For technical reasons 3 we agree to identify price systems as being identical if they agree almost everywhere in the natural measure /-lH induced on houses by the given population measure J-L; viz. the measure J-LH is the first marginal of the population measure J-Ls defined on S == H x [0,1]. Thus, a price system is a member of LOO (/-lH). The trading allocation of houses to buyers and sellers is described by a housing distribution measure y E
M+(I x Ho).
If the support of the housing measure /-lH is a proper subset of H, there is an ambiguity about prices for houses in H which are not in the support of /-lH. It is convenient to insist that prices for such houses be identically 1 so that no buyer ever strictly prefers to buy such a house. Moreover, for the fictional house 0 we extend p by setting p( 0) == O. It ,vas shown in (Gretsky, Ostray, and Zame 1992) in the present case of an equicontinuous family of buyer functions that Walrasian price systems can be chosen to be continuous on the support of /-lH. Given any Walrasian price system p, we will without further mention always reduce to the unique price system p' which agrees with p almost everywhere with respect to /-lH, is continuous on the support of J-LH, and is identically 1 on the complement of the support of /-lH· In order to define Walrasian equilibrium, we introduce an auxiliary concept. Given a price system p for houses, the indirect utility junction at these prices is defined to be V~( ) == {max{suPhEH{b(h) - p(h)}, O} t P max{p(h) - r, O}
if i == b if i == 8 == (h, r)
where the supremum in the first expression is the PH-essential supremum performed in the space LOO(PB) and the second expression is in LOO(ps). This gives the maximum utility attainable by an agent of type i subject to the budget constraint
-p(h)
m
== { p(h)
if i == b if i == s
DEFINITION: A Walrasian equilibrium for the assignment economy P is a pair (p, y) where p E LOO(/-lH) is a price system and y E Jvf+(I x Ho) is a housing distribution such that
(i) Yl == P,
(ii) y(B x G) == y(S x G) for every Borel set G
(Hi)
y (I x
~
H, and
Ho) == y { (b, h) E B x H : Vb ( h) - p (h) == V; (p) } +y{ (8, h) E S x H : vs(h) + p(h) == v;(p)} + y { ( i, 0) : Vi ( 0) == v; (p)}
These conditions state, respectively, that y is population consistent with P; that the houses received by the buyers equal those supplied by the sellers; and that, except possibly for a /-l-null set, all buyers and sellers are maximizing utility subject to the budget constraint defined by prices p. 3If individual point functions are used for price systems then there are several constructions the results of which cannot be guaranteed to be measurable. See (Gretsky, Ostroy, and Zame a) for rnore detail.
116
3
Gretsky et at.
Misrepresentation
A natural question in the study of perfect competition is whether any agent or coalition of agents can favorably manipulate the Walrasian equilibria by misrepresenting their type. We consider a subpopulation of a given ,population to be a measure 1/ E M+ (1) such that o :::; 1) :::; M. Note that, in an assignment economy, trading really occurs only between pairs of rnatched individuals - one buyer and one seller. Consequently, without loss of generality, we may restrict attention to subpopulations consisting solely of buyers or sellers. We first consider subpopulations of buyers only. Thus, we start with a population /-L == (MB, /-Ls) and a subpopulation 1/ of MB for which we will measure the results of misrepresentation. Denote by Ll the diagonal set of B x B. DEFINITION: An announcement of misrepresentations for the subpopulation a measure a on B x B such that
(i)
1/
of buyers is
al == /-LB,
(ii) II a 211
== II,LLB 11, and
(iii) (a 1.6. c ) 1
:::; 1).
The interpretation is that a is a distribution on B x B which describes announcements of types; here, the pair (b, b') means that an agent of type b announces himself to be of type b'. (Notice that an agent may misrepresent his preferences but not his endowment.) The three conditions are, respectively, that the population implicit in the announcement is consistent with the given population, that population mass is conserved, and that the actual misrepresenters are contained in the subpopulation 1/. The misrepresentation leads to the "new" economy given by the population (0:2, ,LLs) and to a resulting Walrasian equilibrium
(yQ, pQ). Given a population measure /-L, a subpopulation of buyers 1/, an announcement 1/, and a choice (yQ, pQ) of a Walrasian equilibrium for the economy given by the population measure (a2' /-Ls), an outcome of the announcement is a measure TJQ E M+(B x B x Ho) such that DEFINITION:
a of misrepresentations for
(i) (ii)
TJ?2 == 0:, and TJ2,3 == yet.
The measure TJQ is a distributional description of house assignments with respect to agents' true and announced types. The total utility of this outcome to all buyers is the aggregate of the outcome that each agent receives in this "new" economy measured in terms of that agent's true preferences. This total utility can be expressed as
r
{b(h) - pQ(h)}d1]Q(b, b', h)
} BxBxHo
Unfortunately, as it stands, the distribution 1] is not very useful for two reasons: (i) TJ lacks descriptive power in that the structure of the misrepresentation is not clear; (ii) 1] lacks
Application of Measure Theory to Perfect Competition
117
technical power in that there is not a convenient 4 way to prove any characterization theorems about manipulation in this formulation. We recall a classical result in the theory of vector measures.
PROPOSITION 1 Let (0, E, j.L) be a finite complete measure space and let X be a Banach space. If a countably additive vector measure F : E ~ X* satisfies IIF(E) 11 S KIj.L(E) I for all E E E for .some constant K, then there exists a weak* -measurable function f : 0 ~ X* such that for all x E X and for all E E 2: (F(E), x)
=
le (J(w),
x)dfl(W)
We call the function f a Gelfand density (or a weak* density) for F with respect to j.L. We write F(E) == J fdfL weak* in X*. This weak* representation theorem is equivalent to the fact that every bounded linear operator T from L 1 (0, E, j.L) to X* has a weak* density. In fact, the vector measure F and the operator T which correspond to each other under the map F(·) == T(X.) have the same weak* density. The Gelfand integral and its properties were introduced in (Gelfand 1936; Gelfand 1938). The representation theorem was proved by Gelfand in the special case that X is a separable Banach space's . The general case is proved by means of a lifting theorem as in (Dinculeanu and Uhl 1973). A more detailed discussion may be found in (Diestel and Uhl 1977). In the present model, it has been assumed that the collection H of houses is a compact Hausdorff space. Consequently, the Banach spaces of continuous functions C(H), C(B), and C(8) are separable and we may apply Gelfand's original theorem in these cases for misrepresentation outcomes. We start with housing allocations. For clarity, we will write the argument of a measure-valued density as a subscript; e.g. for'l/J : I ~ M+(Ho), we will write 'l/Ji(E) instead of 'l/J(i, E) or (1/J(i))(E).
THEOREM 1 Let y E M+(I x Ho) be a Walrasian allocation for the economy given by the population measure j.L. Then there is a weak* -measurable function 'ljJ : I ~ M+(Ho) with II1/Jill == 1 a.e.{/t} such that for any A ~ I y(A,·)
=
i
1jJdfl
weak' in M+(H)
In fact, dy(i, h) == 1/Ji(dh)dj.L(i) Proof: Define F : SI ~ M+(Ho) by (F(E))(G) == y(E, G) for each Borel set E ~ I and each Borel set G ~ Ho. Since Ho is compact, l\1(Ho) is the dual of the space of continuous functions C(Ho). It follows easily from y E M+(I x Ho) that F is a countably additive vector measure. Moreover, since y is a Walrasian allocation for IL, vve have that IIF(E)11 == (F(E))(Ho) == y(E, Ho) == Yl(E) == JL(E). Consequently, by Gelfand's theorem there exists a weak* density 1/J : I ~ M+(Ho) such that for all continuous functions x on Ho and for all Borel E ~ I
4 However, see (Gretsky, Ostroy, and Zame a) for an alternative approach which was developed after this talk was given. 5The speaker is grateful to J.J. Uhl, Jr. for pointing out this historical fact.
Gretsky et al.
118
which can be written as
( y(E, dh)x(h) == { ( x(h)d'l/Ji(h)dj-l(i)
lHo
lE lHo
or, equivalently,
dy(i, h) == 1/)i(dh)dj-l(i) Note that for any A a Borel subset of I we have j-l(A) == y(A, Ho) == II'l/Jill == 1 a.e.[j-l]. 0
J II'l/Jilldj-l(i)
so that
With this theorem there is a complete description in terms of weak* densities for the outcome of a misrepresentation.
THEOREM 2 Let (yQ, pQ) be any Walrasian equilibrium and TJQ be any outcome for a population measure j-l, a buyer subpopulation v, and an announcement a of misrepresentations for v. Then there exist a weak* -measurable junction 1/)Q : B ~ M+ (Ho) UJith 11'l/J~ 11 == 1 a.e.[j-lB] and a weak*-measurable junction a : B ~ M+(B) with Ilabll == 1 a.e.[,LB] such that
yCt(E,·)
=
a.(E,·) =
k k
weak' in M(Ho)
1/J'b- d/LB(b')
weak' in M(B)
abd/LB(b)
and
Proof: Given the announcement a of misrepresentations, we apply Gelfand's theorem to the vector measure (Z(E)) == a(E, F) and theorem 1 to the Walrasian allocation yQ. 0 The total utility to all buyers in the outcome of the announced economy
(
lBxHxHo
{b(h) - pQ(h) }dTJ(b, b' , h)
may be rewritten by Theorem 2 as
Thus, the total utility to the misrepresenting subpopulation v is
{ ( r {b(h) -
lB lB lHo
pQ(h)}'l/J~(dh)ab(db')dv(b)
With no manipulation allowed, a is ILB normalized on the diagonal mass 6b E M(B). In this case, the utility to the subpopulation v is
{ { ( {b(h) - P(h)}'l/Jb (dh)6b(db')dv(b)
lB lB lHo
l
which simplifies to
{ ( {b(h) - P(h)}'l/Jb(dh)dv(b)
lB lHo
~;
i.e. ab is the point
Application of Measure Theory to Perfect Competition
119
Consequently, the increase in utility to the misrepresenting subpopulation v of buyers resulting from the announcement a in the economy J-L is
U~(v) ==;; ;; ;; B
B
{b(h) -
Ho
pO(h)}7/J~(dh)ab(db')dv(b)
-;;;; {b(h) - P(h)}7/Jb(dh)dv(b) B
Ho
Note that the increase in utility may be negative. Moreover, we have in the notation sup(v) on the choices of y O and T); these need not be unique for a pressed the dependence of given J-L, v, and a. A similar expression can be developed for subpopulations of sellers. Now the announcement a is defined on S x S and this leads to a distributional description T) on S x S x Ho of houses as a result of true and announced types. The utility to the misrepresenting subpopulation v is
U:
(
{pO(h) - a(s) }d7J(s, s', h)
iSXSXHo
which can be rewritten in a similar manner to that of Theorem 2 as
where y O (', F) == IF cP~,dJ-Ls(s') weak* in M(Ho) with IlcP~' 11 == 1 a.e.[J-Ls) and a(E,') IEasdJ-LS(s) weak* in M(S) with Ilasll == 1 a.e.[J-Ls). Since a seller type can misrepresent only her preferences, viz. the reservation value for her own house, the measure as which describes how the type s misrepresents is a measure on S == H x [0, 1] which is supported on the set {1f s} X [0, 1] and consequently can be described as a measure Ps on [0,1]. Moreover, if a seller of type s were to announce herself as being type s', then it would have to follow that s' == (h, r') where h == 1f" and r' E [0, 1] since the house endOWlllent cannot be misrepresented. Consequently, cP~' == tlO(r')c5 h + {I - uO(r') }c5o is a measure on Ho in which the latter terrn represents no-trade activity and thus will not contribute to the utility integral. So we may rewrite the utility integral as
({
is i[O,l]
{pQ(1fs) - a(s)}uO(r)dps(r)dv(s).
We can compute the increase in utility to the misrepresenting subpopulation v of sellers resulting frolll the distribution Cl of announcements in the economy J-L to be
U~(v) ==
{
(
is' i[o,l]
{pO(1fs) - a(s)}uO(r)dps(r)dv(s) -
1.' {p(1fs) s
a(s)}u(s)dv(s)
As we did in the case for subpopulations of buyers, we again suppress in the notation the dependence of U/~ (v) on the choices of yO and 7J.
4
Manipulation
DEFINITION: The economy J-L is non-manipulable if there is a Walrasian price p E P(J-L) with the property that given E > 0 there is (~ > 0 such that for any subpopulation v of buyers or sellers satisfying I/vll < 6, any distribution Cl of announcements of misrepresentations for v, any Walrasian equilibrium (yO, pO) for the misrepresented econOlllY, and any outcome T) of the announcement it follows that
120
Gretsky et at.
An economy being non-manipulable is a manifestation of it being perfectly competitive in the sense that asymptotically small coalitions 6 cannot favorably manipulate the economy by misrepresentation. This turns out to be equivalent to a number of other conditions including stability of the Walrasian price correspondence and differentiability of the function totaling the gains from all trades. We give a sample result. The Walrasian price correspondence is a correspondence (a possibly multiply-valued function) P : M + (I) -t L ~ (jj H) which takes each economy described by a population measure to the collection of Walrasian prices for that economy. We will consider the stability question for the subset of population measures which are absolutely continuous with respect to a fixed measure jj. If v is such a measure then it has Radon-Nikodym derivative dv / d/-L E L~ (/-L); moreover, v is a subpopulation if and only if dv/d/-L :S 1 a.e.[jj]. Our restriction gives the correspondence PJl: L~(/-L) -t L~(J-lH) where PJl(h) == P(v) for h == dv/djj.
THEOREM 3 The economy /-L is non-manipulable if and only if the (restricted) price correspondence PJl : L~ (/-L) -t L~ (jjH) is norm-norm continuous at 1. Proof: For the sake of argument, we consider a subpopulation of buyers. Assume that PJl is norm-norm continuous at 1. We need to compare the expression
to the expression
hhHo {b(h) - P(h)}1/Jb(dh)dv(b). . B
The latter may be written as
{ { ( {b(h) - p(h) }1/Jb' (dh)cSb(b')dv(b).
lB lB lHo utility vb is
Note that the indirect continuous with respect to the sup norm topology on prices. Thus, although the Walrasian allocations for nearby prices need not be close in variation to the given allocation for jj, the utility of such allocations is close to the utility of the given allocation, i.e.
IItl {b(h) - pQ(h)}1/JII (dh) is continuous at 0 with respect to
Q.
L
{b(h) - p(h)}'1Pb,(dh) II
Consequently,
UQ(v) lim _Jl_ == 0 II v ll--rO
Ilvll
as desired. Conversely, if PJl is not norm-to-norm continuous at 1, then there exists some E > 0 such that for every cS > 0 there is a subpopulation v and an announcement Q consistent with v 6S mall coalitions serve as proxies for the individual agents which would be the objects of concern in finite economies.
Application of Measure Theory to Perfect Competition
such that IIvll < cS and of v it follows that
IlpQ - pll
Iv;(pQ) - v;(p)1
2: f for some pQ
121
E PtL(l - dv /dJ-L). Hence for b in the support
I sup{b(h) - pQ(h)} - sup{b(h) - p(h)}1 h
h
> IlpQ - pll Consequently,
U:(v)/llvll
need not converge to O. 0
References Diestel, J. and J. J. Uhl, Jr. (1977). Vector Measures. Number 15 in Mathematical Surveys. Providence: American Mathematical Society. Dinculeanu, N. and J. J. Uhl, Jr. (1973). A unifying Radon-Nikodym theorem for vector measures. Journal of Multivariate Analysis 3, 184-203. Gelfand, 1. M. (1936). Sur un lemme de la theorie des espaces lineaires. Comm. Inst. Sci. Math. Mec. Univ. de Kharkoff et Soc. Math. Kharkoff (4) 13,35-40. Gelfand, 1. M. (1938). Abstrakte funktionen und lineaire operatoren. Matematicheskii Sbornik (New Series) 4 46, 235-286. Gretsky, N. E., J. M. astroy, and W. R. Zame. Perfect competition in the nonatomic assignment model: The continuous case. Forthcoming. Gretsky, N. E., J. M. astroy, and W. R. Zame. Perfect competition in the nonatomic assignment model: The discontinuous case. Forthcoming. Gretsky, N. E., J. M. astroy, and W. R. Zame (1992). The nonatomic assignment model. Economic Theory 2, 103-127.
Dilations of Hilbert-Schmidt Class Operator-Valued Measures and Applications YUICHIRO KAKIHARA Department of Mathematics, University of California, Riverside, Riverside, CA 92521-0135, U. S. A. Dedicated to Professor M. M. Rao on the occasion of his 65th birthday
ABSTRACT The space of Hilbert-Schmidt class operators has a gramian structure, i.e., a trace class operator valued inner product. A gramian orthogonally scattered dilation of a Hilbert-Schmidt class operator valued measure is considered. Several new characterizations of it are given. An application to Hilbert space valued second order stochastic processes is made, where some equivalence conditions are given for a process to have an operator stationary dilation.
1. INTRODUCTION The orthogonally scattered dilation of Hilbert space valued measures has been fully studied and, in Section 2, we shall state basic results on it. The purpose of this paper is to consider gramian orthogonally scattered dilation of Hilbert-Schmidt class operator valued measures and its application to Hilbert space valued second order stochastic processes. Let H, K be a pair of complex Hilbert spaces. B(H) denotes the algebra of all bounded linear operators on Hand T(H) the Banach space of all trace class operators on H. S(K,H) denotes the Hilbert space of all Hilbert-Schmidt class operators from K into H. S(K, H) has SOll1e nice properties and among them is a gramian structure. That is, 123
Kakihara
124
S(K, H) is a left B(H)-module with the operator multiplication from left and if we define [x, y] == xy* E T(H) for x, y E S(K, H), then [".] satisfies that (1) [x, x] 2: 0, and [x, x] == 0 if and only if x == 0; (2) [x + y, z] = [x, z] + [y, z]; (3) [ax, y] == a[x, y]; (4) [x, y]* == [y, x], where x, y, z E S(K, H) and a E B(H). The T(H)-valued inner product [', .] is called the gramian in S(K, H). We say that S(K, H) is a normal Hilbert B(H)-module. In Section 3, we characterize those S(K, H)-valued measures which have gramian 01'thogonally scattered dilations. In Section 4, a new necessary and suffifient condition for an H -valued second order stochastic process on a locally compact abelian group to have an operator stationary dilation is given together with known conditions. All the contents of this paper will be included in detail in the monograph [6].
2. ORTHOGONALLY SCATTERED DILATION Let (8,21) be a measurable space. ca(21, K) denotes the set of all K -valued bounded c.a. (countably additive) measures on (8,21). The semivariation II~II (A) of ~ E ca(2t, K) at A E 2l is defined by
II~II(A) = sup {11 ~ Q~~(.6.)t : IQ~I ::; 1, .6. E
7[
E
II(A) },
(2.1)
where I1(A) denotes the set of all finite 2l-measurable partitions of A and II·IIK the norm in K. ~ is said to be orthogonally scattered (0. s.) if (~( A), ~ (B) ) K == 0 for every disjoint pair A, B E QC where (" ')K is the inner product in K. caos(2t, K) denotes the set of all o.s. measures in ca(2t K).
DEFINITION 2.1 (1) ~ E ca(2t, K) is said to have an orthogonally scattered dilation (o.s. d.) if there exist a Hilbert space Jt containing K as a closed subspace and an Tj E caos(21, Jt) such that ~ == JTj, where J : it -t K is the orthogonal projection. The triple {1], Jt, J} is also called an O.s. d. of ~. (2) ~ is said to have a spectral dilation if there exist a Hilbert space .R, a (weakly c.a.) spectral measure E(·) in it, an operator S E B(it,K) and a vector 'l/J E .R such that ~(.) == SE(·)'l/J, where B(.R,K) is the Banach space of all bounded linear operators from .R into K. Let X and Wbe two normed linear spaces. A linear operator T : oX ---+ ~ is said to be absolutely 2-summing if 7f2(T) < 00, where 1r2(T) == inf{C > 0 : (2.2) holds}:
for any n 2: 1 and
Xb' .. ,X n
E
x,
Dilations of Operator-Valued Measures
125
where II'II~ and 11·llx· are norms in !D and X·, respectively. Let LO(8) be the set of all complex valued ~-simple functions on 8. For! E LO(8) consider the sup norm 11/1100 defined by 11/1100 == sup I/(t)l. Then (LO(8), 11.11(0) becomes tEe
a normed linear space. Let ~ E ca(~, K) and define the integral of I ==
n
l: a j 1Aj
E LO(8)
j=1
w.r.t. (with respect to)
~
~
over A E
in an obvious manner by
1 d~ = t Q;j~(Aj f
A
n A),
j=l
where lA is the indicator function of A. Then the following theorem is known (cf. Niemi [10, 11], Rao [13] and Rosenberg [14]):
THEOREM 2.2 Let ~ E ca (21, K). Then the following conditions are equivalent: (1) ~ has an a.s.d. (2) ~ has a spectral dilation. (3) There exists a constant C > 0 such that for any n 2: 1 and 11, ... ,In E LO(8)
till Ji J=1
by
e
2
d1,11
K
~ ell t
J=1
2
(2.3)
IJi' 11 00 .
(4) The operator Se : (LO(8), 11·1100) -t K is absolutely 2-summing, where Se! == fe f d~ for f E LO(8). (5) There exists a positive finite measure v E ca(m, IR+) such that
1vhere
jR+
== (0, (Xl). In this case,
1/
Se
is defined
is called a 2-majarant of ~.
It follows frolll the Grothendieck's inequality (cf. Grothendieck [3] and Lindenstrauss and Pelczyri.ski [7]) that the inequality (2.3) holds with C == Kcll~II(8)2 where Kc is the Grothendieck constant and "~II (8) is the total semivariation of (cf. Rosenberg [14]). Therefore we have:
e
COROLLARY 2.3
Every ~ E ca(21, K) has an o.s.d.
3. GRAMIAN ORTHOGONALLY SCATTERED DILATION We consider S(K, H)-valued c.a. measures on (8,21) and assume that H is separable, so that H has a countable CONS (complete orthonormal system). As in the Introduction, the T(H)-valued gramian [".] in S(K, H) is considered.
DEFINITION 3.1
An S(K, H)-valued measure ~ E ca (21, S(K, H)) is said to be
gramian orthogonally scattered (g.o.s.) if [~(A),~(B)] == 0 for every disjoint A, B E
Kakihara
126
cagos(m,S(K,H)) denotes the set of all g.o.s. measures in ca(m,S(K,H)). ~ E ca(m, S(K, H)) is said to have a gramian orthogonally scattered dilation (g.o.s.d.) if there exist a Hilbert space Jt containing K as a closed subspace and an TJ E cagos(m, S(fi, H)) such that ~ == PTJ, where P : S(.R, H) ---t S(K, H) is the gramian orthogonal projection, i.e., p. is an orthogonal projection which satisfies that
2L
[P 2 x, y] == [Px, y] == [x, Py]'
x,y E S(fi, H).
The triple {TJ, Jt, P} is also called a g. O.s. d. of ~' As we mentioned in Section 2, every Hilbert space valued bounded c.a. rneasure has an o.s.d. But not every Hilbert-Schluidt class operator valued measure has a g.o.s.d. Thus we shall give SOl1le necessary and sufficient conditions for the g.o.s.d. Let ~ E ca(2l, S(K, H)) and LO (8 ; B(H)) be the set of all B(H)-valued 2l-sil1lple n
functions on 8. The integral of ==
L:
a j lA) E LO(8; B(H)) W.r.t. ~ over A E
m is
j=l
defined by
m
which is in S(K,H).
Let F E ca(m,T(H)).
For ==
L:
ajIA), W
j=1
LO(8; B(H)) the integral of (, w) w.r.t. F over A
1
1
(CP, '11) dP =
dF '11*
A
A
=
E
ft
2l is defined by
ajF(Aj
n B k n A)bk·
j=1 k=1
Let us put T+(H) == {a E T(H) : a ~ O}. With these preparations we prove the following proposition which was lllentioned without proof in Kakihara [4, 3.9. Proposition].
PROPOSITION 3.2 Let ~ E ca(m,S(K,H)). Then, ~ has a g.o.s.d. if and only ifii has a T+(H)-valued 2-majorant FE ca(2t,T+(H)). That is, (3.1 )
Proof: Suppose that ~ has a g.o.s.d. {TJ,fi,P}. Put F(-) == [TJ('),TJ(')] == TJ(-)TJe)*, then n
F E ca(2l, T+ (H)). For ==
I:: aj lA
j
E LO (8 ; B(H)) we have that
j=l
~L d7] = since
~
l
d(P7]) =
l
de
== PTJ and P commutes with the module action of B(H) and hence
Dilations of Operator-Valued Measures Conversely~
127
suppose that (3.1) holds and define M : 2! x 2!
n B)
M(A, B) == F(A
M~(A,
-
B),
-7
A, B
T(H) by E
2!,
where M~ (A, B) == [~(A), ~(B)]. Then we see that M is a positive definite kernel on 21 x 21 in the sense that
L
ajM(Aj , Ak)ak 2: 0
j,k for any n ~ 1, aI, ... ,an E B(H) and AI, . .. ,An E 21. Thus there exist a reproducing kernel normal Hilbert B(H)-module Y of M containing S(K, H) as a closed sublnodule and an 1] E cagos(2!, Y) such that ~ == P1], where P : Y -7 S(K, H) is the granlian orthogonal projection (cf. [4]). By the structure theorem (cf. Ozawa [12]) there exists a Hilbert space Jl such that Y ~ S(Jl, H), i.e., Y and S(K, H) are isomorphic as normal Hilbert B(H)-modules, and K can be regarded as a closed subspace of R. Therefore, {1], .R, P} is a g.o.s.d. of ~. Let ~ E ca (2!, S(K, H)). The operator semivariation 11~llo(A) of ~ at A E 21 is defined by
11~lIo(A) = sup {II ~ at.~(Li)L where
: at.
E B(H),
11· Ila is the Hilbert-Schmidt norm.
Ilat.11
~ 1, Li E
1r
E I1(A)},
Now ~* defined by ~*(.) == ~(-)* is In E 21 is defined by
ca (21, S(H, K)). The strong sernivariation II~* lis (A) of ~* at A
IIClls(A) = snp
{II ft
C(Li)1>t.t : 1>t.
E
H,
111>IIH
~ 1}.
Note that ~*4> defined by (~*4»(') == ~*(')4> is in ca(2t, K) for each 4> E H. Then we have the following proposition (cf. Kakihara [5, 5.2 Theorem, 5.3 Corollary and 5.7 Theorenl]). The part (2) ::::;> (1) was proved in Truong-van [17].
PROPOSITION 3.3 (1) ~ has a g.o.s.d. (2) 11~llo(8) < 00. (3) 11~*lls(8) < 00.
For ~ E ca (2t, S(K, H)) the following statements are equivalent:
(4) For some CONS {4>k}r=l in H there exists a family {17k,Rk,J k }r=l of o.s.d.'s of 00
{~*4>k}~I ~ ca(21,K) such that
2: 1117k(8)/Il < 00, k
k=l
where 1I·II.~k is the norm in Rk for
k 2: 1.
°
L (8 ; K) denotes the set of all K -valued 2l-simple functions on 8 and fa (21, K) the Banach space of all K-valued finitely additive (f.a.) measures on (8,2t) with the total semivariation norm 11·11(8), where the semivariation 11(11(·) of ( E fa(2t, K) is defined as in n
(2.1). For (
E
fa(2t, K) and r.p ==
2: 'l/JjlA j=1
( by
j
E LO(8;
K) we define the integral of r.p w.r.t.
Kakihara
128
Moreover, the norm
II'PII*
IIcpll.
is defined by
= sup {
Il
(cp, d() I:(
E
fa(21, K), 11(11 (8) :::; 1}.
It is known (see e.g. Makagon and Salehi [8]) that the dual space of (LO(8; K), 11·11*) can be identified with the Banach space fa(2!, K), where the isomorphism U : fa(2!, K) -+ LO(8 ; K)* is given by
(U()(cp) =
l
(cp, d(),
If ~ E ca (21, S(K, H)), then the integral of 'P w.r.t. ~ is defined by
Now we get other characterizations of g.o.s.d. as follows: For ~ E ca (21, S(K, H)) the following statements are equivalent: (1) ~ has a g.o.s.d. (2) For every CONS {4>k }~=l in H there exists a family {1]k, .Rk' Jk}k=l of o.s.d. 's of
PROPOSITION 3.4
00
{~*4>k}k=l ~ ca(21,K) such that
k
~
1.
L
k=l
II1]k(e)ll~k
kll(8)2
kll(8)2 ~
L k=l
00
11~*'ljJkll(8)2 ~
L k=l
II"lk(8)II~k
k)12 :'P EL O(8;K), II'PII. ~ I}
:s z= sup {1Ik}k=l ~ ca(21, K) such that
k
~
00
L:
k=l
II1Jk(8)II~k
0 and A E 21 there exist a compact set C and an open set 0 such that C ~ A ~ 0 and II~II(O\C) < c. Regularity of X-valued measures is defined sinlilarly. rca(Q3c,L5(O)), rcao8(Q38,L5(0)), rca(Q3 c ' X) and rcagos(Q3c' X) denote the sets of all regular measures in ca(Q3 c , L6(O)), caos(Q3 c ' L5(O)), ca(Q3c' X) and cagos(Q3c' X), respectively. The strong semivariation 11~lls(') of ~ E ca(Q3c' X) is defined to be that of Tee.) E ca(~c' S(H, L5(0))) (cf. (4.1)). Let {x(t)} be an L~(O)-valued process on G. That is, x(·) : G -+ L6(O) is a mapping. Then, {x( t)} is said to be weakly stationary if its (scalar) covariance function l' (8, t) == (x(s), x(t))2 (s, t E G) depends only on the difference 8t- 1 and if, putting 1'(8t- 1) == 1'(8, t),
Dilations of Operator-Valued Measures
131
:y is continuous on G, where (', ')2 is the inner product in L6(0). In this case, there exists a unique ~ E rcaos(~G' L6(0)) such that x(t)
=
fc (t, X) ~(dX),
t
E G,
(4.2)
where (', .) is the duality pair of G and G. ~ is called the representing measure of the process {x (t) }. {x (t)} is said to be 'weakly harmonizable if it has an integral representation (4.2) for some' E rca(2t, L6(0)).
DEFINITION 4.1 Let {x(t)} be an X-valued process on G, Le., x(·) : G -+ X is a mapping. (1) {x(t)} is said to be scalarly weakly harmonizable if {(x(t), 4»H} is an L6(0)-valued weakly harmonizable process for each cP EH. (2) {x (t)} is said to be weakly harmonizable if it has an integral representation (4.2) for some' E rca(~G' X). (3) {x (t)} is said to be weakly operator harmonizable if it has an integral representation (4.2) for some ~ E 1~ca(~G' X) of bounded operator semivariation, i.e., 11~llo(8) < 00. (4) {x(t)} is said to be operator stationary if its operator covariance junction r(s, t) == [x(s),x(t)] (s, t E G) depends only on the difference st- 1 and, putting f(st- 1 ) == f(s, t), f is a T(H)-valued weakly continuous function on G, i.e., Tr(af(.)) is continuous for every a E B(H). Here Tr(·) is the trace in T(H). In this case, {x(t)} has an integral representation (4.2) for a unique ~ E 1'cagos(~G' X). We are interested in stationary dilations of L6(0)-valued or X-valued processes on G. An L6(O)-valued process {x(t)} is said to have a weakly stationary dilation if there exist a Hilbert space .ft containing L6(0) as a closed subspace and a .ft-valued weakly stationary process {y(t)} on G such that x(t). = Jy(t) for t E G, where J : Jt -+ L6(0) is the orthogonal projection. The triple { {y(t)},.R, J} is also called a weakly stationary dilation of {x(t)}. Note that .R can be taken as an L 2-space L5(0) for some probability measure space (O,~, jj). Moreover, if "7 E rcaos(~G'.R) is the representing measure of {y(t)}, then {x(t)} is weakly harmonizable with the representing measure' E rca(~G' L6(0)) for which "7 is an o.s.d. of ~. Since every ~ E ca(~G' L6(0)) has an o.s.d. by Corollary 2.3, every L6(O)-valued weakly harmonizable process has a weakly stationary dilation, which was proved by Niemi [9] (see also Chang and Rao [1] and Rao [13]). If {x(t)} is an X-valued scalarly weakly harmonizable process, then there exists a family {{Yet>(t)}, L5(0et», J} et>EH of weakly stationary dilations of L5(0)-valued processes {(x(t), 4»H} et>EH' That is, for each cP E H, (0et>' ~et>' Met» is a probability measure space, L5(0Yet>(t) for t E G, where J : L6(0et» -+ L5(0) is the orthogonal projection. In this case, {{Yet>(t)}, L6(0et», Jet>} et>EH is called a scalarly weakly stationary dilation of {x (t) }.
DEFINITION 4.2 An X-valued process {x(t)} on G is said to have an operator stationary dilation if there exist a normal Hilbert B(H)-module Y = L6(0; H) containing X as a closed submodule and a V-valued operator stationary process {y(t)} such that x(t) = Py(t) for t E G, where P : Y -t X is the gramian orthogonal projection and
Kakihara
132
(O,~, it) is a probability 111easure space. The triple { {y(t)}, Y, stationary dilation of {x (t) }.
p} is also called an
operator
DEFINITION 4.3 An L6(O)-valued process {x(t)} on G is said to be V-bounded if a) x ( .) : G -+ L6 (0) is norIll continuous, b) {x(t)} is bounded, i.e., sup {llx(t)112: t E G} < 00, c) There exists a constant C > 0 such that (4.3) where L 1 (G) is the Ltgroup algebra of G with the Haar measure dt, 11 0 is the constant in (4.3) for
(6) {x(t)} is scalarly V-bounded and G E Hand t E G we put ~r;<jJ(t) = (x(t),4»H' Then, {x<jJ(t)} is an L6(0)-valued weakly harmonizable process with the representing measure ~<jJ(') = (~(.), cP) H E Tca(fJ3 c ' L6(0)). It is well-known that an L5(0)-valued process is weakly harmonizable if and only if it is V-bounded (see e.g. [13]). Hence, {x(t)} is scalarly V-bounded. Let {ePk}k=l be any ON sequence in Hand {7Pk}k=l be a CONS in H containing {4>k} k=l' Then we get for each k ~ 1
sup
{II fa
(1): Let {'l/Jk}k=l be a CONS in H. Since {x(t)} is scalarly V-bounded, it is scalarly weakly harmonizable. For each k ~ 1 let ~1/Jk E rca(fJ3 c ' L6(0)) be the representing measure of the L6(0)-valued process {(x(t), 'l/Jk)H}. Then, we have by (4.4) that
since 77(')
_
11~1/Jkll(G)
00
:S Clj1k' Define
~(.) =
L
k=l
~1/Jk(')'l/Jk' Then ~ is well-defined and in
rca(fJ3 c 'X)
Kakihara
134
since (cf. Diestel and Uhl [2, p. 4]) 00
Ilell(G)2
~ 16 sup Ile(A)II~ == 16 sup AE~a
S 16
L Ile1/Jk(A)II~
AE~ak=l
L lI~l/Jk
11 (0)2
S 16
k=l
L C~k
.(t) =
(_t)n
~ r(l + An)'
t
~ 0,
Characterization of Hida Measures
149
where r is the Gamma function. The function L).("15), , E S, is positive definite. Hence by the Minlos theorem there exists a unique probability measure v). such that
It can be shown that v). is a Hida measure of order 1 - A. For details, see Kondratiev . and Streit [1] or Kuo [3].
2. A CHARACTERIZATION THEOREM
!.
Note that the inclusion L 2 (IR) C S~ is a Hilbert-Schmidt operator for any p > Hence (L 2 (IR), S~) is an abstract Wiener space for such p (see Kuo [2]). Thus the standard Gaussian Ineasure J.-l is supported on S~ for any p > ~. Then by the Fernique theoren1 (see Kuo [2]), there exists SOll1e constant a > 0 such that
This type of integrability can be used to characterize Hida measures on SI.
THEOREM. A Borel measure v on S' is a Hida nleasure of order {3 if and only if v is supported in S~ for some p ~ 1 and
L, exp [~(1 + fJ)lxl~]
dv(x)
0: 00 ( un ) r eTu / 4 < e3r / 2 ~ -n 'Vu 2 o. L....J 2 n=O n.' ' 2
By letting r == 1 + (3 and u == Ixl~, we get
1+,6
2]
exp [ -4-lxl~ ~
e
3
x E S~,c·
(3)
(l+,B)/2 v ( VJ).
(4)
(1+,8)/21/J(x),
It follows from equations (2) and (3) that
l,
exp
[1 : fJ Ixl ~] dv( x) ~ e
3
Now note that the inverse operator of A == -d 2/ dx 2 + x 2 + 1 acting on L 2(IR) is continuous and its operator norm is given by /lA -1/1 == 2- 1. Hence for any p 2 q 2 1 we have Ixl- p == lA -Pxlo == IA-(p-q)A-qxlo
::; 2-(p-q) lA -qx/o
== 2-(p-q) Ix I_q'
Characterization of Hida Measures
151
We can choose large p such that p > q + ~' Then
L,
exp
[~(1 + iJ)lxl~] dll(x) ~
L,
exp
[1 :iJlxl~] dll(x).
Thus by equation (4) we have
L, e~p [~(1 + iJ)lxl~]
dll(x)
~ e3 (l+,B)/2 v('lP) < 00.
But Ixl-p == 00 for any x E 5' \ S~' Thus the last inequality implies that the measure v is supported on S~ and
L,
exp
[~(1 + iJ)lxl~] dll(x)
.(x) < 00.
p
REFERENCES [1] Kondratiev, Vu. G. and Streit, L.: Spaces of white noise distributions: Constructions, Descriptions, Applications. I; Reports on Math. Phys. 33 (1993) 341-366 [2] Kuo, H.-H.: Gaussian Measures in Banach Spaces. Lecture Notes in Math. 463, Springer-Verlag, 1975 [3] Kuo, H.-H.: White Noise Distribution Theory. CRC Press, 1996 [4] Lee, Y.-J.: Analytic version of test functionals, Fourier transfOrlTI and a characterization of measures in white noise calculus; J. Funct. Anal. 100 (1991) 359-380 [5] Reed M. and Simon, B.: Methods of Modern Mathematical Physics I: Functional Analysis. Academic Press, 1972
New Results in the Simplex Method in Linear Programming ROGER N. PEDERSEN Pittsburgh, PA 15213
Department of Mathematics, Carnegie Mellon University,
"Notation is important. It can even solve problems. But, at some point, you must do some work yourself." K. O. Friedrichs.
1. INTRODUCTION AND STATEMENT OF THE PROBLEM. Without using any symbols at all, we can give a precise statement of the problem by saying that it is to find the maximum, if it exists, of a linear function of a finite number of real variables on a convex plane polyhedron of the same variables. The simplex method of solving the problem is then to find a vertex of the polyhedron and then to proceed along edges from one vertex to the next, in a manner that the linear function increases, until the maximunl is reached. All the data needed to state and solve the problem can be stored in an (m + 1) x (n + 1) matrix A. The analytical statement of the problem then is to find the maximum of the objective function n
L
A m + 1,jXj
+ A m + 1,n+1
(1.1)
2: 0, i = 1, ... , m.
(1.2)
j=1
subject to the constraints n
L AijXj + A i
,n+1
j=1
153
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154
By defining A to be the matrix comprising the first m rows and first n columns of A and b to be the transpose of A1,n+l, .. ,Am,n+l, the constraint (1.2) takes the simpler form Ax
+b2
0,
(1.3)
meaning, of course, that each component of the column vector is non-negative. The vector x is superfluous for the purpose of applying the simplex algorithm. But, working only with
the matrix A, can lead to misconceptions as we shall see in the next section. But first, let us find another notation for the constraint set by using Ai to denote the rows of A. Then (1.3) can be replaced by (1.4) where bi is the i th coordinate of band (,) represents the canonical inner product.
2. BUT, THOSE SLACK VARIABLES ARE UNNECESSARY. Let us re-write (1.3) as
(AC)(C-- 1 X)+b2 0
(2.1)
where C is any non-singular n x n matrix, noting that this does not require an equality. Now, assuming A has rank n, we may apply elementary column operations to reduced echelon form. If C is the product of the corresponding elementary column matrices and y == C- 1 X, the first n coordinate of (2.1) are (2.2) Then, by making the translation Zi == Yi + bi , we may assume the constraint set to be in, what is commonly called, canonical form. Furthermore, if for one j, 1 ::; j ::; n, we put Xj == Zj - fJj in (1.1),(1.2) we see that this corresponds to multiplying the jth column of the full matrix
A by fJi and subtracting it from the (n + l)th column; that is, it is an elementary column operation. I prefer doing elementary row operations on the transpose. Thus the simplex method reduces to transposing the matrix A and applying elementary row operations until the first n column are in reduced echelon form, with the restriction that the pivots are to be picked from the first n rows of AT. The only question that remains is when to start using the simplex pivoting strategy. After the system is in canonical form, we must use the simplex strategy; before that we may use instead the standard Gaussian Elimination Strategy. Note that the simplex strategy requires picking the maximum positive element of the current column and hence is a partial pivoting strategy. We shall have more to say about this in Section 5.
3. EMPTY SETS, REDUNDANT CONSTRAINTS AND LOWER DIMENSIONAL SETS. Let us now suppose that the normals of the first n constraints form a linearly independent set. Then, for any k > n, n
Ak
==2: DkiAi
(3.1)
i=l
and hence
n
Lk(x)
==2: DkiLi(X) + 6 k i=l
(3.2)
155
Simplex Method in Linear Programming
with
n
D.k == A k,n+l -
L akiAi,n+l.
(3.3)
i::::::;l
It follows from (1.4) and (3.2) that if (akl,., ... , ak,n, D.k) are all non-negative, the k th constraint is redundant and that if they are all negative the set is empty. If for some i :S; n, aki > 0, akj :S; 0 for j :# i and D.k < 0, then the i th constraint is redundant. In all other cases where none of the numbers (akl,., ... , ak,n, D.k) is zero it is easily shown that the set forrned for the first n and the k th , is non-empty. The other important special case occurs when D.k == 0 and aki :S; 0 for i == 1, ... , n. Then the entire constraint set is contained in the set where Lk(x) == O. Hence, we may use this constraint to eliminate a variable and obtain a lower dimensional set. This means that, by reducing the number of dimensions, we may assume that this case does not occur. We note from (3.2) and (3.3) that, when the constraint set is in canonical form, Ai,n+l == 0, i == 1, ... , n, so the aki's and D.k are just the coefficients of the constraint equation. From this point on \ve shall assume that the set is in canonical form. The origin will be called the basic vertex, the first n constraints the basic constraints and the rest of the constraints the non-basic constraints.
4.
THE SIMPLEX ALGORITHM WITH A NON-DEGENERATE BASIC VERTEX.
A vertex which is the intersection of more than n-planes is called a degenerate vertex. This means that, when the basic vertex is non-degenerate, all of the non-basic constraints have non-zero constants. The simplex strategy then is to increase by one the number of positive constants among these until they are all positive and then to increase the constant in the objective function. Let us assume that the constraints are ordered so that Ai,n+l
> 0, i < ]J
Ai,n+l
< 0,
and if]J < m
(4.1) Our first objective is to increase k to maximize
]J
]J:S;
i :S; m.
by one when it is less than m. The first step is to choose
{Ap,j : Ap,j
> 0, 1:S; j :S; n} .
(4.2)
When p :S; rn, the results of Section 3 insure that we may assume the above set to be non-empty; when p == m + 1, it is only empty when we have found the maximum. Next, we choose £ to maximize the negative numbers Av,n+l : v :S; p - 1, A v,k { ~ v,k
0 there is no maximum while if A m + 1,k < 0 we may set Xk == 0 and continue in one less dimension. If p :S; m we simply set == p, observing that the silnplex method requires only one step. Next, we interchange the eth non-basic constraint with the k th basic constraint and put the constraint set back into canonical form. This requires applying Gaussian elimination to the k th colurnn of AT. The new elements of the matrix then are
e
I A ek
,
1 == - -
Ae,k
(4.4)
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156
A' - - A ej £,j -
and when i
=1=
Af,k'
j
I- k
(4.5)
f, A' i,k -
A~· == A·· 'J
Ai,k Af,k'
(4.6)
- AikA£j
Ae,k' j
'J
=1=
k
.
(4.7)
In particular, A' - - A£,n+l f,n+l Af,k
>0
(4.8)
since, whether f == p or f < p, A f ,n+l and A£k have opposite signs. If i =1= f and f < P - 1, we see from (4.7) that A~,n+l is the sum of two positive numbers when A ik > 0 and that when A ik < 0 it is positive as a consequence of the choice (4.3) of f. Hence, in any case, the first p - 1 constants remain positive and if f == p, A p,n+l is also positive and we have increased p by one. But we also see from (4.7) that if f < p,
A~,n+l > Ap,n+l
(4.9)
Since the constraint set has only a finite number of vertices, we shall, in a finite number of steps either find the set to be empty, prove that A~,n+l > or arrive at a degenerate vertex.
°
5. THE CASE OF A DEGENERATE VERTEX. The case of a degenerate vertex occurs when there are zero constants Ai,n+l == O. Suppose that we apply the previous strategy to the basic constraints and the non-basic constraints with non-zero constants. Then we see from (4.7) that when Ai,n+l == 0, A 'i,n+l
== -
(Afj) A ik A fk
(5.1)
°
and since A fj < 0, A fk > 0, we have A~,n+l > whenever Ai,k > 0. There is no reason that this should be the case, but, by applying the simplex strategy to the first n columns of A, with the k th playing the roll of the constants, we can use the simplex strategy to achieve this. Because the algorithm is slightly more complicated when the degeneracy is of higher order, it is convenient to introduce constants ak, (3k satisfying, after reordering the constraints and variables Ai,k == 0, n + 1 :s; i < ak
(5.2) < 0, {3k:S; i < ak+l with a n+2 == m. The cases ak == n + 1, 13k == CYk and {3k == ak-t 1 are used to indicate that the corresponding set is empty. Now we apply the following algorithm to the constraint set in canonical form.
[1] k == n + 1
Simplex Method in Linear Programming
157
[2] Reorder the constraints so that (5.2) is satisfied. Now, we are ready to pick the current constraint indexed by p. The choice agrees with (4.1) when k == n + 1. [3] If k
== n + 1 or
13k
P and is multiplied by (_l)r-£ when j == £. Hence by rnoving the £th ratio to the end of the list and decreasing the order of those indexed by k, £ + 1 :::; k :::; T, we obtain a valid induction proof. Similarly for the l's. • Theorem 8.2. In applying the empty set or redundant constraint test, it is sufficient to scan (8.7), (8.8) for all permutation (k 1, ... , kr ) and (£}, ... , £r) in increasing order of T. Proof. In proceeding from r to T + 1 we interchange the constraints indexed by k r + l and Pr + 1 • A simple computation shows that in an (n + 1) constraint set in canonical form, an interchange of the (n + 1)st constraint with a basic constraint can't change the sign test indicating an empty set or redundant constraint l . But, by Theorem 8.1, we may assume that any k i and Pi were interchanged. 1
See Section 13, #2.
160
Pedersen
9. THE RECURSION FORMULA. Assuming that we have computed the matrix Aij' the matrix A~j+l is obtained by interchanging the constraints indexed by k r + 1 , f r + 1 and updating the matrix as in [1]. The result is (9.1) (9.2) and for i
#- kr + 1 , (9.3) (9.4)
Note, in particular, that (9.4) is the ratio of a 2 x 2 minor and a 1 x 1 minor and when r == 0, it agrees with (8.10). Also, when r == 0, (9.2), (9.3) agree with (8.8) (8.9). In order to make (9.1) agree with (8.6) we make the convention fo == 1. Before proving the general result, we shall develop some lemmas on determinants.
10. SOME LEMMAS ON DETERMINANTS. Let us use the usual convention that Bij is the co-factor of bij . Then our first and main lemma is:
Lemma 10.1. Let B == (b ij ) be a k x k matrix and let C be the (k - 1) x (k - 1) matrix j C == (b t).. - bi'kbk b kk
)
< ,i J. < k - 1. ' 1-
(10.1 )
Then
det C == det B/b kk .
Proof.
(10.2)
Define: (10.3)
Now we use the fact that the derivative of a determinant is the sum of the determinants obtained by differentiating one row of the matrix. When we differentiate the i th row of C, the new i th row is (10.4)
•
If we interchange this row with each of those indexed by i + 1, ... , k - 1, we have the matrix obtained by deleting the i th row from the first k columns of B. Hence, when we take the determinant, we obtain (10.5) It follows that
k-l
n, the new coefficients for the r th constraint are (12.13)
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164
A~j == A~j - A~f A~j / A~f' j
-I- f
(12.14)
After substituting from (12.1) - (12.4), these become (12.15) A~j ==
A rj - A rf Arj / A k £, j
-I- f
(12.16)
which are just the coefficient obtained after interchanging the r th constraint with the fth in the original matrix. This together with the remarks following (12.8) and (12.12) yields a proof of the following theorem.
Theorem 12.1. Interchanging the i th non-basic constraint with the fth, updating and then interchanging the k th with the fth and updating is equivalent to merely interchanging the k th with the fth in the original matrix, updating and then interchanging the i th and k th . Now let us determine the effect of interchanging one non-basic constraint with two different basic constraints. If after obtaining the formulas (12.1) - (12.4), we interchange the i th constraint with the qth basic constraint, q f i, the new parameter for the i th constraint are (12.17) A~~ == 1/ A~q, A~~ == -A~f / A~q,
(12.18)
and A~j == -A~j
/ A~q, j f q, f.
(12.19)
The formulas (12.17) - (12.19), after substituting from (12.1) - (12.4) are just the formulas obtained after interchanging the i th with the qth in the original matrix. For k f q, (12.20) (12.21) and A~j == A~j - A~q A~j / A~q, j
f
q, f.
(12.22)
Again, after substituting from (12.1) - (12.4), these are just the formula for the k th constraint after interchanging the k th with the qth in the original matrix except that the qth and Rth variables have been interchanged.
Theorem 12.2. IEwe interchange the i th non-basic constraint with the fth basic constraint, update and then interchange the new i th constraint with the qth, q f i, and update, this is equivalent to merely interchanging the i th with the qth updating and permuting the qth and fth variables.
Simplex Method in Linear Programming
13. THE CASE OF
(n
165
+ 2) CONSTRAINTS.
Let us assume the constraint set to be in canonical form. If er is any subset of the non-basic indices, we shall denote by So. the corresponding set of non-basic constraints and by So. the set So. together with the basic constraints. For a single index i we define
st == {j :S n : A ij > O}
(13.1)
and (13.2) The cardinality of set S shall be denoted by ISI. We shall assume that our constraint set contains no degenerate vertices and that minor determinants used in counting are always non-zero. Our (n + 1) constraint set Si is empty when ai,n+l < 0 and == n, hence == O. It
Istl
ISi-1
1st I == n or ai,n+l < 0 and 1st I == 1. 1st I == a, 0 :S a :S n. If a < n there exists an f :S n such that < o. If we
contains a redundant constraint when ai,n+l > 0 and Now suppose that
Ail
interchange the i th and fth constraints and put the set back into canonical form we obtain the constraint set
Si with ISi I =
If IJ > 0 there exists an index {' ::;
IJ.
nwith ai,f > O. After
interchanging the i th and fth constraint and putting the set back into canonical form the set
Si has ISi I = n+ 1-
IJ.
It follows that interchanging two constraints in an
(n + 1) - constraint
set cannot change its status relative to being empty, or having a redundant constraint. Hence, if neither Si nor Sj has this property, we can find an empty set or redundant constraint in an (n + 1) constraint set only by interchanging Si with a basic constraint and examining S j, j i- i or conversely. In particular, after making this interchange, the new constant term is Aj,n+l
== -
~; (Ai,n+l - ~;: Aj,n+! )
(13.3)
Hence, that constant term in the i th constraint, after interchanging the jth and the £th, has the same or opposite sign as the jth constant, after interchanging the i th and the gth , according to whether A jl and A if have the opposite or the same signs. Let us no\v study the jth constraint after interchanging the i th and the fth with A if < 0 and Aj,i > O. This requires analyzing the signs of (13.4) and
A 'j,k == A j,k
-
jiA
A -A
ik·
(13.5)
if
Since A jl and Ail have opposite signs, it follows from (13.4) that (13.6)
166
Pedersen
and from (13.5) that Ajk > 0,
k E Si n Si
(13.7)
Ajk < 0,
k E Si- n Sj-'
(13.8)
and For k E Si- n
st - {£} , we may make the signs of A"k J
== (Ajk _ Ajl) . A ik A Ail
(13.9)
ik
all negative or all positive, without violating (13.6), by choosing £ to minimize or maximize the ratios A jk ,k E Si_ n Sj+ . ( 13.10 ) -A ik Since, by (13.6), Ajl < 0, we can't achieve an empty set or redundant constraint unless Aj,n+l < O. This rules out the possibility A i ,n+l > 0 and A j ,n+l > O. When A i ,n+l < 0, A j ,n+l > 0, this cannot be the case unless it is true for either Si or Sj. There remains the cases where A i ,n+l and Aj,n+l have opposite signs. By applying the results (13.6) - (13.10), we see that Sij is empty or contains a redundant constraint if £ minimizes the ratios (13.10), (13.11) and either
A· 1 < _JfA'n > 0, ~ · 0 A 1.,n+l A j,n+l 0, n + 1 ::; i ::; p - 1, Ap,n+l < O.
(14.1 )
Definition 14.1. For each k E S:' we define pk ,,+ .. A i,k < 0, A p,n+l - A 'k -- { 1,. A A i,n+l > 0 }
(14.2)
ik
and
7.:-
=
{i :
A ik
< 0,
Ap,n+l -
~: Ai,n+l
0, Ap2 > 0, A p3 < 0, A p4 < O.
S;
(14.4)
S;,
If there is a constraint indexed by i < p for which 1 E and 2 E it follows from (13.11) - (13.13) that there is a redundant constraint. Let us then assume that there exist two non-basic constraints indexed by i and j for which i E ~-
n ~+,
j E ~+
n ~- .
(14.5)
The constraints indexed by i, j, p have the following sign configuration 1
e j k
234
+ +
(14.6)
e + +
+ +
The circled and uncircled minuses referring to
0.:-
and
0.:+
respectively. We then have
A p1 Ap4 - TAi4 < 0,
(14.7)
i1
A p2 A p4 - TAi4 > 0,
(14.8)
i2
A p2 -A· A j2 J 4 < 0
(14.9)
A p1 A p4 - TAj4 > O.
(14.10)
A p4 and
-
jl
These are equivalent to
f2 (i, p : 1, 4)
> 0, 12 (i, p : 2, 4) < 0, 12 (j, p : 2, 4) > 0,
12 (j, p : 1, 4)
< O.
(14.11) It follows from (14.7), (14.8); (14.9), (14.10) that
12 (i, p : 1, 2) < 0, 12 (j, p : 1, 2) > O.
(14.12)
By writing (14.12) in the form
A p2Ai1 < 0, A pI - -A A p2A j1 > 0, A p1 - -A i2
(14.13)
j2
we have
12 (i, j
: 1, 2) > O.
(14.14)
Similarly, it is a consequence of (14.7), (14.10) and (14.8), (14.9) that
12 (i, j
: 1,4) < 0,
12 (i, j
: 2,4) > O.
(14.15)
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168
After interchanging the i th and first constraints and updating, we have the coefficient matrix
1
fdi
(1 : 1)
-11 (i,2) -11(i:3) -11(i:4) ) : 1, 2) 12 (i, j : 1, 3) 12 (i, j : 1, 4) 11 (p : 1) 12 (z, p : 1, 2) 12 (z, p : 1, 3) 12 (z, p : 1, 4)
f dj, 1)
12 (i, j
(14.16)
It follows from the imposed signs (14.6) - (14.15) that the matrix (14.16) has the sign configuration 1 234 J p
+
+ + ± +
(14.17)
+ ± -
If the coefficient indexed by p, 3 were negative the third basic constraint would be redundant. Therefore, we impose the sign (14.18) l(i,p: 1,3) < 0 leaving the configuration
( ~- +=:+ :) -
(14.19)
with only the 2,3 element having an arbitrary sign. In any case the interchange of the jth and second constraints is admissible. After this interchange, we have the matrix with D 2 == 12 (i,j: 1,2) > 0
1 (11 (j : 2) - 11 (i : 2) 12 (i, j : 2, 3) 12 (i, j : 2, 4) ) - 11 (j : 1) 11 (i : 1) - 12 (i, j : 1, 3) - 12 (i, j, 1, 4) D 2 -12(j,p:1,2) 12(i,p:1,2) 13(i,j,p:1,2,3) 13(i,j,p:1,2,4)
(14.20)
Now let the coefficients of the pth constraint be denoted by A~j' It follows from (14.12) that
(14.21 ) Hence, if A~4 < 0 the set is either empty or there is a redundant constraint. If A~4 > 0, this configuration does not contribute to the promoted version of ~+. Of course, this statement does not apply if the interchange is made with respect to some other constraint. Let us now examine the other admissible exchanges within the present matrix. From (14.19) it appears that the interchange of the i th and second variables is one such possibility. But this follows the interchange of the i th and the first. But this is, by Theorem 12.2, the interchange of the i th and second followed by a permutation. From (14.19) we see that the only other admissible interchange is the interchange of the jth and third constraints under the condition
12 (i,j: 1,3) > O.
(14.22)
This interchange gives the matrix
1 (11 (j : 3) - 11 (i : 3) 12 (i, j : 3, 2) 12 (i, j : 3, 4) ) -11 (~: I? 11 (~: 11 -12 (i,j: 1,2) -12 (i,j: 1,4) D 2 - 12 (J, P . 1, 3) 12 (z, p . 1, 3) 13 (i, j, p : 1, 3, 2) F (i, j, p : 1, 3, 4) 2
(14.23)
Simplex Method in Linear Programming
169
with D 2 == 12 (i,j : 1,3) which by (14.22) is positive. By (14.18) we have 12 (i,p: 1,3) < O. This configuration appears to have insufficient information to resolve the sign of 12 (j, p : 1,3) . However, if the assumption (14.22) leads to a legitimate simplex step it does impose the additional sign 12 Ci, j : 3, 4) > O. (14.24) In any case, the previous configuration \vas sufficient to resolve the case of the constraints in three variables. When there are more constraints the additional condition (14.24) may be helpful in analyzing the interaction of various sets of three non-basic constraints combined with the basic constraints. We remark, also, that if the same constraints i and j solve the maximum problem determining the next simplex step for two steps in a row, the analysis of (14.20) is sufficient to produce either a complete simplex step or to find a redundant constraint. That this be the case when both maximums are achieved by the i th constraint \vould require the interchange i - 3. By (14.19) this is impossible since both the 1,3 and 3,3 elements are positive. Finally, we consider the sign configuration 234
1 8
j p
+ + + 8 + + + +
(14.25)
The interchange of the i th and first constraints leads to
+ + + + + +
(14.26)
instead of (14.19). Some of these signs are determined as before and the others are consequences of Theorem 10.2. Now we notice that the interchange of the jth constraint with the second is the only admissible simplex interchange. Now to apply the preceding analysis to (14.20), we need only (14.21). This is again a consequence of Theorem 10.2.
15. THE CASE OF SIX CONSTRAINTS IN THREE VARIABLES. The analysis of the preceding section yields the following Theorem.
Theorem 15.1. Let us consider a set of Six Constraints in Three variables which is in Canonical form and with only one constraint not satisfying the basic vertex. If completing a simplex step or finding a redundant constraint or finding the set to be ernpty requires more than three steps then up to a permutation of the first three columns we may assume the configuration of the non-basic constraints 8
+ ± +
J:+8±+ p :
(15.1)
+ +
We leave open the question of whether the number of steps can be reduced from three to two by starting with the configuration
+ -t- +for the pth constraint.
(15.2)
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170
References [1] Dantzig, Linear Programming and Extensions, Princeton Univ. Press. [4] Polya, G., Szego, G., Problems and Theorems in Analysis, Springer-Verlag, New York, Heidelberg, Berlin, Berlin, 1972. [5] Strang, G., Linear Algebra and its Applications (3rd Ed.), Harcourt, Brace, Jovanovich, San Diego. [6] Wu, S. and Coppins, R., Linear Programming and Extension, McGraw Hill. Acknowledgement
I would like to thank Jenny Bourne Wahl for criticizing an earlier version of Sections 1 7 of this manuscript.
An Estimate of the Semi-Stable Measure of Small Balls in Banach Spaces BALRAM S. RAJPUT Knoxville, TN 37923
Department of Mathematics, The University of Tennessee,
Abstract. Let (lE:, 11 . 11) be a separable Banach space. Let J.L be a symmetric r-semistable probability measure of index 0 < a :S 2 on lE:, and let 0 < q < a. It is proven that q if fIE IIxll dJ.L == 1 then J.L{llxll :S t} :S const. t a / 2 , for all t > 0, where const. depends only on r, q and a (and not lE: or J.L). This result compliments similar known results for symmetric Gaussian and a-stable probability measures on lE. Two other related results are also proved; these are needed for the proof of the above main result.
1.
INTRODUCTION AND PRELIMINARIES
Let (lE:, 11 . 11) be a separable Banach space. Let J.L be a symmetric Borel probability measure on lE:. In a recent paper, M. Lewandowski, M. Ryzner, and T. Zak (1992) showed that, if J.L is a-stable, satisfying fIE IIXllqdJ.L == 1 with 0 < q < a, then J-L{ Ilxll :S t} :S const. t, where const. depends only on a and q (and not lE: or J.L). In the case when J.L is centered Gaussian, a similar result is proved earlier by S. Szarek (1991) and also by X. Fernique and by J. Sawa; Sawa requires in addition that lE: be a Hilbert space. (For a discussion and references of the Fernique and Sawa contribution, we refer the reader to Lewandowski, Ryzner, and Zak (1992)).
This research is partially supported by the University of Tennessee Science Alliance, a State of Tennessee Center of Excellence.
171
172
Rajput
The main effort of this paper is aimed at proving a version of the above result of Lewandowski, Ryzner, and Zak (1992) for the larger class of semi-stable probability measures. Specifically, we prove the result stated in the abstract. The proof of Lewandowski, Ryzner, and Zak (1992) in the stable case is based on the fact that every E-valued symmetric a-stable random variable is conditionally Gaussian and on the well known Anderson Inequality for Gaussian nleasures. Since a serni-stable random variable in general is not conditionally Gaussian (Rosinski (1991), p.32), the methods used in Lewandowski, Ryzner, and Zak (1992) do not apply in the more general semi-stable case; a similar situation seems to prevail with regard to the methods of proof used by Szarek and Sawa. Our proof, like the one due to Fernique in the Gaussian case (see Lewandowski, Ryzner, and Zak (1992)), is based on the well known Kantor Inequality. In the Gaussian case (a == 2), this approach yields the same upper bound for J.L{llxll ::; t} as obtained in Lewandowski, Ryzner, and Zak (1992) in the stable case (namely, const. t ). In the proper semi-stable case, on the other hand, this approach yields the upper bound for J.L{ Ilxll ::; t} as const. t Ci / 2 . which, in the interesting case, i.e., when t is close to 0, is worse than const. t . (For more on this point see Concluding Remark). For our proof of the main result, in addition to the Kantor Inequality, we also need an estimate for the lower bound of the tail of symmetric semi-stable probability measures on IE; this is obtained in Lemma 1. This lower bound is obtained by using the PaleyZygmund Inequality and another result which provides a comparison between moments of a semi-stable probability measure and a related F-norm (Proposition 1). Throughout, r and a will denote real numbers satisfying 0 < r < 1 and 0 < a < 2; and the notation r - SS (a) will mean " r-semi-stable index a". Further, throughout IE will denote a real separable Banach space. By a measure on IE, we shall always mean that it is defined on its Borel a-algebra. For the sake of brevity, we refer the reader to Chung, Rajput, and Tortrat (1982) and Rajput and R,ama-Murthy (1987) for the definition and properties of E-valued r - SS(a) random variables and r - SS(a) probability measures on IE. A fact regarding these which will be important for us is the following: Let X be a symmetric E-valued random variable and let J.L == £(X), the law of X; then X is an r - SS(a) random variable (equivalently, J.L is an r - SS(a) probability measure) {::} J.L is n infinitely divisible and J.Lr == rr/Ci • J.L, for all n == ±1, ±2, ... , where a . J.L == £(aX) for a s real number a and J.L , s > 0, denotes the sth root of J.L (see Chung, Rajput, and Tortrat (1982) and Rajput and Rama-Murthy (1987)). Note also that if J.L is centered Gaussian then J.L is r - SS(2) measure, for all 0 < r < 1. Before we end this Section, we introduce a few more notation: Let 0 < P < q < a, then we set
C(r, a,p, q) == Let C
(
nJ u;
a_
q) ~ (2-r~_+:~ ~) .
== C(r, a, ~,q), then we put K(r,a,q)-=-
1 ) ((2C)~) (23) (Jr(l-r) 2~-1;
we note that both C(r, a, p, q) and K(r, a, q) are greater than 1. For a non-negative random variable ~ and p > 0, we shall use the notation Ap(~) for SUPt>o t (P{ ~ > t} )l/ P ; and, for an lE- valued random variable X and q > 0, we shall use the notation IIXllq for (Ellxllq) ~.
Semi-Stable Measure of Small Balls in Hanach Spaces
2.
173
STATEMENTS AND PROOFS OF RESULTS The main result of this paper is the following:
THEOREM 1. Let J-L be a symmetric r-SS(a) probability measure on E and let 0 and K == K(r. a, q). If fIE IIXllqdJ-L == 1, then J-L{llxll ::; t} ::; K tCl./2, for all t > O.
CE, we have
P{IIXII > E} 2 in particular, if
IIXllq 2
2CE, then
Proof of Proposition 1: The nontrivial part here is the right inequality. A proof of this in the stable case was given by Linde (see Linde (1986), p. 137 and Linde (1982)). The proof given in Linde (1982) and Linde (1986) is based on functional analytic methods and uses certain results of De-Acosta (1977), in particular, the fact that the counterpart of
Rajput
174
An (IIXII) (in the stable case) is finite. Later another proof of this inequality in the stable case was given by Gine, Marcus, and Zinn (1985). This proof is probabilistic and based on an idea of Pisier (see Gine, Marcus, and Zinn (1985)). We adapt this proof to the present semi-stable case. For every n == 1, 2, ... , let k n == [r~ ] ' the integral part of 1/ r n . Then r~ == k n + rn Cn, 0 :::; Cn < 1. Fix n, and let Xj's be iid random variables with [,(X j ) == M , j == s 1, ... ,kn , and let Yn be independent of Xj's with [,{Yn } == Ml-rnkn, where J-L , S > 0, denotes the sth root of J-L == ['(X). Clearly, we have
P{ max (IIYn ll, IIX.ill,j == 1, ... ,kn ) > t} ==1 - P{ max (1IYnll, IIXjll,j == 1, ... ,kn ) ~ t} =1 -
(il
t}) P(llYnll s: t)
P{IIXjll s:
21 - (1 - P{IIX1 II > t} )k n
==1 -
(
1 - J-Lr n
{llxll > t}
)
k
n
(1)
. n
Denote the left side of (1) by L(t) and let v(t) == J-Lr {llxll > t} == J-l{llxll > r-n/nt}. Then (1) becomes L(t) 2 1 - (1 - v(t))k n ; equivalently, v(t) ::; 1 - (1 - L(t))l/k n . Then, using the fact 1 - (1 -
x)l/rn
s:
~ (l':X) ,OS: x < 1, for all m = 1,2, ... , one finally gets v(t)
s:
(k1) n
L(t)
(2)
1 - L(t) .
1ft> 0 is such that P{IIXII > t} < 1/2, then (2) and Levy's Inequality (Araujo and Gine (1980), p. 57) yield
P{IIXII > r-n/nt} < -
(~) kn
2P{IIXII > t}
1 - 2P{IIXII > t}
.
(3)
Set t q == 4~ IIXllq; then, by Chybeshev's Inequality, we have P{IIXII > t q } 1/4. Thus, by (3), we have
::;
Ellt~lIq == q
(4) for every n
(t)
get we have
a
= 1,2, ... Set Sn = tq/rn/a; then (t) a = r~ = k n + Cn' Hence, using (4), we P{IIXII > Sn} = (k n + cn)P{IIXII > Sn} s: 2. Therefore, for every n = 1,2, ... , s~P{IIXII
Now let Sn < S < Sn+!, then sa P{IIXII sn}::; (2/r)t~. Thus, we have
> s}
sup snp{IIXII S>SI
> sn}
s:
> s}
~ 2t~.
s~+lP{IIXII > ~ (2/r)t~.
(5) Sn}
= (~) . P{IIXII >
175
Semi-Stable Measure of Small Balls in Banach Spaces
But since, clearly, 8a p{IIXII > 8}:::;
Showing
~,ifO < 8:::; 81 ==~, we have ro
) IIXllq.
Aa(IIXII) ::; (2:i~
Another proof of this part of the inequality is also possible; this proof uses the analog of the above inequality for the stable case (Linde, 1986), and a comparison result of the tail probabilities of symmetric stable and semi-stable JE.-valued random variables due to Rosinski (1987). The proof of this comparision result in turn depends on the theory of single stochastic integrals (Rosinski, 1987). The above proof is direct and does not depend on any of these facts. The left side inequality is standard; we include a simple proof for completeness. For simplicity of notation, Set ~ == 11 X 11. From above, we have A == Aa (~) < 00; and, clearly, by the definition of A,P{~ > t} :::; min{~:,l}. Using this we get E(~q) ==
qJoOOuq-lP(~ > u)du:::; qJoOOuq-lInin{~:,l}du == qJoAuq-ldu+qJ;uq-l(~)adu
= Aq
+ (~)
Aq
=
(a~q) Aq.
This yields
(~)~ IIXllq ::; Aa(IIXII)·
This completes
the proof of the Proposition. The proof of the Corollary is immediate from the Proposition. Proof of Lemma 1: Let ~
!!
== IIXII and A == 5. Then, since IIXII!! E~2 2
~ C- 1 11 X
llq >
C- C£, 0 < A < 1. Therefore, by the well-known Paley-Zygmund Inequality, 1
Since
~~; f.,
;:::
~. C2
This proves the first inequality; the second is now immediate from the
first. Proof of Theorem 1: As in the definition of K, Set C == C(r, a, 1, q) and let Y == 2CX where X is an JE.-valued randolll variable with L:(X) == J.-l. Let n and m be any positive rn integers satisfying 0 < rnm :::; 1. Let Y1 , ... ,Ym be iid's with L:{Y1 } == v == r n / a .v, th where, as before vs, 8 > 0, is the 8 roof v == L:(Y); and let Zm be independent of Yj's . (n with L:{Zm} == v I - r n m. Then uSing v r ) *m * V 1 - r nm == V T nm * V 1 - T nm == V, we have
176
Rajput
v == £(Y) == £ (2:::1 Yi + Zm) . (Here * denotes the usual convolution). Hence, it follows, form Kantor's Inequality (Araujo and Gine (1980), p. 136), that
m
i=l
S;
(D
~
(3) [mP{IIY
=
(~) [mP{11Y11 > 1}]-1/2;
2
[
-1/2
~P{IIY;II 1 11
> rn/a} + P{IIZmll
> rn/o}
provided
IIYllq ~
]
]
in the last step we have used the fact that v inequality of Lemma 1, we have
P {II YII> 1}-
> r n/ a
1/2
-1/2
(6)
rn
== r n / o . v. Taking c == 1, in the second C~
S; (1 - 2 _!l)' 2
2C. Therefore, (6) yields
(7) where D
== (~) 2
(--2L) = (;!) (2C)~ 1-2-1 2 (21-1)
.
Let kn be the integral part of l/r n , then 7'~
== kn + Cn, 0
~
Cn
< 1. Taking m == kn in
(7), we get
Thus, since (l-rncn) ~ 1-r and n was arbitrary, we have P{IIYII ~ rn/o} ~ D(r n / o )0/2(1r)-1/2, for all n == 1,2, .... Now let t be any positive real number satisfying r~ < t < r~; then the preceeding inequality yields that
D ( n/a)0/2
~
P{IIYII S; t} S;P{IIYII S; rn/a} S; D
n+l
(
r~
)0/2
----;::::=========:--
Jr(l - r) if
t}
IIYllq < -
r!;, then
D to./
2
~ V r(1-7')
>
D
(
,
IIYllq
~ 2C, then
P{IIYII
~
1) 0./2
ra
v~ r(l-r)
==
D
~-r v J.-'"
> 1 ; thus, th e
Semi-Stable Measure of Small Balls in Banach Spaces
177
preceeding probability inequality is valid for all t > O. Therefore, recalling that Y == 2CX 2 2C 2D / and observing that K(r,Q,q) = ~' we have P{IIXII -s: t} -s: K t a 2, provided Q
Q
/
/
r(l-r)
IIXllq ~
1. This completes the proof.
CONCLUDING REMARK Let IE == lR, the real line; then it is easy to show that, for any symmetric r - SS(o:) probability measure J-L on lR satisfying .fIR Ixlqdpl == 1 with 0 < q < 0:, J-L{lxl ~ t} ~ const. t where the const. depends only on T,O: (and not on J-L). To see this, we proceed as follows: Let a be the (finite) symmetric spectral measure (on 6 == {r i- < Is I ~ I}) of the given measure 11, then cp(y), the characteristic function of J-L, is given by
where ko(t) == Itl-O L:~=-oo T- n (l - cosrn/ot), t :f- 0, ko(O) == 0 (Rajput and RamaMurthy, 1987). Now, if f denotes the probability density function of J-L, then, using the fact ko (t) ~ do (r, 0:), t -# 0 (the constant do depending only on T and 0:) (Rajput and Rama-Murthy, 1987), we have, for t > 0,
Now recalling the fact that (.f~ IsIOa(ds))i- ~ d1(r, 0:, q)·(fIR Ix,qdJ-L)~ (where d 1 depends only on r, 0: and q) (Linde, 1986), we get J-L{lxl ~ t} ~ const.(T, 0:, q) t, provided fIR IxlqdJ-L == 1, where const.(r, 0:, q) ==
2f(i).
r/ ell
This fact notwithstanding, the question whether t a / 2
7rod
can be replaced by t in the statement of TheoreIIl 1 relnains open. The bet here seems to be that the answer to this is affirnlative!
178
Rajput
REFERENCES 1 A. Araujo and E. Gine (1980). The CLT for Real and Banach Valued Random Variables, J. Wiley New York. 2 D. M. Chung, B. S. Rajput, and A. Tortrat (1982). Semi-stable laws on topological vector spaces, Z. Wahrsch. verw. Geb, 60: 209- 218. 3 A. De-Acosta (1977). Asynlptotic behavior of stable measures, Ann. of Probab., Q: 494-499. 4 E. Gine, M. B. Marcus, and J. Zinn (1985). A version of Chevet's Theorem for stable processes, J. Functional Anal., 63: 47-73. 5 M. Lewandowski, M. Ryzner, and T. Zak (1992). Stable measure of a small ball, Proc. Amer. Math. Soc., .f: 489-494. 6 W. Linde (1982). Operators generating stable measures on Banach spaces, Z. Wahrsch. verw. Geb., 60: 171-184. 7 W. Linde (1986). Probability in Banach Spaces, J. Wiley, New York. 8 B. S. Rajput and K. Rama-Murthy (1987). Spectral representations of semi-stable processes, and semi-stable laws on Banach spaces, J. Multi. Anal., 21: 141-159. 9 J. Rosinski (1987). Bilinear random integrals, Dissertations Mathematicae, CCLIX. 10 J. Rosinski (1991). On a class of infinitely divisible processes as mixtures of Gaussian processes, Stable Processes and Related Topics (S. Cambanis, et al), Birkhauser, Boston, 27-41. 11 S. Szarek (1991). Condition numbers of random matrices, J. of Complexity, 1: 131-149.
Nonsquare Constants of Orlicz Spaces ZHONGDAU REN 92521
Department of Mathematics, University of California, Riverside, CA
Dedicated to Professor M. M. Rao on the occasion of his 65th birthday.
Abstract. Estimation of nonsquare constants, in the sense of James, of Orlicz spaces is given. Clarkson's inequalities for LP space have been generalized for Orlicz space by using M. M. Rao's interpolation theorem. The exact values of nonsquare constants of a class of reflexive Orlicz spaces are also obtained by using a new quantitative index of N-functions and the inequalities of Clarkson type for Orlicz spaces. 1993 Mathematical Subject Classification: 46B30.
1
Introduction
Let X be a Banach space and let S(X) == {x EX: IIxll == I} he the unit sphere of X. In 1964, Jarnes[9] called X uniformly nonsquare if there exists a 6 > 0 such that for any x,y E S(..Y), either 11~(x + y)11 :S 1 - 6 or 11~(x - y)11 :S 1 - 6. In 1990, Gao and Lau[4] introduced the following. Definition 1.1 The pararneter J(X) of a Banach space X, which will be called nonsquare constant in the sense of J ames in this paper, is defined by
J(X) == sup{ min(llx
+ YII, Ilx - yll) : x, y
E
S(X)}.
(1)
Gao and Lau[4] proved that ..Y is uniformly nonsquare in the sense of James if and only 179
180
if J(X)
Ren
< 2.
Remark 1.2(See [4]) SchaJfer[16] called ..Y uniformly nonsquare if there exists an a > 1 such that max(llx + yll, Ilx - yiD ~ a for any x, y E S(X). Nonsquare constant g(X) of a Banach space .LY, in the sense of Schaffer, is defined by
g(X) == inf{max(II.T + yll,
Ilx -
yll) : x,
yE
S(X)}.
If dim X ~ 2, then 1 :S g(X) :S V2 :S J(X) :S 2 and g(X)J(X) == 2. Therefore, 1 < g(X) if and only if J(X) < 2, i. e., .LY is uniformly nonsquare in the sense of Schaffer if and only if X is uniformly nonsquare in the sense of James(see also Gao and Lau[5]). In this paper, we only deal with J(X) when X is an Orlicz space. Let lvl lul (u) == la 4J(t)dt and \lJ(v) == la ljJ(s)ds
r
r
be a pair of complementary N-functions, i.e., 4J(t) / 00 as t / 00. The Orlicz function space L cl> (0) on ~l == [0,1] or [0,(0) is defined to be the set {x : x is Lebesgue measurable on 0 and p O}. The gauge norm and the Orlicz norm are defined by Ilxll(4))
= inf { C > 0 : P4>(~)
and
IIXII4>
= sup
t > 0, by (3), there exists 0 < Uo < ~ such that [-1(UO)/-1( 2uO)] + E or, equivalently, [(a~ + E)-l (2uo)] > Uo.
or, equivalently
E
> 0, by (3), there exists m. '¥
Let k o ==
°< Vo
/3g-~
[2-1(VO)] '3 0 > 2vo· 2{
-
E
[21\0]' Then ko :::; 2~o < k o + 1. Choose t ~ ko
~ such that [-l(vo)/-l (2vo)]
°
such that 2kov o + (t) == 1. Put
ko A
== (4)- 1(vo), . . . , - 1 ( Vo )', 4>- 1(Vo ), . . . , - 1 ( Vo )', t, 0, 0, . . .)
184
Reo
and Y
Then Ilxll(4))
==
ko
ko
A
A
== (~-l(vo),"" -l(vo)','--l(vo),"" --l(vo)', 0, t, 0,"
IIYII(4))
.).
== 1 since P4>(x) == P4>(Y) == 1 and
x- Y
P4> [ (1 - E) (2 f3~ - E)
]
+Y
x
]
P4> [ (1 - E) (2f3~ - E) 1 [ -x + Y- ] > --P4> I -
2f3~
E
E
{k [2-1(V 2f3~ -
1
1-
-
f
0
2kovo l-E
O)] f
(
+2
t 2f3~ -
)} E
1 - 2vo 1-E
> -->-->1. Therefore,
+ YII(4))) 2: (1 - E) (2f3g
min(llx - YII(4))' Ilx
- E).
Since E is arbitrary, we obtain (18). Finally, (14) follows from (17) and (18).0 Next we deal with another three classical Orlicz spaces equipped with Orlicz norm. Theorem 2.2 Let be an N-function. Then nonsquare constants of L4>[O, 1] == (LcI>[O, 114», L4>[O, 00) and £cI>, in the sense of James, satisfy respectively max
(2,B\jJ'
l
a
J :;
J(£[O,
ID,
1], 11·
(19)
(20) and max
(
0 2,B\jJ,
1)
a~
::; J(£
f
> 0, there exist
'Un
~
°such that for all n
W- I (v n ) W-I (2v n ) >
We may assume 2v n
::;
1 for all n ~ 1. Let kn
0
fJ'It -
==
1 -k- - < 2v n n +1 'It-I
(~ll
Since ~ /'
and
00
as v ~ 0, we have
~
f
2'
[2~n]' Then ::;
1 -k . n
1
Reo
186
Put
and Cn
Then bn
~
= 2(kn + 1)'11 -1
[
2(k
1] + 1) - 2k '1l n
-1 (
n
1)
2k
n
'
0 and 1 Cn < 2(k n + 1)W- 1 (-k ) - 2kn w- 1 2 n
as n --+ 00. Choose no ~ 1 such that bncn < k o == k no , Co == Cno and bo == bno ' Put
E
(_1_) == 2w- (_1_) --+ 0 2kn 2kn 1
for all n
~
no. For simplicity, we set Vo == v no '
ko ~
X == (b o, bo, ... , bo, 0, 0, ... , 0, ... ) and ko
ko
~~
Y == (0, 0, ... , 0, bo, bo, ... , bo, 0, 0, ... ). We have Ilxll«I> == Ilyll«I> == bok ow- 1 (t) == 1 and
Ilx + yll«I>
Ilx - yll«I>
bo2ko'1l-
1
C~J
bo { 2(k o +
1)'1I~1
[2(k 1+ 1)] - Co } o
> bo ['11-:~ Vo ) - co] > bo [l~O
~
((J~ -
D
1 '11- ( 2v o) - co]
i) boko'1l~l CJ - boco 2 (f1~ - i) - boco 2
((J~ -
> 2(/3~ - E). Since
E
is arbitrary, we obtain (24). Finally, we prove 1
-0
Gw
For any given 1 >
E
> 0, there exist
~
>
~
'Un
«I> J(€ ). ~ 0 such that for all n ~ 1
(25)
Nonsquare Constants of Orlicz Spaces
°
1) -
== (k n + 1) W-1 ( k + 1
Sn
Since t n ~ and Sn ~ and so, for n 2:: no
187
n
°
as n -+
00,
11,0
==
Uno'
to == t no and
So
==
(
1)
k
'
n
there is an no 2:: 1 such that E
E
2
1+f
2t n s n < - < - - < Let us set
k n W-1
-
tns n
== t o2koW- 1 (2k o ) == 1 and
Ilx -
YII1>
Ilx + yll1> 2t okoql-l
(:J
2t o [(k o + 1)q1-1 (k
o
~ 1) - 8 0 ]
1 > 2t o [W- ( 2u o) _ so]
2uo
q1~I~'UO))
> 2t o[
2uo
D:\lJ
+f
-
so]
- W - 1 ( - 1 ) - 2toso > -io2ko D:~ + f 2ko 1-
t
> D:~ + f ' Since E is arbitrary, we have proved (25). Thus, (21) follows from (24) and (25). 0 Some exarnples will be given in Section 4. Remark 2.3 James[9] proved that every uniformly nonsquare Banach space is reflexive. For the above six classical Orlicz spaces, this can be easily proved. For instance, by Theorem 1.3 and Theorem 2.1 "ve have that
Moreover, Chen[l], Hudzik[7] and \\Tang and Chen[17] proved that uniform nonsquareness coincides with reflexivity for these ()rlicz spaces(see also [6]). Some relations between nonsquare constants and other geometric coefficients of Banach space can be found in [5, Theorem 5.4] and [18, Theorern 3.2].
188
3
Reo
A Generalization of Clarkson's Inequalities
Clarkson[2] is the first mathematician to study geometry of Banach space. His results, called Clarkson's inequalities in these later days, deal only with LP spaces(see also Corollary 3.4 in this section). In 1966, Rao[12] first obtained Riesz-Thorin type interpolation theorem between Orlicz spaces equipped with Orlicz norm(see also [14, p. 226]). In 1972, Cleaver[3] generalized Rao's interpolation theorem for fP-product of Orlicz spaces(see also [14, p. 240]). In 1985, the author proved that these theorems are still valid for Orlicz spaces equipped with gauge norm(see [14, p.226, p.256]). In this section, by using Rao's theorem with its generalization, we generalize Clarkson's inequalities for the case of Orlicz spaces. The main result of this section is Theorm 3.2, which will be used in Section 4. Let us start with the following. Lemma 3.1 Let
0 (27) we have
_ _ l-s ( fJ~s = (fJ~) and
1) (1) v'2
v'2
s
~
s
(v'2)S 1 - > -. v ' 2-2 2
__ (_all> )l-S( - 1
all> s
Therefore, the conclusion follows from Theorem 1.3(iii). 0 Theorem 3.2 Let
0 if 0 < f :S 2, i. e., "\8 is uniformly convex. On the other hand, if Ilxll(s) == 1 and IIVII(
0 -
u
a\.l S
1
b/oo
== lul b and
(39)
we get (37), again by (28). 0 Remark 3.5 In view of (:34), (35), (38) and (39), one has cS ( LP (D),
and
E)
~
q
{ 1 - ~ (2 1 - ~ (2 P -
f.
q) : '
fP)
P,
(!(LP(D), T) S { (1 + TP)~ - 1, (1
+ Tq) 4 -
1,
~~
~
1< p 2 If 2 ~ p < 00
~f 1 < p ~ 2 If 2 ~ p
d·]
is a functional called the unnormJalized optirnal .filter.
In the particular case
ft(x) == f(x(t)), there are two approaches to computing 4>df]. In the first approach (Lo and Ng (1983), Mikulevicius and Rozovskii (1995), Ocone (1983)), the functional 4>df] is expanded in a series of multiple integrals with respect to the observation process. This approach can be used to obtain representations of general functionals, but these representations are not recursive in time. In fact, there is no closed form differential equation satisfied by rPt Lt]. In the second approach (Kallianpur (1980), Liptser and Shiryayev (1992), Rozovskii (1990)), it is proved that, under certain regularity assumptions, the functional 4>t[f] can be written as
4>tUJ =
Jj(x)u(t,x)dx
(1.1)
for some function u( t, x), called the unnormJalized .filtering density. Even though the computation of u( t, x) can be organized recursively in time, and there are many numerical algorithms to do this (Budhiraja and Kallianpur (1995), Elliott and Glowinski (1989), Florchinger and LeGland (1991), Ito (1996), Lototsky et al. (1996), etc.), these algorithms are time consumingbecause they involve evaluation of u(t, :r) at many spatial points. Moreover, computation of 4>df] using this approach requires subsequent evaluation of the integral (1.1). The objective of the current work is to develop a recursive in time algorithm for computing 4>df] without computing u(t, x). The analysis is based on the multiple integral representation of the unnormalized filtering density (Lototsky et al. (1996), Mikulevicius and Rozovskii (1995), Ocone, (1983)) with subsequent Fourier series expansion in the spatial domain. For simplicity, in this paper we consider a one-dimensional diffusion model with uncorrelated noises. In the proposed algorithm, the computations involving the parameters of the model can be done separately from those involving the observation process. If the parameters of the model are known in advance, this separation can substantially increase the on-line speed of the algorithm.
2
REPRESENTATION OF THE UNNORMALIZED OPTIMAL FILTER
Let (O,:F, P) be a complete probability space, on which standard one-dimensional Wiener processes (V(t))t~O and (W(t))t;::o are given. R,andom processes (x(t))t20 and (y(t))t~O are defined by the equations
t t r b(x(s))ds + r a(x(s))dV(s), lo ./0
x(t) == xo
+
y(t) =
h(;r;(s))ds + W(t).
l
(2.1)
In applications, x(t) represents the unobserved state process subject to estimation from the observations y(s), s ::s t. The a - algebra generated by y(s), s ::s t, will be denoted by:Ff. The following is assumed about the model (2.1):
(AI) The Wiener processes
(V(t))t~O
and (W(t))t~O are independent of xo and of each
other; (A2) The functions b( x), a (x), and h (x) are infinitely differentiable and bounded with all the derivatives;
Nonlinear Filtering of Diffusion Processes
201
Xo has a density p(x), x E R, so that the function p == p(x) is infinitely differentiable and, together with all the derivatives, decays at infinity faster than any power of x.
(A3) The random variable
Let j == j (x) be a measurable function such that (2.2) for some ko 2: 0 and L > O. A.ssumptions (A2) and (A3) imply that Elj(x(t))1 2 < 00 for all t 2: 0 (Liptser and Shiryayev, 1992). Suppose that T > 0 is fixed. It is known (Kallianpur (1980), Liptser and Shiryayev (1992)) that the best mean square estimate of j(x(t)) given y(s), S' :::; t :::; T, is j(x(t)) == E[f(x(t))\Ff], and this estimate can be written by the Kallianpur-Striebel forrnula as follows:
.f(x(t))
= E[j~x(t) )p(t) IFl] E[p(t)IFf]
where
p(t)
= exp { l h(x(s))dy(s)
(2.3)
,
~ ~ llh(x(s)Wds},
and E is the expectation with respect to measure P(.) :== J. (p( T) ) -1 dP. Moreover, under measure P, the observation process (y( t) )O~t~T is a \Vicner process independent of
(x(t))Odf], it now seems natural, according to (2.4), to multiply both sides of (2.10) by f(x) and integrate, but this cannot be done in general because (2.10) is an equality in L 2 (R) and f need not he square integrable. The difficulty is resolved by choosing a special basis {en} so that integral JR 1(.r) en (x) dx can be defined for every function 1 satisfying (2.2). Specifically, let {en} be the Hermite basis in L 2 (R) (Gottlieb and Orszag (1977), Hille and Phillips (1957)):
en(x) ==
1
V2
n Jr1/2 n !
2
e- x /2Hn(x),
(2.11)
where Hn(x) is the nth Hermite polynornial defined by
Then the following result is valid. THEOREM 2.3. then
If assumptions (At) - (A3) and (2.2) hold and en is defined by (2.11),
1Jti Lt] ==
L In1Pn( i),
P - a.s.,
(2.12)
n20
where In == JR I(x)en(x)dx and 1f)n(i) is given by (2.9). Proof. Condition (2.2) and fast decay of en(.r) at infinity imply that In is well defined for all n. Then (2.12) will follow frorn (2.4) and (2.10) if the series l:n~o In'l/Jn(i) is P - a.s. absolutely convergent for all i == 0, ... , M. Since measures P and P are equivalent, it suffices to show that (2.13) IInl EI1/Jn(i) I < 00.
L
n20
Arguments sirnilar to those in Hille and Phillips (1957), paragraph (21.3.3), show that
which implies that
Ilk I ~
Cn( 2k o+1)/4.
(2.14)
On the other hand, it follows froIll the proof of Theorem 2.6 in Lototsky et al. (1996) that for every integer ! there exists a constant C (J) such that (2.15) Taking ! sufficiently large and cornbining (2.14) and (2.15 ) results in (2.13). REMARK. It is known (Hille and Phillips (1957), paragraph (21.3.2)) that su Px Ien (x) I ~ en -1/12. Together with (2.15), this inequality implies that, for the Hermite basis, the series in (2.10) converges uniformly in :r E R, P - a.s.
204
3
Lotosky and Rosovskii
RECURSIVE APPROXIMATION OF THE UNNORMALIZED OPTIMAL FILTER
It was already mentioned that the infinite sums in (2.9) must be approximated by truncating the number of terms, if the formula is to be used for practical computations. Multiple integrals in (2.6) must also be approximated. The effects of these approximations are studied below. For simplicity, it is assumed that the partition of [0, T] is uniform (~i == ~ for all i == 1, ... , M). With obvious modifications, the results remain valid for an arbitrary partition. Given a positive integer 1'\;, define random variables ljJn,K (i), n == 0, ... ,I'\;, i == 0, ... , M, by
1Pn,K(O)
==
'l/Jn,,,(i) =
(p, en)o,
t
((PLlCI, cn)o
+ (PLlhcl, cn)o[y(t i ) - y(ti~l)]+
(3.1)
[=0
(1/2)(P~h2e[,en)o[(y(t i ) - y(t i_d)2 - ~])7jJn,K(i - 1), i == 1, ... , M. Then the corresponding approximations to u (t i , x) and ko + 1/2. On the other hand, assumptions (A2) and (A3) imply that u(t,') E L 2 (R, r) for all r E R (Rozovskii, 1990, Theorem 4.3.2), and the same is true for U1(t i , '), U1(t i , '), and UK(t i , .). Fix an even integer r > k o + 1/2 and define iJ(x) :== (1 + x 2 )r/2. Notation Ill· IIlr :==
JEll' 11;
will also be used. By the Cauchy inequality,
JElcPti,K[fJ - cPtiLfJI2 == JE(uK(t i ,·)
J
u(t i , '), f)5 ~
-
11 f 11 ~ rill n t i , .) - U ( t i , .) Ill; ~ 11 f 11- r ( Illu(t i, .) Illu1(ti ,·) - U1(t i , ·)lllr + Illn,l(i-i") - UK(t i , ')lllr)' K (
t
'U 1 ( i, .)
II1 r +
(3.13)
Since the operator Pt is linear bounded from Hn(R, r) to itself (Ladyzhenskaia et al. (1968), Rozovskii (1990)), the arguments of steps 1 and 2 can be rcpeated to conclude that (3.14)
Next, it follows from the proof of Theorem 2.6 in Lototsky et al. (1996) that for every positive integer, there exists C (,) such that for all i == 0, ... , AI - -1 2 C(,) E(u (t i , '), en)o ~ n 2'Y+ r '
(3.15)
Similarly, by (3.12), there is C(,) so that
Illu1(ti ,·) ~ n,,;(t i , ')III~ :::; K2'Y~~!;fl2' On the other hand, repeated application of the relations e~ == (Viien-l - vn+1"e n +l) / V2 and -e~ + (1 + x 2 )e n == 2( n
+ 1 )e n
(3.16)
shows that
n+r/2
(g,e n );/2 ~ C
L
mr(g,em)~
m=n-r/2
(if m < 0, the corresponding term in the sum is set to be zero), and consequently
L (g, e
n
);/2 ~ C
n~O
L
n (g, e n )6· r
n~O
Combining the last inequality with the identities
I/gll; ==
Ilgj)ll~ == L(gj), e n )6 == L(g, e n );/2 n
n
Nonlinear Filtering of Diffusion Processes
207
results in
Illu 1 (t i ,') -
UK(t i , ·)111; ==
L
E(n~l(ti") - UK(t i , '), en );/2 :s;
n2::0
CL n 'ECu 1
K
1
(t i ,·) - 'lLK(t i , '), en )6 ==
n>O
C
I: nTE( ill (t
CL n ECu (t i ,·) T
1
UK(t i , '), en )6+
n=O i , '),
en )6·
n>,',
Now, (3.15) and (3.16) imply
Together with (3.13) and (3.14), the last inequality implies (3.4). REMARK. The constants in (3.3) and (3.4) are determined by the bounds on the functions b, a, 12, and p and their derivatives and by the length T of the time interval. The constants in (3.4) also depend on Land ko from (2.2). The error bounds in (3.3) and (3.4) involve two asymptotic parameters: ~ (the size of the partition of the time interval) and K (the number of the spatial basis functions). With the appropriate choice of these parameters, the errors can be made arbitrarily small. In Lototsky et al. (1996), the multiple integrals (2.6) were approximated using the Cameron-Martin version of the Wiener chaos decomposition. The analysis was carried out only for the unnormalized filtering density, but the results can be extended to the unnormalized optimal filter q>dfJ in the same way as it is done in the present work. The overall error of approximation from Lototsky et al. (1996) has the same order in ~ and K as (3.3), but the approximation formulas are more complicated. Formulas (3.1) and (3.2) provide an effective numerical algorithm for computing both the unnormalized filtering density u( t, x) and the unnormalized optimal filter 1Jt [1] independently of each other. If the ultimate goal is an estimate of i(x(t i )) (e.g. estimation of moments of x(t i )), it can b~ achieved with a given precision recursively in time without computing u(t i , x) as an intermediate step. This approach looks especially promising if the paranleters of the model (i.e. functions b, a, h and the initial density p) are known in advance. In this case, the values of (P~el, en)o, (P~hel, en)o, (1/2)(P~h2el, en)o, and in == (!, Cn)o, n,l == 1, ... ,K, can be pre-conlputed and stored. When the observations become available, the coefficients 1Pn(i) are computed according to (3.1) and'then 1J t il K [!J is COlllputed according to (3.2). As a result, the algorithm avoids performing on line the time consuming operations of solving partial differential equations and computing integrals. Moreover, only increments of the observation process are required at each step of the algorithm.
REFERENCES Budhiraja, A. and Kallianpur, G. (1995). Approximations to the Solution of the Zakai Equations using Multiple vViener and Stratonovich Integral Expansions, Technical Re-
Lotosky and Rosovskii
208
port 447, Center for Stochastic Processes, University of North Carolina, Chapel Hill, NC 27599-3260. Elliott, R. J. and Glowinski, R. (1989). Approximations to solutions of the Zakai filtering equation, Stoch. Anal. Appl., 7(2):145-168. Florchinger, P. and LeGland, F. (1991). Time discretization of the Zakai equation for diffusion processes observed in correlated noise, Stoch. and Stoch. Rep., 35(4):233256. Gottlieb, D. and Orszag, S. A. (1977). Numerical Analysis of Spectral Methods: Theory and Applications, CBMS-NSF Regional Conference, Series in Applied Mathematics, Vo!.26. Hille, E. and Phillips, R. S. (1957). Functional Analysis and Semigroups, Amer. Math. Soc. Colloq. Publ., Vo!. XXXI. Ito, K. (1951). Multiple Wiener integral, J. Math. Soc. Japan, 3:157-169. Ito, K. (1996). Approximation of the Zakai equation for nonlinear filtering, SIAM J. Cont. Opt. ( to appear). Kallianpur, G. (1980). Stochastic Filtering Theory, Springer. Ladyzhenskaia, O. A., Solonikov, V. A., and Ural'tseva, N. N. (1968). Linear and quasilinear equations of parabolic type, American Mathematical Society, Providence, Rhode Island. Liptser, R. S. and Shiryayev, A. N. (1992). Statistics of Random Processes, Springer. Lo, J. T.-H. and Ng, S.-K. (1983). Optimal orthogonal expansion for estimation I: Signal in white Gaussian noise, Nonlinear Stochastic Problems (Bucy, R" and Moura, J., ed.), D. Reidel Pub!. Company, pp. 291-309. Lototsky, S., Mikulevicius, R., and Rozovskii, B. L. (1996). Nonlinear Filtering Revisited: A Spectral Approach, SIAM Journal on Control and Optimization, to appear. Mikulevicius, R. and R,Ozovskii, B. L. (1995). Fourier-Hermite Expansion for Nonlinear Filtering, Festschrift in honor of A. N. Shiryayev. Ocone, D. (1983). Multiple integral expansions for nonlinear filtering, Stochastics, 10: 1-30. Rozovskii, B. L. (1990). Stochastic Evolution Systems, Kluwer Academic Publishers.
A Berry-Esseen Type Estimate for Hilbert Space Valued V-Statistics and On Bootstrapping Von Mises Statistics V.V. SAZONC)V Steklov Mathematical Institute, Moscow, and Hong Kong University of Science and Technology
This paper consists of two parts related to each other only by employing a common approach. This approach consists in using the technique developed for the proof of Berry-Esseen type estimates and Edgeworth type expansions for Hilbert space valued independent random variables. A number of researches contributed to this area and a rather complete account of the related work up to 1990 can be found in the survey paper by Bentkus et al. (1990). Here we will nlention only papers by G()tze (1979), Yurinskii (1982), and Sazonov, Ulyanov and Zalesskii (1988, 1991), which are most closely related to the present work. First consider Hilbert space valued V-statistics. Let Xl, ... ,..:Yn be independent identically distributed (i.i.d.)
random
vari~bles
with values in a measurable space (X, X).
Denote P the distribution of Xl: P(X I E A), A E X.
Let be a map defined on
(X x X, X x X) with values in a separable Hilbert space H such that (XI, X2)
:=:
Xl, X2 E X. The inner product and norm in }f will be denoted (.,.) and
respectively.
11.11
(X2' Xl),
Assume that E (Xl, X 2) == 0 (this assumption is not essential and is made for simplicity) and EII(X 1 , ..:Y2 ) \ I
0 such that I\B(u) - B(v)11 ~ Cllu - vii and IIB(u)11 2 ~ 0 2 (1 + Ilu11 2 ) for all u,v E H. (A3) We assume the Wiener process w(t) takes values in its own separable Hilbert space U. (A4) G : U --+ H is a bounded linear transformation and does not depend on t or u. We will also assume that w (t) has a trace class covariance operator ([DZ92], ch 4.1). That is, Ew(s) ® w(t) == (t /\ s)Q where Q : U --+ U is a positive definite, self-adjoint, bounded linear operator with finite trace. If we denote by Vi and Vi the eigenvalues and (complete orthonormal set of) eigenfunctions of Q then the trace of Q is I:~l Vi and we may write ([DZ92], ch 4.1) 00
w(t) ==
L vfiJif3i(t)Vi
(12)
i=l
as an expansion of w(t) where the f3i's are independent standard real valued Wiener processes. The convergence is in V'. In order to derive our results we will generally restrict the functions f and h to being of class Ct (notation below) but explicit conditions will be given in the theorems below. For the purpose of expressing (and proving) the Zakai and FKK filtering equations, it is useful to model the noise process w (t) (or more precisely Gw(t)) as the sum
(13) where G 2 : R m ~ H is a bounded, full rank linear operator. This is possible by setting G 1 dWl (t) equal to the difference between the first and last terms. The Wiener process Wl (t) takes its values in a separable Hilbert space U 1 (Ut == H is one possibility) and G l : U l --+ H bounded linear. If we use (13) and adjoin a random initial condition, we get
du(t) == [Au
+ B(u)] dt + G1dwl(t) + G2dw2(t) u(O) == uo.
(14)
(15)
We will fix throughout T > 0 and work on the time interval [0, T]. The results of the paper also hold on [0,00), except that in this case convergence,
Stochastic Reaction-Diffusion Equations
223
e.g., in C([O, 00), H), is in the topology of uniform convergence on compact subsets of [0,00). There will be several filtrations Ft of increasing a-fields C F. Usually these filtrations are generated by one or more processes with independent increments; for such processes v(t) we will denote by the completion of a{v(s) : 0 < s ~ t}, and we include in every F~ all P-null sets.
Fr
SOLVABILITY OF THE STOCHASTIC REACTION DIFFUSION EQUATION
3
We can take the following basic probability space:
o == H
X
C([O? T]; U),
F == B(H x C([O, T]; U))
m == Ita x ,\ where B(.) is the Borel algebra, Ita is the distribution of the initial data and ,\ is the Wiener measure. On the basic probability space (0, F, m) given above we will take as our normal filtration F:0'w the complete a-fields generated by Uo and w(·) and all P-null sets ([DZ92], ch 3.3 and 7.1). Since Uo is nO-measurable, and E F, the solution u(t) of (14) and (15) will be a predictable process with respect to this a-field.
no,w
Definition 1 For any H -valued
(14)
no,w
-measurable random variable Uo, a predictable H -valued process u(t), 0 ~ t ~ T is a mild solution of and (15) if, for all t E [0, T]
F:0'w
m{l lIu(s)Wds < oo} = t
i) and
ii) u(t)
= S(t)uo +
I
I
1
t
S(t - s)B(u(s))ds
t
+
S(t - s)Gdw(s), m-a.s..
224
Hobbs and Sritharan
The condition that the process u(t) be predictable with respect to the filtration F;"0'w is important for it will play a role in our main result. This means ([DZ92], ch 3.3) that u(t) = u(t, w) is measurable with respect to PT the (completed) a-field generated by all subsets of [O,T] x n which have the form (s, t] x F where s; S < t S; T and F E F:o,w.
°
Theorem 1 Let assumptions (A 1) to (A4) hold and the initial value Uo be an
H-valued random variable which is independent ofw(·), and with Elluollq < 00 for some q 2 2. Then the initial value problem (14) and (15) has a unique (up to equivalence) mild solution u(t). Further, u(t) has a version whose trajectories are continuous a.s., i.e., u(·) E C([O, T]; H), and there exists q C > 0 (depending on T) such that SUPtE[o,T]Ellu(t) IIq S; C(l + Elluoll ). Denoting by X q the Banach space of ,H -valued predictable processes v(t) such that the norm (suPo~t~TEllv(t)"q)llq< 00, proof of the existence of u(t) is obtained by taking u(t) as the limit of successive approximations of the mapping !( : X q ~ X q defined by
Kv(t)
= S(t)uQ +
I S(t - s)B(v(s))ds + I S(t - s)Gdw(s). t
t
(16)
!{ is a contraction mapping on sufficiently small subintervals of [0, T] ([DZ92], Theorem 7.4 or [Ich82]). We note three things that we will need for Theorems 2 and 3, the main results of this paper: First, although the solution u( t) only satisfies (14) in a mild sense (it need not take values in D(A)), it is H-valued and not just a distributional solution. Second, as a mild solution, u(t) is predictable and hence adapted to the filtration F;"0'w ([DZ92], ch 7.1). An examination of the proof shows that F;"0,w is the smallest a-field that can be used for Ft, so u(t) is indeed F;"0'w adapted. Third, u(t) is a measurable function ofuo and w(s) for S S; t. For on the subinterval (ti, ti+l), u(t) is the limit of a sequence of the form ui(t) = !(ui-1(t) where I{v(t) is the limit of sums of the form
S(t - ti)V(ti)
+ ES(t -
sk)B(v(Sk))(Sk - Sk-l)
k
+ E S(t - Sk)G(W(Sk) - W(Sk-l)) k
(17)
225
Stochastic Reaction-Diffusion Equations
and the sums are taken over a partition {Sk} of the subinterval. We see that each term in these sums is clearly measurable with respect to the starting data (S(t) is continuous) and the O"-field generated by w(t) on the relevant subinterval only.
DERIVATION OF THE FKK AND ZAKAI EQUATIONS
4
In this section we will derive evolution equations for the conditional expectation IIt(f) = E [f(u(·, t))IFtZ]. Let us define the innovation process Y(t) = {Y1(t),···, Yn(t)} as,
Y(t)
= z(t)
-it
IIr(h)dr.
(18)
Lemma 1 [Fl(/(72} Let u(·) be the solution of (9) (Theorem 1), z(t) be the observation process defined in (10) and h(·) E Cb(Rm ). Then (Y(t),Ft,m) is an m-vector standard Wiener process. Moreover, the two sigma fields {Y(r) - Y(s), t ::; s < r ::; T} and Ft are independent.
0"
The following martingale representation result due to Fujisaki-KallianpurKunita [FKK72] is the key to the derivation of the nonlinear filtering equation.
Lemma 2 Every square integrable martingale (M (t), Ft, m) is sample continuous and can be represented as a stochastic integral with respect to the innovation process:
M(t) = £[M(O)] where
E
it
+ ~(s) · dY(s),
s E [0, T],
iT 1~(t)12dt < +00
and «P(t) is jointly measurable in (0, T) x
(20)
n and adapted to Ft.
Definition 2 The class of cylindrical test functions COY
=
{f(·,·) : [-a, T] x H ei E D(A),i
(19)
~ R;
CCY
as follows:
f(t, u) = 4>(t, (u, el),···, (u, en)),
= 1,··· ,n;4> E Cgo((-a;T)
x Rn)} ,a> O.
(21)
226
Hobbs and Sritharan
We now define (22) where
f
E
CCY
and £ given by (5).
Lemma 3 For all gale in [0, T].
f
E CCY ,
(M f (t), Ft, 'In) is a square integrable martin-
This follows from the fact that for the mild solution, combining the results in [DZ92] and [Vio76] we can conclude that, for f E CCY ,
Mf(t):= j(t,u(t)) - j(O,u(O))
-It(~~(s,u(s)) +£j(s,u(s)))ds
(23)
is a square integrable ~,UO-martingale (see also [HSS95] for details). We n.ow note the following estimate,
E IT Ij(u(t)h(u(t))1 2 dt < +00,
(24)
Under the condition (24) we can follow the method in [FKK72] to obtain the explicit form of tP(t) in (19) using lemmas (2) and (3): (25) where M is defined in (6). We thus get the :Fujisaki-Kallianpur-Kunita equation (4) for
f
E
CCY.
We have due to the boundedness· of h,
E IT Ills(h)1 2 ds < 00. Define Bt(f) for f E
et(J)
CCY
(26)
as,
= llt(J) exp {It lls(h) · dz(s) - ~ It Ills(h)1 2 ds} .
(27)
Then by Ito formula (see [HSS95]) we get the Zakai equation (7) for f E CCY.
227
Stochastic Reaction-Diffusion Equations
5
KOLMOGOROV'S BACKWARD EQUATION
The proof of the uniqueness of measure valued solutions in Theorem 3 will be based on having a unique solution of Kolmogorov's backward equation,
+ h(v)· ~(t)(t,v),
t > T,V E D(A),
(28)
and
(29)
(T,V)=='l1(V), vEH.
Definition 3 A strict solution to (28)-(29) is a function
qT} = fj; [fj; [w( u(T, 0, v) )PTIFT]qT] .
(48)
Since processes of the form qt defined above are dense in L 2 (0, Ft , m) [Roz90], we conclude that
< eT, \IJ >= E [\IJ( u(T, 0, v) )PTIF~], m-a.s.
(49)
Since for an arbitrary measure-valued solution TIt of the FKK equation, et defined by (39) satisfies (36), we have thus established the uniqueness of TIt and et in the interval [0, T].
References [Ahm94] N. U. Ahmed. Nonlinear filtering for stochastic differential equations in Hilbert spaces. In W. F. Ames, editor, 14th IMACS World conference on computational and applied mathematics, pages 5-8, 1994. [DZ92]
G. DaPrato and J. Zabczyk. Stochastic Equations in Infinite Dimensions. Cambridge University Press, New York, 1992.
[FKK72] M. Fujisaki, G. Kallianpur, and H. Kunita. Stochastic differential equations for the nonlinear filtering problem. Osaka J. Math., 9:1940, 1972. [Get75]
R.K. Getoor. On the construction of kernels. In P.A. Meyer, editor, Seminaire de Probabilites IX. Lecture Notes in Mathematics, vol465. Springer-Verlag, 1975.
234
Hobbs and Sritharan
[HSS95] S. L. Hobbs, G. Sobko, and S. S. Sritharan. Nonlinear filtering theory of stochastic semilinear partial differential equations. To be published, 1995. [Ich82]
A. Ichikawa. Stability of semilinear stochastic evolution equations. J. Math. Anal. Appl., 90:12-44, 1982.
[Roz90]
B.L. Rozovskii. Lecture notes on linear stochastic partial differential equations. Lecture Notes 25, Dept. Math., University of North Carolina, 1990.
[Roz91]
B.L. Rozovskii. A simple proof of uniqueness for Kushner and Zakai equations. Stochastic Analysis, ed. E. Mayer-Wolf and E. Merzbach and A. Schwartz:449-458, 1991.
[Sri94]
S. S. Sritharan. Nonlinear filtering of stochastic Navier-Stokes equation. In T. Funaki and W. A. Woycznski, editors, Nonlinear Methods on Stochastic Partial Differential Equations: Burgers Turbulence and Hydrodynamic Limit. Springer-Verlag, 1994.
[Vio76]
M. Viot. Solution faibles D 'equations aux derivees partielles stochastique nonlineaires. These, tJniversite Pierre et Marie Curie, Paris, 1976.
[VK84]
M. J. Vishik and A. I. Komech. On Kolmogorov's equations corresponding to the two dimensional stochastic Navier-Stokes system. Trans. Moscow Math. Soc., pages 1-42, 1984.
[Zak69]
M. Zakai. On the optimal filtering of diffusion processes. Wahrscheinlichkeitstheorie. Verw. Geb., 11 :230-243, 1969.
Z.
An Operator Characterization of Oscillatory Harmonizable Processes RANDALL J. SWIFT Department of Mathematics, Western Kentucky University, Bowling Green, Kentucky
Dedicated to Professor M.M. Rao, advisor and friend, on the occasion of his 65th birthday.
1
INTRODUCTION
A class of nonstationary stochastic processes which are encountered in some applications is the class of modulated stationary processes X(t). These processes are obtained when a stationary process Xo(t) is multiplied by some nonrandom modulating function A(t):
X(t)
==
A(t)Xo(t).
This class of processes has been investigated by Joyeux (1987) and Priestley (1981). The book by Yaglom (1987) provides a nice treatment of these processes. In particular, if A(t) admits a generalized Fourier transform, the class of oscillatory processes, studied by Priestley (1981) is obtained. In sorne physical situations, the assumption of stationarity for the process X o(t) is unrealistic R,ao (1982). If this condition is relaxed, and Xo(t) is assumed to be harmonizable and if A(t) admits a generalized Fourier transform, the process X(t) is not oscillatory, but is oscillatory harmonizable. This paper investigates the properties of oscillatory harmonizable processes. Section 2 recalls the basic theory of harmonizable processes required for the subsequent analysis. Section 3 introduces and develops the class of oscillatory harmonizable processes. In this section, the spectral representation of oscillatory harmonizable processes is obtained. This representation is used to deduce relationships between the oscillatory harmonizable processes and 235
Swift
236
other classes of nonstationary processes. Section 4 obtains an important and useful operator characterization for oscillatory harmonizable processes.
2
PRELIMINARIES
In the following work, there is always an underlying probability space, (0,2::, P), whether this is explicitly stated or not. DEFINITION 2.1 For P 2: 1, define Lf;(P) to be the set of all complex valued f E LP(O, 2::, P) such that E(f) == 0, where E(f) == In f(w)dP(w) is the expectation. In this paper, we will consider second order stochastic processes. More specifically, mappings X : IR ~ L6(P), DEFINITION 2.2 A stochastic process X : lR ~ L6(P) is stationary (stationary in the wide or Khintchine sense ) if its covariance r (s, t) == E (JY" (s )..(Y" (t)) is continuous and is a function of the difference of its arguments, so that
r(s, t) == f(s - t). An equivalent definition of a stationary process is one whose covariance function can be represented as
(1) for a unique non-negative bounded Borel measure F(-). This alternate definition is a consequence of a classical theorem of Bochner's (Gihman and Skorohod, 1974), and motivates the following definition. DEFINITION 2.3 A stochastic process X : IR ~ L6(P) is weakly harmonizable if its covariance r(·,·) is expressible as
r(s, t)
=
fIR fIR ei>..s-i>"'tdF(A, A')
(2)
where F : IR x IR ~ C is a positive semi-definite bimeasure, hence of finite Frechet variation. The integrals in (2) are strict Morse-Transue, (Chang and Rao, 1986). A stochastic process, X(·), is strongly harmonizable if the bimeasure F(·,·) in (2) extends to a complex measure and hence is of bounded Vitali variation. In either case, F(·,·) is termed the spectral bi-measure (or spectral measure) of the harmonizable process. Comparison of equation (2) with equation (1) shows that when F(·,·) concentrates on the diagonal A == A', both the weak and strong harmonizability concepts reduce to the stationary concept. Harmonizable processes retain the powerful Fourier analytic methods inherent with stationary processes, as seen in Bochner's theorem, (1); but they relax the requirement of stationarity. The structure and properties of harmonizable processes has been investigated and developed extensively by M.M. Rao and others. The following sources are listed here to provide a partial summary of the literature. The papers by Rao (1978, 1982, 1989, 1991, 1994) provide a basis for the theory. Chang and Rao (1986) develop the necessary bi-measure theory. A study of sample path behavior for harmonizable processes is considered by Swift (1996b). Some results on moving average representations were obtained by Mehlman (1992). The
Oscillatory Harmonizable Processes
237
structure of harmonizable isotropic random fields and some applications has been consid~red by Swift (1994, 1995, 1996a). Second order processes with harmonizable increments has been investigated also by Swift (1 996c). The forthcoming book by Kakihara gives a general treatment of multidimensional second order processes which include the harmonizable class.
3
OSCILLATORY HARMONIZABLE PROCESSES
M.B. Priestley (1981), introduced and studied a generalization of the class of stationary processes. This generalization is given by: DEFINITION 3.1 A stochastic process X : 1R -t L6(P) is oscillatory if it has representation
X(t)
=
fR A(t, A)ei>.tdZ(A)
where Z (.) is a stochastic measure with orthogonal increments and
A(t, A)
=
h
eitx H(A, dx)
with H(·, B) a Borel function on JR, H(>..,·) a signed measure and A(t, >..) having an absolute maximum at >.. == 0 independent of t. Using this representation the covariance of an oscillatory process is
The idea of definition 2.3 provides the motivation for the following definition: DEFINITION 3.2 A stochastic process X : JR -t L6(P) is oscillatory weakly harmonizable, if its covariance has representation
r(8, t) ==
r r A(s, >")A(t, >"')eiAS-iA'tdF(>.., >..')
.J IR .JlR
where F(·, .) is a function of bounded Frechet variation, and
A(t, A)
= fm eitx H(A, dx)
with H(·, B) a Borel function on JR, H(>.., .) a signed measure and A(t, >..) having an absolute maximum at >.. == 0 independent of t. Note that if A(t, >..) == 1, this class coincides with the weakly harmonizable processes. As Priestley's definition provides an extension to the class of stationary processes, definition 3.2 provides an extension to the class of weakly harmonizable processes. Observe, further, that in this definition, for F(·,·) concentrating on the diagonal, >.. == >"', the oscillatory processes are obtained. Thus the oscillatory harmonizable processes also provide an extension to the class introduced by Priestley, which we will now term oscillatory stationary.
Using this definition, it is possible to obtain the spectral representation of an oscillatory harmonizable process X(·).
Swift
238
THEOREM 3.1 The spectral representation of an oscillatory weakly harmonizable stochastic process is:
X(t)
= fm A(t, .>-)eiAtdZ('>-)
where Z (.) is a stochastic mJeasure satisfying
with F(·,·) a function of bounded Frechet variation.
Proof: Let X(·) be an oscillatory weakly harmonizable process. Then, the covariance r(·,·) has representation
r(s, t) ==
r r A(s, A)A(t, A')ei>.s-i>"tdF(A, A').
JmJm
Applying a form of Karhunen's theorem, (Yaglorn, 1987, volume 2, pages 33 - 41) gives the spectral representation of X (.) as
X(t)
= fIR A(t, .>-)eiAtdZ('>-) ,
which is the desired result. 0 The following condition on the signed measure H, for oscillatory strongly harmonizable processes show these processes are actually a subclass of the strongly harmonizable processes. A similar result was obtained by R. Joyeux (1987), for the oscillatory stationary processes. THEOREM 3.2 If X(·) is an oscillatory strongly harmonizable process with
j~ 1H('>-, d:r) 1< CXl uniformly in A E lR, then X (.) is strongly harmonizable.
Proof: Let Z(A) = where A is a Borel set of
fm H('>-, A -
'>-)dZ('>-)
m and A - A == {x - A : x EA} .
.2(.) is a stochastic measure since H(A,') is a signed measure, and uniformly bounded by K. Now set
-,Y(t) ==
r
eiAtdZ(A) . .fm Claim: X(.) is a strongly harmonizable process. If one lets .2(A, B) == E(Z(A)Z(B)) A, B Borel sets of lR, it must be shown that
r r I F(dw, dw') I
-))E(Z(d.>-)Z(d.>-))
fm fm H(.>-, d(w -
.>-))H(.>-', d(w' - .>-))F(d.>-, d.>-')
Oscillatory Harmonizable Processes
239
where F(A, B) == E(Z(A)Z(B)) is of finite \!itali variation since -"Y(t) is strongly harmonizable. Thus,
fIR fIR
I
F(dw, dw')
fIR fIR fIR fIR H(>', d(w I
1
< since Now
I H I (A, JR)
>'))H(N, d(w' - N))F(d>', dA')
1
00
is bounded, proving the claim.
== X(t). So X (t) is strongly harmonizable~ which completes the proof of the theorem. 0 An additional class of processes related to the oscillatory processes is given by: DEFINITION 3.3 An oscillatory weakly harmonizable stochastic process X : 1R --* L6(P) is c-slowly changing weakly harmonizable if
B(>.) =
fIR I x I I H I (>., d:r)
::;
E,
V >.
E
JR.
Slowly changing stationary processes where first considered by Priestley (1981) and are of interest not only in engineering but also in economics. Priestley showed that it is possible to define a spectral measure for these processes. The class of slowly changing harmonizable processes introduced above extend the class of slowly changing stationary processes. The following corollary shows that it is possible to consider a similar concept for the slowly changing harmonizable class.
COROLLARY 3.1 Slowly changing strongly harmonizable processes form a subclass of strongly harmonizable processes.
Proof: The assumption is
.fIR I x I I H I (A, dx)
::; cV>. E JR.
Claim:
.fIR I H
[ (>., dx)
'Pv(F) == 0]
=>
ev
'Pv(F) == 0
( 2 ) Because F
E
pO implies
IFI E
'Pv(F) == 0
(VF E pO).
pO,
(VF E pO)
=>
'Pv(IFI) == 0
=> => =>
['P Jj ( IF I) == 0 (V J-L [fJ.L == 0 (VJ-L E P) ev
=>
v E P.
==
(VF E pO) E
'Pv(IFI) == 0]
P)
=>
=*
Iv == 0]
0
D
Let £(P) be the linear span of P. Then we have the following.
Lemma 3.
£(P) == {v E ca(fl,F) :-'Pv(F) == 0
== {v
(VF E pO)}
E ca(fl, F) : ~J-Ll,' .. ' J-Ln E
3Cl, ... , Cn
> 0, Ivl :S
CIJ.Ll
P,
+ ... + cnJ.Ln} -,
where Ivl means the total variation measure of v and - is the closure in norm topology. Proof. The first equality follows from the previous lemma and the Hahn-Jordan decomposition. l,From this, £(P) is a closed subspace of ca(n, F). Put
x
== {v
E
ca(n, F) : ~J-Ll, ... , J.Ln
E
P, 3Cl,·.·, en > 0,
Ivl :::; CIJ.Ll + ... + CnJ-Ln}'
£(P) 2 X is trivial. Now suppose that there exists v E £(P) \ X. By the HahnBanach theorem and Proposition 1 there exists F E .:\11 (0, F) such that
'Pv(F) =I- 0
rpJj(F) == 0
.and
This contradicts the first equality.
D
(VJ.L E £(P)).
Operator Algebraic Aspects for Sufficiency
249
Proposition 4. M(O, F)/po is a von Neumann algebra and its predual is isometrically isomorphic to £(P).
Proof. This is a direct consequence of
pO == {F
E M(O,F):
'Pv(F) == 0
("'Iv E £(P))}.
0
We denote M(O, F)/po by M(O, F, P) and call it a set of observables. This space can also be constructed as follows. The direct sum
EB LOO(n, F, J-L) == {{f~}~E1' : f~
E
LOO(O, F, J-L)
(VJ-L E P)
~E1'
sup J-L -ess. sup
and
~E1'
If~ I < oo}
is a von Neumann algebra and its predual is
EB L
1
(0, F, J-L) == {{f~}~E1' : fJ1. E L 1 (0, F, J-L)
J1.E1' and
L J1.E1'
JIf,..ldfJ.
" is g-measurable. In Example 2, 9 is also sufficient for P, because M(rl, Q, P) is equal to [00[0,1] EB C l and E(·IQ) maps f E B(O, F) to f ffi f(x)dx .
fo
°
4. Gibbs states on a countable set. Let S be a countable set and the power set of S (the set of all subsets of S ). For each s E S , we define a {O, 1}-function as on 0 by (X E n) where Ix is the indicator function of X on S . It is well known that the weakest topology on 0 induced by {as} sES is totally disconnected, compact, and metrizable. The space C(O) of all complex valued continuous functions defined on 0 is a C*algebra. The Borel field on 0 is denoted by F , which is the smallest a-field on 0 generated by {as} sES' It coincides with the Baire field on 0, which is generated by C(O). Every probability measure is identified with a state on C(O) (i.e., positive linear functional J-L with J.L(I) == 1). EBJ.tEca(f2,F) £oo(n, F, J.L) is known as the enveloping von Neumann algebra which is the second dual Banach space C(O)** of C(O) . For any A ~ S , we denote by FA the a-subfield of F which is generated by {as}sEA. Clearly F 0 == {0,n} and Fs ==:F . We put
[A, A] == {Y EO: Y n A == A}
(A
~
X).
Then :FA is the smallest a-subfield containing {[A, A] : A ~ A} . Let C be the set of all finite subsets of S . A subset {fA} AEC of C(O) indexed by C is called a local specification if:
fA(X) ~ 0
L
(X EO),
fA(AUB) == 1
(B ~ A C )
A~A
for all A E C , and
/1'1 2 (A U B) == fA 1 (A U B)
L
/1'1
2
(A' U B)
A'~Al
for all A ~ Al ~ A2 E C and B with specification {fA} AEC if
~ Al C
•
We say that J.L E pr(O, F) is a Gibbs state
Tsukada
252 Ell (l[A,A] IFAC) (X) == fA(A U
(X n AC))
for all A ~ A E C and X EO. Let P be the set of Gibbs states with specification {fA} AEC . It is known that P is a non-empty compact convex subset of pr(O, F) in the vague topology (see, for example, [Preston, 1974]). For any J.L E P , if A ~ A', EIL(l[A,A/) IFAc) (X) == EIL(l[AnA,A)n[An(A'\A),A'\A) IFAc) (X)
== E tL (l[AnA,A] l[An(A/\A),A'\A) IFAc) (X) == l[An(A'\A),A'\A) (X)EIL (1 [AnA,A] IFAc) (X) == l[An(A'\A),A'\A](X)jA((A n A) U (X
for all A
C
nA
))
~
A' and X E f2. This says that FAc is sufficient for P. We put FAc . Since M(f2, F oo , P) == nAEC M(f2, .rAc, P) , using the martingale convergence theorem on von Neumann algebras (see,[Tsukada,1985]), we conclude the following theorem.
F oo ==
nAEC
Theorem.
.roo
is sufficient for P. REFERENCES
1. P.R. Halmos & L.J. Savage, Application of the Radon-Nikodym theorem to the theory of sufficient statistics, Ann. Math. Statist. 20 (1949), 225-241. 2. F. Hiai, M. Ohya & M. Tsukada, Sufficiency, KMS condition and relative entropy in von Neumann algebras, Pacific J. Math. 96 (1981), 99-109. 3. F. Hiai, M. Ohya & M. Tsukada, Sufficiency and relative entropy in *-algebras with applications in quantum systems, Pacific J. Math. 107 (1983), 117-140. 4. L. LeCam, Asymptotic Methods in Statistical Decision Theory, Springer, 1986. 5. H. Luschgy & D. Mussmann, Equivalent properties and completion of statistical experimaents, Sankya: Indian J. Stat. 47 (1985), 174-195. 6. T.S. Pitcher, A more general property than domination for sets of probability measures, Pacific J. Math. 15 (1965), 597-611. 7. D. Petz, Sufficient subalgebras and the relative entropy of states on a von Neumann algebra, Commun. Math. Phys. 105 (1986), 123-131. 8. C. Preston, Gibbs States on Countable Sets, Cambridge Univ. Press, 1974. 9. M. Tsdukada, Convergence of closed convex sets and a-fields, Z. Wahrsch. verw. Geb. 62 (1983), 137-146. 10. , The strong limit of von Neumann subalgebras with conditional expectations, Proc. Amer. tvIath. Soc. 94 (1985), 259-264. 11. H. U megaki, Conditional expectation in an operator algebra Ill, Kodai Math. Sem. Rep. 11 (1959), 51-64.
Nonlinear Parabolic Equations, Favard Classes, and Regularity GISELE RUIZ GOLDSTEIN t Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, and CERI and Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152
1. INTRODUCTION
Let A be an m-dissipative operator (not necessarily linear) on a Banach space X. By the Crandall-Liggett theorem A deternlincs a contraction semigroup T on (V(A)). The Favard class (or the generalized domain) V( A) is defined to be V(A) = {f E (D(A)) : ~fo IIAAfl1 < oo}.
Here AA is the Yosida approximation of A, namely AA == )..-1 (I - (I - )"A)-l) for).. is not difficult to show that the Favard class can be equivalently defined to be
V(A) == {I ==
E ('0( ..4 )) :
IIT(t)I -
III
:S Mft for some M f > 0 and 0 < t < I}
{I E ('O(A)): for sonIe sequence {gn} E 'O(A) with gn -t
I, Ag n is bounded as
n -t oo}.
Clearly, 'O(A) C 13(A) C ('O(A)),
Partially supported by an NSF grant. 253
>
o. It
Goldstein
254
and one can show V(A) == V(A) if X is reflexive. From our perspective, the most important aspect of the Favard class is the property T(t)(V(A)) c V(A) for each t
> 0,
that is, the Favard class is an invariant set for the semigroup. Hence, the Favard class contains information on spatial regularity of a problem. For example if we can show that Wok,P(f!) C V(A) c Wk,P(f!), (1.1) says the solution u(t) will have spatial derivatives up to and including order k, each of which is in LP(f!). The problem with this method is that V( A) is very difficult to compute explicitly. Our purpose in this paper is to calculate the Favard class explicitly in the case of a nonlinear parabolic problem with degeneracy and to draw some conclusions about regularity.
°
The problem of calculating the Favard class for this problem in the case 'ljJ == with either Dirichlet or nonlinear boundary conditions was studied in (4), (5). In this paper we consider a more general operator with several different types of boundary conditions, so that even in the case where no lower order terms are present this paper gives new results. The main result is Theorem 2. It is stated in Section 2 and proved in Section 3. Section 4 contains some extensions, while Section 5 contains concluding remarks and directions for future research. 2. A SINGULAR NONLINEAR PARABOLIC PROBLEM
We consider the problem
(2.1 ) for x E [0,1] and t E [0,00). Let X
== e[O, 1]; we assume the initial condition u(O, x) == uo(x).
We allow several types of boundary conditions at j
u(t,j)
== 0,1.
== 0
(-l)ju x (t,j) E (3j(u(t,j))
((BC. i )D) ((BC j )N)
Here f3 j is a strictly increasing maximal monotone graph in IR? containing the origin. Thus o E (3j(O), and if Yi E !3(Xi) for i == 1,2 and Xl < X2, then YI < yz. Note that (BC.i)N includes the linear boundary conditions
255
Parabolic Equations, Favard Classes, Regularity
for
0: j
> 0.
We also allow for periodic-like boundary conditions
u(t, 0) ux(t,O)
= u(t, 1)
= ux(t, 1).
Regarding 'P and 'l/J we assume 'P(x,q)
> 0 for
0
< x < 1,'P(x,q) 2: 'Po(:r) where
'P 0 E C [0, 1J, 'Po (x) > 0 for x E [0, 1] \ S and 'P~l E L l
(2.2)
[0, 1].
Here
s=
{x E [0,1] : 'Po(x) = O}.
(Hence meas S == 0.) There exist positive constants L, AI,and N such that
Iv'(x, p, q) 1'l/J(x,p,q)1
- VJ (x,
p,q) l:s Lip - p I
(2.3)
:s M(lpl)(l + 'P(x,q))M(l + Iq!)
(2.4)
:s N(l + Ipl)·
(2.5)
and
IVJ(x,p,O)1
where M : [0, 00) -+ [0, (0) is a continuous nondecreasing function.
In fact the constant L can be replaced by a continuous nondecreasing function £( Iq I), so that (2.3) holds only locally, and our theorems still remain valid. For such extensions see [2]. Let X be the Banach space C[O, 1] with the sup norm. We define the operator A on . .X" by
A.u = 'P(', u')u"
+ 1/J(x, u, u').
Choose one boundary condition at j = 0 and one at j (BCj),(BCj)N, or choose (BC)P. Then we define the set
yr
_
BC -
C[O, 1] C[O,l]nC I [O,l) 1] n CI(O, 1] { Cf[O, I C [O,l]
if if if if
1 from the conditions
(BC!?), (BCl)D hold (BC){j and (BC})D hold (BCo)N and (BCI)D hold (BCj)N holds for j == 1,2, or if (BC)? hold.
256
Goldstein
We define the domain V(A) of the operator A by
D(A) == {u E YBC n C 2 (0, 1) : Au E C[O, 1) and u satisfies the chosen boundary conditions at x == 0,1}.
Theorem 1: A is m-dissipative on X.
°
This result is due to J.A. Goldstein and C.Y. Lin [8] in the special case tP == 0, <po > and
°
such that
+ a) 1/ (x, y, ~ + a) and for all n == 1,2···. Let Mo == Ilcpo-llll. Then for all n 2: 1,
°
°
(3.1)
Parabolic Equations, Favard Classes, Regularity
257
Define two quantities
wL(f; J) = sup{JE I f(x)ldx : E is a subinterval of [0,1] with
IEI < 8},
and
wc(f;J)
=
sup{lf(x) - f(y)1 : x,y E [0,1] with Ix -
yf S; J}.
[1]) that for J E (o,~], and if
It is not difficult to show (cf. fIt E LI[O, 1], we have
IIf'lloo
~
f
E YBC
n G2(0, 1)
411fll00 + III"11t
with
(3.2)
111' 1100 s:: ~ 111' 1100 + w df"; J).
(3.3)
°
Also, notice that if {fn} ~ L 1 [0, 1], then the statement that for every E > 0 there is a such that WL(!n; J) < E for all n is equivalent to saying that the sequence {In} is uniformly integrable on [0,1]. Similarly, the statement that for each E > there is a fJ > such that wc(fn; 8) < E for all n is equivalent to saying that the sequence {fn} is equicontinuous on
°
°
[0,1]. In our problem if we write
"() Un X
it follows that, given
E
~
0 and all n since is integrable. Together the estimates (3.1) and (3.2) show that {u~} is a pointwise bounded sequence in G[O, 1], while (3.4) and (3.5) show that {un} is an equicontinuous sequence in GI[O, 1]. Hence by the Arzela-Ascoli theorem there is a subsequence, which we again denote by {un} which converges uniformly to a function u E GI[O, 1]. lYsing the boundedness of {Au n }, we have, at least for some subsequence,
and
in X,
Un
-+
u~
-+ u' in . .X",
u~
-+ u" a.e.
U
Goldstein
258
for some
cS
sufficiently small. It also follows that
where Au = c.p(x, u')u" + ~(x, u, u'). Thus it remains to show that the boundary conditions hold. In the case of Dirichlet (BCj)D or periodic-like boundary conditions (BC)?, the result follows by the uniform convergence of Un to u. In the case of the nonlinear boundary conditions (BCj)N, the result follows from the closedness of the graph (3j.
U
This completes half of our proof. Next, we show that D ~ V(A). Let can be written uniquely in the form
{X (Y
= a + bx + 10 10
u(x)
U
E D. Clearly,
(3.6)
u"(s)dsdy.
Since the continuous functions are dense in L 1 [0,1], we can choose a sequence {In} ~ C[O, 1] with (a) in -+ u" a.e. and in L 1 (0,1) (b)
lin(x)l:S 2I u"(x)1 + 1 a.e. for
+ ~(x,
(c) sup Ilc.p(x, u~)u~~ where
Un
Un,
n
u~)lloo
2: 1 :S lV
O. Let Ui - AAui == hi for i == 1,2 when A > O. To prove dissipativity we must show IIUt - u2110c> :S IIh I - h 2 110c>' Choose XQ E [0,1] such that IluI - uzll oo == (UI - U2)(XQ). When 0 < XQ < 1, the proof proceeds as usual by the first and second derivative tests. Now consider the case XQ == O. (The case XQ == 1 is similar.) Evaluate Uj - AAui == hi at 0 and use the boundary condition AUj(O) + bjUi(O) == 0 where ba 2: O. Then ui(O)(l + AbQ ) == hi(O); whence,
11 U 1
-
U
2)(0) == (1
211 co == (u 1
-
U
Ilh I
-
hzll oo .
:S
+ Abo) -1 ( h 1 ( 0) -
h2(0) )
This implies the dissipativity of the operator. Favard classes associated with this type of boundary condition seem quite difficult to classify. We plan to study these objects in the future.
References 1. J.R. Dorroh~ and G.R. Rieder, A singular quasilinear parabolic problem in one space dimension, J. Diff. Eqns. 91 (1991), 1-23. 2. J.R. Dorroh, and G.R. Goldstein, Existence and regularity for singular parabolic problems, in preparation. 3. G. R. Goldstein, Nonlinear singular diffusion with nonlinear boundary conditions, Math. Meth. Appl. Sci. 20 (1993), 1-20. 4. G.R. Goldstein, J.A. Goldstein, and S. Oharu, The Favard class for a nonlinear parabolic problem, in Evolution Equations (ed. by A. C. McBride and G.F. Roach), Longman, Pitrnan Notes, Harlow (1995), 134-147. 5. G.R. Goldstein, J.A. Goldstein and S. Oharu, in preparation. 6. G.R. Goldstein, J.A. Goldstcin and S. Romanelli, in preparation. 7. J.A. Goldstein, Semigroups of Nonlinear Operators, in preparation. 8. J. A. Goldstein and C. Y. Lin, Singular nonlinear parabolic boundary value problems in one space dimension, J. Diff. Eqns. 68 (1987), 429-43. 9. J. A. Goldstein and C. Y. Lin, Highly degenerate parabolic boundary value problems, Diff. Int. Eqns. 2 (1989), 216-227. 10. 1. Rada, K. Hashimoto and S. Oharu, On the duality map of (1979), 71-97.
fCXJ,
Tokyo J. Math 2
11. K. Hashimoto and S. Oharu, On the duality mapping of LCO(O, 1), to appear.
Parabolic Equations, Favard Classes, Regularity
263
12. G. R. Rieder~ Spatially degenerate diffusion with periodic-like boundary conditions~ in Differential Equations with Applications in Biology, Physics, and Engineering (J. A. Goldstein~ F. Kappel, W. Schappacher, eds.), Lecture Notes in Pure and Applied Math., Marcel Dekker, New York (1991),301-312.
Index
absolutely 2·sumnting 124
configuration 40
abstrad Wiener space 149
constant conditional variances 49
agents 114
constraint 153, 154, 155, 158
amplitude-frequency modula.tion 22
convex pla.ne polyhedron 153
analytic sentigroup 86
coordinate 153, 154
a.ssignment economy 114
Cramer·Jtao inequalities 2 Crandell.Liggett theorem lOS, 253
Bana.ch function space 4 basic:
constrain~
Crandall.Liggett.8enilAn theorem 105 155, 163 164 current constraint 156
ba.sic vertex 155
current pivot row 156, 157
Bayesia.n boatstrap 211
current pivot column 157 Berry-Esseen type 209, 213 cylindrical test function 225, 232
Besicovitch-Orlicz space 5 Birkhofl' normal form 22
Dj)«loc) 57
bond 40
1l2-condjtion 180
bootstrap 209 211
degenera.te evolution equation 85
boundary conditions 254,255
degenerate vertex 155, 156 det,erminant 160, 161, 164
CaJIonica.l form 158, 165 differenti~le semigroup 94
Cantor distributed 56
diffusion approximation 85 centrallimJt theorem diffusion matrix 42
in Hilbert Space 210 Clarkson
inequaljt~
diffusion process 199
188, 191
direct limit 14 classical multivariate Dirich1el form 40, 41 normal distribution 46, 49 dj&&ipath-e 106 cla.ssical normal model 57 CIement-Timmerrnl\.lls theorem 99, 103
Eckart-Young theorem 71
conditional expectation 219,220,221
Edgeworth type expansion 209 265
Index
266 elli ptically con toured 57
generalized Cramer-Rao inequalities 2
elementary column matrix 154
generalized random fields 11
elementary column operation 154
Gibbs measure 41
elementary row operation 154
Gibbs state 251, 252
embedded Markov chain 138
Girsanov transform 232, 233
t-slowly changing weakly
Giitze lemma 211
harmonizable 239
gramian 130
evolution equation 85, 225
gramian orthogonally scattered 124
experiment 248
gramian orthogonally
explosive process 8
scattered dila.tion 125
Faber theorem 217
Haar subspace 61
Favard class 253
Hamiltonian perturba.tion 17
Fenchel-Orlicz space 5
Hardy-Orliez space 5
Fernique theorem 149
harmonizable 9, 237
Feyman-Kac formula 227
harmonizable process 9, 10
filtering 219, 220
Hida measure 147
F'richet space 148
Hilbert B(H)-module 124
Frechet variation 237
llilbert matrix 152
FrobeniuB norm 72
Hilbert-Schmidt operator 123
Fujisaki-Kallianpur-Kunita
Hilbert space valued
(FKK) filtering equation 220,226,228,229,233 GCS2(k) 50, 53
GauBsian conditional structure of second order 45,50,52,54,57,58
U-statistic 211 Hoeffding decomposition 209 hydrodynamic scaling 41 idem potent operator 2 indirect utility function 115
Gaussian measure 171
ill-conditioned matrix 157
Gelfand density 117
inference 13
Gelfand weak • density 117
inner product 153
Index
167
innovation process 225
ma.x.imal monotone gTaph 254
in~elading par~icles
measure-valued solution
39
infinitely di vi si ble 57
219,220,228,229,234
invariant 159
Melnikov condition 22
Ita formula 232
Melnikov theorem 17
James' constant 179, 180, 181, 184
microscopic p.icture 39
mild solution 223, 224, 226 Kagan cla.ss 57
minimal project.ion 61
Kallianpur·Kunita
minimal LI projection 61
filtering equation 219 KAM
~heory
17
MJnlos theorem 148 misrepresenta.tion 116
Kantor' inequality 176
MHtag-Leffier function 148
KdV equation 23
mixing c.onditionJl 41
Kolmogorov's backward equa.tion 227,232 Kompanee~s equation
102
Kwapien's example 51, 58
Loo factorization 215 lattice gas model 39 linear regressions 46,48, 49 linear structure 46 local density function 40
mixtures 55 module 124 modulus of convexity 190 modulus of smoothne s 190 Mors~Tra.nsue
237
multiple Wienel integral 199 multivariMe normal distribution 46 V,-condition 180
local specification 250
nonhasic constraint 155, 156, 164, 169
Lusln space 230
nonlinear d1ifulrion 42
Lyapunov-Sc.hmidt decomposition 25
nonlinear filtering 199,219 nonlinear prediction 5
M/M{l 137 macroscopic para.mder 39
nonlJnear wave equa.tion 31
majorant 125 manipula~ion
nonlinear SchrOdinger equation 32
119
non-manipula.ble 119
268
Index
nonsquare constant 179
qoasi-Gaussian distribution 45
nonnaJ Hilbert B·modulc 124
qoasi.peri~c
normal conditionals distribution 48
queueing system 137
normal conditionals model 47 normaJs 154 n-step transition probabilities 138
evolution 17
randomization 138 random number generator 157 range condition 106
objective function 153, 157
aaa's interpolation theorem 189
observable 249
real variables 153
operator semi variation 127
recursive filter 199
operator stationary 131
recursi ve formula 161
operator stationary dilation] 31
redundant constraint
Orlicz space 3 179, 180
158, ]59, 164, 165, 166. 168, 169
orthogonal invariance 74
reduced echclon form 163, 165
orthogonally scattered 124
reflection principle 138
orthogonally scattered dilation 124
regularity 261
oscillatory stationary 239
representing measures 114
oscillatory weakly barmonizable 23
reservation vaJ ue 114
parabolic 254 partial pivoting strategy 154 perfect competition 114 periodic lattice 39 permutation 159, 168, 169
resonant set 2] Riesz space 4 r·semi stable measure of index a 172 semivariation ] 27
persistency 17
scalarly weakly barmonizable 131
population measure 114
scalar1y weakly
predictable 223, 224 projective limit 14 projective limit topology 148
stationary dilation 131 Schaffer constant 180 Schwartz space 147 second order IIQ process 7
Index
269
semi·stabl 171
U-statistic 209
simplex method 153, 155
utility function 114
simplex pivoting strategy 154 simplex strategy 154, 155, 156, 166 singular site 27
vector measure 113 Vitali variation 237 von Mises w 2 statistic 211
singular value decomposition 76 small balls 171
Walrasian allocation 117
spectral bi-measure 237
Walrasian equilibrium 115
spectral dilation 124
Walrasian prices 115
stable probability measure 171
weak convergence 56
stationary 236
weakly harmonjzable 131, 237
stochastic reaction-diffusion
weakly operator harmonizable 131
equation 219
weakly of class (C) 240
strongly continuous semigroup 85
weakly stationary 130
strongly harmonizable 237
weakly stationary dilation 131
sufficiency 12, 245
well-localized 18
symmetric semi·stable measure 171
Wentzell boundary condition 85, 102
trace class covariance 222 trace class operators 123 trace class whi le noise 220 lransient probabilities 138
Wiener chaos 199 Wiener measure 224 Wiener process 220, 225 Yosida approximation 254
translation 154 transpose 154 two-majoriant 125 uniform marginals 56 uniformization 138 uniformly nonsquare 179, 180 updating subroutine 157
Zakai equation 221,226,228,229
about the book Covering the are of modem analy i and probability theory, thi e citing Fe 1chrift pre eOl ,coll lion of paper gi en at the conference held in honor of the 65th birthda of . M. Rao, h prolifj publi hed re ear h loin lude th well-r cei ed Marcel Dekker, Inc. book Theory of Orlicz Space and Conditional
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