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0, is said to be irreducible at time to > 0 if, for arbitrary non empty open set r and all x E F,
Pt.(x,F) = PtoXr(x) > 0. Again by the semigroup property if a Markovian semigroup Pt, t > 0, is a strongly Feller (resp. irreducible) semigroup at time to > 0 then it is a strongly Feller (resp. irreducible) semigroup at arbitrary time
t > to. The following result is due to R.Z. Khas'minskii. Proposition 4.1.1 If a Markovian semigroup Pt, t > 0, is a strongly Feller semigroup at time to > 0 and irreducible at time so > 0, then it is regular at time to + so.
Proof - Assume that for some xo E E and F E E, Pto+so (xo, F) > 0. Since Pto+so (xo, r) =
Pso (xo, dy)Pto (y, F),
JE for some yo E Ewe have Pto (yo, r) > 0. Since Pto (y, r) = Pto (Xr)(yo) and Pto(Xr) E Cb(E), because the semigroup Pt, t _> 0 is a strongly
Feller semigroup at time to > 0, there exists ro > 0 such that
Pto(y,F)>0forallyEB(yo,ro) Consequently, for arbitrary x E E,
Pto+so(x,F) = JE0(0) >
J (yo,ro) PO (x, dy)Pto (y, F) > 0.
We used here the fact that Pso(x, B(yo, ro)) > 0 and Pto(y, F) > 0 for all y E B(yo, ro). Thus if Pt0, +,0(xo, F) > 0 for some xo E E then Pto+so (x, r) > 0 for all x E E and the (to + so)-regularity follows.
Regular Markovian systems
4.2
43
Doob's theorem
The main result concerned with regular semigroups is due to J. L. Doob [56] and can be formulated as follows.
Theorem 4.2.1 Let Pt, t > 0, be a stochastically continuous Markovian semigroup and µ an invariant measure with respect to Pt, t > 0. If Pt, t > 0, is to-regular for some to > 0, then (i) p is strongly mixing and for arbitrary x E E and F E E
lim Pt(x, F) = µ(r).
t-++o3
(ii) p is the unique invariant probability measure for the semigroup Pt, t > 0. (iii) p is equivalent to all measures Pt(x, ), for all x E E and all t > to.
Proof Step 1 - µ is ergodic Let r E 9 be an invariant set PtXr = Xr, for all t > 0,
(4.2.1)
such that p(F) > 0; we have to show that µ(F) = 1. From (4.2.1) Pt0, (x, F) = 1, x E F, p-a.s., and for µ(F) > 0 there exists xo E I' such that Pt0, (xo, F) = 1. Since all probabilities Ptp(x, ) are equivalent, we have Pt0, (x, F) = 1, d x, E E. It follows that
p(F) = JEo() = 1. Step 2- p is the unique invariant measure If v is another invariant probability measure for Pt, t > 0, then v is ergodic by Step 1. Since the two measures µ and v are equivalent with respect to any measure P(to, x, ), x E E, they are equivalent. By Proposition 3.2.5 we have p = v.
Step 3- Pt, t > 0 has no angle variable
Chapter 4
44
Assume that for some A E R, A # 0, we have
Pt f_eaatf, t > 0.
(4.2.2)
We will show first that If is constant. Since µ is ergodic, it is enough to prove that
PtIfI = If I, µ-a.s.
(4.2.3)
By (4.2.2) it follows that
If(x)I = IPtf(x)I 0. Then, taking into account that µ is invariant,
4 If(x)Ii(dx) = 1K If(x)Ip(dx)
+f
C
If(x)Ii(dx)
< fh PtIfl(x)µ(dx) + I Ptifl(x)µ(dx) =
f EPtl fl (x)jt(dx) = f EI f(x)I u(dx),
a contradiction. So I f Now we show that
f(X(t)) =
is constant.
f (X(0)), t > 0, Pl''-a.s.,
(4.2.4)
where X (t), t E R, is the canonical process. We have in fact, using (4.2.3),
E'`[I f(X(t)) - e%at
f(X(0))11] = E [I f(X(t))I2] + Eµ[I f(X(O))I2]
-e-'AtE, µ[f (X (t)) f (X (0))]
- ezatEµ[f(X (t)) f (X (0))]. (4.2.5)
Since the process X (t), t E R, is stationary, we have
=1µ[If(X(0))I2],
Regular Markovian systems
45
and since it is a Markov process
E"If(X(t)) f(X(O))l = i" {E"[f(X(t)) f(X(o))I.Fol}
= i" {f(X(o)) E"[f(X (t))I .Fo]} = EE" {f(X(o)) Ptf(X (o))} = e:'tE" {If(X(o))12} . Now the conclusion (4.2.4) follows on substituting this in (4.2.5). Let r be a Borel set on the circumference {A : JAI = 11. We show now that if ]P"(f (X (t)) E r) > 0, then we have
Pt(X(0), f-1(r)) = 1, P"-a.s. In fact
P"(f(X(t)) E r) = P"(X(t) E f-1(r))
= )E" [Pt(X(0),f-1(r))] and so, by (4.2.4), P"(f(X (t)) E F) = 1P"(eiat f(X (o)) E r)
= ]"(f(X (t)) E r, e'at f(X (0)) E r)
J e'atf(X(0))EF} Pt(X(O), f-1(r))dµ. So if ]P"(f(X(t)) E r) > 0 we have Pt(X(0), f-1(r)) = 1, P"-a.s., as required.
To complete the proof of Step 3 set v = C(f(X(t))). Then v is concentrated on the circle {A E C : IAI = 1}. Let ei° belong to the support of v. Then for all n E N P
(f(x(t)) E S(c- n,c+
n> 0,
where
= {AEC: c-1 < argA 0, x E H, and so P(t, x, f-1(c))µ(dx) = 1 p(f-1(c)) = JE and f is a.s. constant. Step 4 - For arbitrary x E E and r E E, we have
t
iim Pt(x, r) = µ(r) 00
(4.2.6)
From the absolute continuity of P(x, dy), u > to, x E E, with respect to y there exists a density p(u, x, z) such that
PP (x, I') = fp(n,x,z)(dz), V F E , u > to. On the other hand, for t > u > to and r E E Pt(x, r)
JE Pu(x, dy)Pt-u(y, r)
f EPt-u(y, r)p(u, x, y)µ(dy) ]Eµ
(Pt-u(X (u), r)p(u, x, x (u))) = 1E (/3 Uta), (4.2.7)
where a = Xr (X (0)), 0 = p(u, x, X (U)).
Regular Markovian systems
47
We have in fact
]EI(QUt(a)) = E"( Q Xr(X (t))) = ]Ea [,QIEa (Xr(X (t))ITu)] = ]Eµ [0 Pt-ux (X (u))] = El [3 Pt-u(X (u), r)] .
By Proposition 3.2.1 and Step 3, the canonical dynamical system is weakly mixing. Thus there exists a set I C [0, +oo[ of relative measure 1 such that
lim Pt(x, r) = ,t(r),
ItI- +00 tEI
for arbitrary x E E, r E 8(E). We show finally that the limit in (4.2.6) does exist without any restriction on t. Let t - +oo, then there exists s = s(t) such that 3 < s(t) < 3 and s(t) E I, t - s(t) E I. This is true for sufficiently large t. Since Pt (x, F)
=
JE P5(t)(x, dy)Pt-s(t)(y, A),
the result follows from the previous considerations and by Lemma 4.2.2 below.
Lemma 4.2.2 Let r be a Bored set in [0, 1] with A(r) > a where A is the Lebesgue measure. Then there exists s E IF such that 1 - s E F.
Proof - Let
r={1-s:sEr}.
Since A(r) = A(r) > 2 we have A (r n r) >0. Remark 4.2.3 Using different methods J. Seidler [136] and L. Stettner [147] were able to show that, under the conditions of Theorem 4.2.1, the convergence in (i) can be replaced by the convergence in the variation norm.
Part II
Invariant measures for stochastic evolution equations
49
Chapter 5
Stochastic Differential Equations Part II of these lecture notes deals with general methods of studying invariant measures with respect to Markovian semigroups Pt, t >_ 0, determined by solutions of stochastic evolution equations on a Hilbert or a Banach space. In this chapter we summarize basic results on stochastic integration and on stochastic evolution equations on infinite dimensional spaces. We recall important inequalities for stochastic integrals and stochastic convolutions and prove the existence of a regular solution to a class of evolution equations with Lipschitz or locally Lipschitz drift and diffusion coefficients. Then dissipative systems are discussed in some detail. Regular dependence of solutions on initial data are established as well.
5.1
Introduction
We deal here with equations of the form:
dX = (AX + F(X))dt + B(X)dW(t), (5.1.1)
X(0)=
,
51
Chapter 5
52
As a rule the process W(t), t > 0, will be a cylindrical Wiener process on a Hilbert space U. Moreover F and B are nonlinear transformations and A the infinitesimal generator of a strongly continuous
semigroup S(t), t > 0. Precise definitions of all the data related to problem (5.1.1), and to the solution X to (5.1.1), will be given in §5.2 and in §5.3.
If X(t,x), t >_ 0, x E E, denotes the unique solution to (5.1.1) with = x constant, then the transition semigroup Pt, given by
Ptw(x) = IE[cp(X (t, x))], t > 0, x E E, cp E Bb(E).
t > 0, is (5.1.2)
If there exists an invariant measure it for Pt, t > 0, given by (5.1.2) then the solution to (5.1.1) corresponding to a random variable such that C(e) = µ is a stationary solution to (5.1.1), and this solution has important limit properties, see §3.5. In the following chapters we shall concentrate on basic principles leading to the existence of invariant measures for Pt, t > 0, given by (5.1.2) and to other related properties. These include uniqueness, ergodicity, mixing and regularity of the density of the invariant measure with respect to a natural reference measure. The present chapter, however, contains a selection of basic results on stochastic evolution equations to which we will often refer.
5.2
Wiener and Ornstein-Uhlenbeck processes
Let U be a separable Hilbert space and it a Gaussian measure on U. The measure y is completely determined by its mean value m E U and its covariance operator Q which is a trace-class nonnegative linear operator. Such a measure will be denoted by N(m,Q). A U-valued stochastic process W(t), t > 0, defined on a probability space (St, 1', P) is called a Q-Wiener process if it has the following properties.
(i) W(.) has continuous paths and W(0) = 0. (ii) W() has independent increments.
Stochastic Differential Equations
53
(iii) For arbitrary t > 0 and h > 0
G(h-112(W(t + h) - W(t))) = N(m, Q). We will always assume that the probability space is equipped with
a right-continuous filtration {.Ft}t>o such that FO contains all sets of IP-measure zero. The Wiener process is assumed to be adapted
to {Ft}t>o and for every t > s the increments W(t) - W(s) are independent of Ft. Let Uo = Image Q1/2 be a Hilbert space equipped with the scalar product (u,V)o =
(Q-1/2u,Q-ll2v),
u,v E Uo,
where Q-1/2 is the pseudo-inverse of Q1/2, and let {ek} be an arbitrary, complete, orthonormal basis in U. Then the formula W(t) = E00 k(t)ek, k=1
t > 0,
(5.2.1)
where /3k, k = 1, ..., are independent real processes defines a QWiener process on U. The series (5.2.1) converges in U, IF-a.s. uniformly on an arbitrary finite time interval [0, T]. Note that for arbitrary g, h E Uo and t, s > 0 E(g, W(t))o(h, W(s))o = (h,g)o t A S.
This is why the process W is also called a cylindrical Wiener process
on Uo. Its covariance operator with respect to Uo is the identity operator. If W is a Wiener process on U with the covariance operator different from the identity then it is called a coloured Wiener process.
Example 5.2.1 An important example of a coloured Wiener process is a process on U = L2(Rd) used to describe "Random environment", see D. A. Dawson and H. Salehi [53]. The covariance operator is defined as the convolution operator
Qu(a) = fd q( - 77)u(rl)drl, u E U,
(5.2.2)
Chapter 5
54
where q is a continuous, positive definite function: N
_ E Rd, a1i...,AN E C,N E N.
0, 1,..., Moreover by Bochner's theorem
q(\) = fd e-(A,')ir(dx), A
E ][Yd
where a is a nonnegative, finite measure, the so-called spectral measure. The Wiener process W(t, e), t _> 0, E Rd, corresponding to
Q has the property that for fixed t, W(t, ) is a stationary Gaussian field with the correlation function EW(t, e)W (t,,1) =
Vtiq( - 1)), e, rl E Rd.
Typical examples: q(e) = e-1E1°, E Rd, for arbitrary a E ]0, 2]. The corresponding spectral measures are then isotropic a-stable distribu-
tions, and in particular for a = 2 the spectral measure is Gaussian. Note that the operator Q is never compact on U and thus never of trace class. A natural generalization of this example consists in replacing Rd by a group G with a translation invariant measure v and in considering covariance operators Q on U = L2(G, v) as generalized convolutions
Qu(e) = J q( - r))u(r))v(di),
E G, U E L2(G, v).
It is sometimes convenient to consider Wiener processes represented in a form more general than (5.2.1):
W(t) = E00akek0k(t), t > 0,
(5.2.3)
k=1
where {ak} C R and the basis {ek} consists of eigenvectors of a self-adjoint operator A.
Example 5.2.2 The sequence cos ne,
sin n
n E N,
E [0, 2r],
Stochastic Differential Equations
55
forms a complete orthonormal basis in U = L2(0, 21r). Its elements are eigenfunctions of the operator A=
,
D(A) _ {x E H2(0, 2ir) : x(O) = x(2ir), x'(0) = x'(2zr)}.
2
In this case the series (5.2.3) can be equivalently written as follows: +00
W(t,e) _
(a.,3.1 (t) cos ne + bn,Q,n(t) sin ne), t > 0, E [0, 1],
n.o (5.2.4) where {an} and {bn} are sequences of real numbers and {,@j}, {,32} are mutually independent Wiener processes. Denote by G the interval [0, 27r] treated as a group with addition calculated mod 21r. Covariance operators Q on L2(G) are of convolution type with a kernel q if and only if, by Bochner's Theorem, the positive definite function q is of the form 00
q(C) = yo +
with 7n
yn, cos of , t: E [0, 21r],
(5.2.5)
n-1
0, n E N and F_°_1 -fn < +oo.
It is easy to see that if an = bn = y;., n E N, then the covariance operator Q of (5.2.4) is of convolution type with the kernel q of the form (5.2.5). Thus, in some cases, Wiener processes defined by (5.2.3) can have a physical meaning. Similar considerations can be made if G is a d dimensional torus and A is either a Laplace-Beltrami operator on G or a Stokes operator.
Assume that U and H are separable Hilbert spaces, A is the infinitesimal generator of a Co-semigroup S(t), t > 0, on H and B is a bounded linear operator from U into H. Let W(t),t > 0, be a cylindrical Wiener process on U and an H-valued To-measurable random variable.
Chapter 5
56
An H-valued Ft-adapted stochastic process Z(t), t > 0, is said to be a weak solution to the equation
dZ = AZdt + BdW, Z(0) _ e,
(5.2.6)
if for arbitrary h E D(A*) and all t > 0, P-a.s.
(h, Z(t)) _ (h, ) + j(A*h, Z(s))ds + (B*h, W(t)).
(5.2.7)
Note that all terms in (5.2.7) have well defined meanings. One can show that there exists a solution to (5.2.6) if and only if the operators t
Qt = f S(r)BB*S*(r)dr, t > 0,
(5.2.8)
are of trace class. In this case the solution is given by the formula t
Z(t) = S(t)d + f S(t - s)BdW(s), t > 0.
(5.2.9)
For the definition of stochastic integral see next section. The process Z is called an Ornstein-Uhlenbeck process and the process WA(t) =
f
t
S(t - s)BdW(s), t > 0,
-
o
(5.2.10)
a stochastic convolution. The process Z is Gaussian and Markovian with transition semigroup Rt, t > 0, of the form
RtV(x) = f(y)H(S(t)x,Qt)(dy), V E Bb(H).
(5.2.11)
The operator BB* will often be denoted by Q:
BB* = Q.
5.2.1
Stochastic integrals and convolutions
We assume, see earlier, that W(t), t > 0, is a cylindrical Wiener process on a separable Hilbert space U, given by a formal expansion
W(t) _
/3,,,(t)e,,, n=1
t > 0,
Stochastic Differential Equations
57
where en,, n E N, is an orthonormal basis on U. Very often the space U will be identical with H. The random variable is assumed to be Fo-measurable.
By IIRIIHs or IIRII2 we denote the Hilbert-Schmidt norm of the operator R E L(U, H). The space of all Hilbert-Schmidt operators
from U into H (endowed with the Hilbert-Schmidt norm) will be denoted by L2(U, H). This is again a separable Hilbert space. The stochastic integral
I (s)dW(s), t > 0, 0t
0
is well defined, see G. Da Prato and J. Zabczyk [44], for any L2(U, H)valued adapted process such that II,t(S)I12
Clot
ds < +oo, t > 0) = 1.
Moreover
ift'(s)dW(S) ]E
2 1
< ]E Jot I I , ( s ) I I Hs ds, t > 0,
0
(5.2.12)
with the equality applying in (5.2.12) if the right hand side is finite. The following generalization of (5.2.12) will be used, see G. Da Prato and J. Zabczyk [44, Lemma 7.7, page 194]: P
sup lE IJa (a)dW(o,) < cp SE[O,t]
(jui)P/2
0
(5.2.13)
where t E [0, T] and cp = (p(p - 1)/2)p/2, valid for arbitrary p > 2, t > 0 and an arbitrary predictable process s E [0,T], with values in L2(U, H). We will also need the so-called maximal inequalities formulated in the following two theorems.
Theorem 5.2.3 (Doob's inequalities) Assume that
El' IIp(S)IIHSds < +oo.
Chapter 5
58
(i) For arbitrary p > 1 and A > 0, P
sup 1JO § (s)dW(s)A t 1, lE
I
sup t 1, define the integral operator Ra,7w(t) =
f 0t(t - o,)a-1(-A)-YS(t - a)cp(a)di, t E [0,T],
on the space LP(O,T;H). Let us choose p > I.. By Proposition A.1.1 and Theorem 5.2.5 on the factorization, it is enough to show that the process Y.(s) =
J0
s
(s - or)`S(s - a)BdW(v), s E [0,T],
has p-integrable trajectories. By Theorem 5.2.4 ]IYa(s)I" < cp (10s II(s - o)-°S(s - o )BII2do,
p/2
, s E [O, T],
'The assumption w > 0 is not essential in Theorems 5.2.6 and 5.2.7 2(-A)" means the fractional power of -A, see e.g. A. Lunardi [108, §2.2.2]
Chapter 5
60
and therefore E
IT 0
IYa(s)Ipds < c,
T JO
(f
s(o)_2aII
ds < +oo.
The proof is complete.
With exactly the same proof the following result can be established.
Theorem 5.2.7 Assume that (i) The operator A generates an analytic semigroup S(t), t > 0, on H such that IIS(t)II < Me-Wt, t > 0, with w and M positive numbers,
(ii) there exists a E ]0,1/2[ such that for arbitrary T > 0 0T
t-2a II S(t)II Hsdt < +oo,
(5.2.17)
Jo
(iii) there exists a constant C > 0 such that P-a. s. II(b(t)II < C for all t > 0.
Then for arbitrary y E [0, a[ there exists a version of WA which is Holder continuous with values in D((-A)") with arbitrary exponent smaller than a - y.
Example 5.2.8 Let H = L2(0,1) = U, B = I, 2
A=
2,
D(A) = H2(0,1) n Hol (0, 1).
The functions =V2-sin
n E N,
E ]0, 1[,
form an orthonormal sequence of eigenfunctions of the operator A corresponding to the eigenvalues
-7 2n, 2n E Isl.
Stochastic Differential Equations
61
The stochastic convolution Z(t) = WA(t), t > 0, is in this case a weak solution of the equation
dZ(t) = AZ(t)dt + dW(t), Z(O) = 0. Note that e-2a2n2t
II S(t)II HS = n=1
The condition (5.2.16) of Theorem 5.2.6 is
JrT t- 2a
°O
e-2"2n2tdt < +oo. n=1
0
It is fulfilled if and only if a < 9. Consequently the process Z is Holder continuous in H with any exponent smaller than 4.
We will now prove a more general result concerned with the stochastic convolution on L2(O) where 0 is a bounded domain in Rd and the semigroup S(t), t > 0, is given by
S(t)x = E e-a"t (en, x) en, x E H,
(5.2.18)
n=1
and {en} is a complete orthonormal set on L2(0). Moreover we will assume that
W(t) =
Xn/3n(t)en, t > 0,
(5.2.19)
n=1
n E N, are independent Wiener processes. As far as the domain 0 is concerned we will assume that where
k(0) > 0
(5.2.20)
where the constant r,(0) is defined in Appendix B, by (B.1.1). Concerning the functions {en} we shall assume that
ek E Co(), I ek(e)I
C, I Dkek(6)I C
ak,
(5.2.21)
E 0, k E N.
(5.2.22)
Chapter 5
62
In the formulation of the theorem below II IIa stands for the Holder
semi-norm on 0:
WO - xWI
IIXIIa. = sup
f#7iEO
14 - 771°
Theorem 5.2.9 Assume that (5.2.18) and (5.2.19)-(5.2.22) hold and for some y E ]0, 1[ 00
Ak 1--Y < -boo.
(5.2.23)
k=1 ak
Then there exists a continuous version of the process with values in Co(?). aMoreover, if r > 0, y E ]0, 1[, a > 0 are numbers such that
,Q-2d
d
2r > 2r > 0' then the version has the additional property that 00
E
Ak 1--y
*/Z
k=1 ak
Proof - Fix T > 0 and define N
t
ZN(t,e) = E an J S(t 0
n=1
N
t
,\nen(b) f n=1
e-ant-8)d%n(s),
t E [0,T].
0
Set also
ZN,M = ZM - ZN, M, N E N.
It is clear that we can assume that ZN is a continuous function on [0, T] x 0 and that ZN(t, ) = 0 for t E [0, T] and belonging to the boundary 00 of 0. By a straightforward interpolation argument it follows that I ek(e) - ek(ll)I 0, andr> 1, r
r > we obtain that there exists a finite constant C such that 2
M EIJIZN,MIII aa_2d < C E
)
Ak 1-
r (k=N+1 ak
ry
r/2
Stochastic Differential Equations
65
r
So the sequence of random variables {ZN} satisfies the Cauchy condition in the space L of random variables Z whose values are -2d Holder continuous functions on [0, T] x vanishing on [0, T] x 00.
Therefore {ZN} has a limit Z on this Banach space which is the desired version. In a similar way one proves the second part of the theorem using (5.2.23) instead of (5.2.24).
Remark 5.2.10 It follows from the proof that if y
d
2
r >
a-2d >0,andr>1, r
then
:5 C
T II r EIIIWA(., -)t
(
M Ak
aa_2d
r
5.3
1-ry
k=N+1 ak
Stochastic evolution equations
In this section we will recall basic existence and uniqueness results on stochastic evolution equations of the form
dX(t) = (AX + F(X))dt + B(X)dW(t) (5.3.1)
on a separable Hilbert space H. We will also study regular dependence of solutions on initial data and Kolmogorov's equation for transition functions.
We will recall now existence results for problem (5.3.1), under suitable Lipschitz continuous conditions on F and B. It is convenient to introduce the following assumptions:
Hypothesis 5.1 (i) A is the infinitesimal generator of a strongly continuous semigroup S(t), t > 0, on H. (ii) F is a mapping from H into H and there exists a constant co > 0 such that I F(x)I 0 and x E H, S(t)B(x) belongs to L2(U; H), and there exists a locally square integrable mapping
K : [0,+oo[-* [0,+oo[, t -* K(t), such that 1IS(t)B(x)IlHs < K(t)(1 + IxI), t > 0, x E H,
1IS(t)B(x) - S(t)B(y)IIHS < K(t)lx - yl, t > 0, x, y E H. An 17t-adapted process X(t), t > 0, is said to be a mild solution of (5.3.1) if it satisfies the following integral equation,
X(t) = S(t)e +
t
J
S(t - s)F(X(s))ds (5.3.2)
+
It t
S(t - s)B(X (s))dW (s), t E [0, T].
0
We shall denote by xp,T the Banach space of all (equivalence classes)
of predictable H-valued processes Y(t), t > 0, such that IIYiip,T = sup (]EIY(t)Ip)1/p < +oo. tE[0,T]
We have the following result.
Theorem 5.3.1 Assume Hypothesis 5.1 and let p > 2. Then for an arbitrary Fo-measurable initial condition e such that +oo there exists a unique mild solution X of (5.3.1) in xp,T and there exists a constant CT, independent of , such that sup IEIX(t)Ip < CT(1 +
(5.3.3)
tE[0,T]
Finally, if there exists a E ]0,1/2[ such that 1
10
s-2"K2(s)ds < +oo,
(5.3.4)
where K is the function from Hypothesis 5.1-(iii), then the solution X(.) is continuous P-a.s. 'Means that for any u E U the mapping x continuous.
B(x)u from H into H is
Stochastic Differential Equations We shall denote by
67
the mild solution of (5.3.1).
The proof of Theorem 5.3.1 is similar to that in G. Da Prato and J. Zabczyk [44, Theorem 7.4, page 186]. We will sketch it here for the sake of completeness. A more general result, needed in some applications, see S. Peszat and J. Zabczyk [124], will be given in Part III.
Proof of Theorem 5.3.1 - For arbitrary E LP(ft, H) and X E fp,T define a process Y = K(6, X) by the formula
Y(t) = S(t)e +
jt
S(t - s)F(X(s))ds (5.3.5)
+
jS(t - s)B(X(s))dW(s), t E [0,T].
We will note first that, by inequality (5.2.13), we have K(e,X) E 7{p,T for arbitrary X E 1p,T. Moreover, setting MT = suptE[o,T] IIS(t)II, we have
EIY(t)IP
0 there exists a sequence {
C Cl ([O, T]; E) n C([O, T]; D(A)),
such that
Y. -+
y - Ay. - F(yn) -+ 0, in
C([O, T]; E).
Chapter 5
80
5.5.3
Existence of solutions for stochastic equations in Hilbert spaces
In this section we present a method which implies existence and uniqueness of the solutions to the problem
dX = (AX + F(X))dt + BdW(t), (5.5.18)
where A, F satisfy some dissipativity assumptions on appropriate spaces and B is a bounded operator. As a byproduct we will obtain a proof of Proposition 5.5.6 with a slightly weaker concept of solution.
Let H be a Hilbert space and let K be a reflexive Banach space included in H. We assume that K is a dense Borel subset of H and such that the embedding of K in H is continuous. On the mappings A and F we impose the following condition. Hypothesis 5.4 (i) There exists q E R such that the operators A- 77 and F - rt are m-dissipative on H. (ii) The parts on K of A -,q and F -,q are m-dissipative on K. (iii) D(F) D K and F maps bounded sets in K into bounded sets of H.
We shall denote by AK and FK the parts of A and F respectively, that is D(AK) = {x E D(A) n K : AKX E K}, AKX = Ax, X E D(AK),
D(FK) = {x E D(F) n K : FKX E K}, FK(x) = F(x), X E D(FK). Let Z(t) = WA(t), t > 0, be the solution to the linear equation dZ = AZdt + BdW(t), (5.5.19)
X(0) = 0, given by
WA(t) = J0 S(t - s)BdW(s), t > 0, 0
where S(t), t > 0, is the semigroup generated by A in H. We will assume the following.
Stochastic Differential Equations
81
Hypothesis 5.5 The process WA(t), t > 0, is continuous in H, takes values in the domain D(FK) of the part of F in K, and for any T > 0 we have sup (II WA(t)II K + II F(WA(t))II K) < +oo, ? a.s.
tE[O,T]
Remark 5.5.7 Notice that Hypotheses 5.4 and 5.5 are satisfied if
(i) K = H. (ii) A is a linear operator almost m-dissipative on H. (iii) F is either Lipschitz continuous, or continuous and monotone with linear growth. (iv) WA() is an H-continuous process.
An H-continuous, adapted process X(t), t > 0, is said to be a mild solution to (5.5.18) if it satisfies lP'-a.s. the integral equation
X(t) = S(t)x +
J
t
S(t - s)F(X(s))ds + WA(t), t > 0.
(5.5.20)
If, for an H-valued process X, there exists a sequence {X,,,} of mild solutions of (5.5.18) such that ]P-a.s., X,,(.) -f X(.) uniformly on any interval [0, T], then X is said to be a generalized solution to (5.5.18). Note that each mild solution is also a generalized solution.
Theorem 5.5.8 Assume that Hypotheses 5.4 and 5.5 are fulfilled. Then for arbitrary x E K there exists a unique mild solution of (5.5.18) and for arbitrary x E H there exists a unique generalized solution of (5.5.18).
Remark 5.5.9 It will follow from the proof that generalized solutions X(t, x), t E [0, T],x E H, of (5.5.18) are all Markov processes in H with a Feller transition semigroup Pt, t > 0, given by PtW(x) = E[p(X (t, x))], x E H, p E Bb(H).
Proof of Theorem - We will show that the initial value problem ye(t) = Ay(t) + F(y(t) + WA(t)), y(O) = x,
(5.5.21)
Chapter 5
82
has a mild solution. That is there exists equation:
satisfying the following
y(t) = S(t)x + f S(t - s)F(y(s) + WA(s))ds, t E [0,T]. (5.5.22) t
Then the solution X to (5.5.18) is given by X (t) = y(t) + WA(t), t E [0,T].
For arbitrary a > 0 we consider the approximating problem
ya(t) = Aya(t) + Fa(ya(t) + WA(t)), y.(0) = x,
(5.5.23)
where Fa are Yosida approximations of F. Since F,,, are Lipschitz continuous, equation (5.5.23) has a unique H-continuous solution. We will show that lima-o ya(t) = y(t) exists uniformly for t E [0, T] and is the required solution of (5.5.22). The proof will be done in several steps. For simplicity we assume 7 = 0.
Step 1- A priori estimate in H. We have, by dissipativity of A and Fa if necessary replacing A by Yosida approximations A,, of A), that dt l y«(t)I H
= (Aya(t), ya(t))H + (FF(ya(t) + WA(t)), ya(t))H
= (Aya(t), ya(t))H + (F.(WA(t)), ya(t))H
+ (F.(ya(t) + WA(t)) - F.(WA(t)), ya(t))H
0, depending on cp, such that II ya(t)II K 0, has a K-continuous version.
Theorem 5.5.11 Under Hypothesis 5.6 for arbitrary x E K there exists a unique continuous solution of (5.5.18) in K which determines
a Feller transition semigroup on K. If in addition for some w E R the mapping F - w is dissipative in H then for arbitrary x E H there exists a unique H-continuous generalized solution X(t, x), t > 0, of (5.5.18) with Feller transition semigroup on H.
Proof - Write y(t) = X(t) - WA(t), t E [0,T] and let SK() be the semigroup generated by AK. Then
y(t) = SK(t)x+10 SK(t-s)F(y(s)+WA(s))ds, t E [0,T], (5.5.29) and for x E K equation (5.5.18) is equivalent to (5.5.29). It is therefore enough to show that for an arbitrary Ii -continuous function z(.) the equation
y(t) = SK(t)x + 1 SK(t - s)F(y(s) + z(s))ds, t E [0,T], (5.5.30) t
has a unique global solution y. Local existence follows by a contraction mapping principle. To obtain global existence it is sufficient to deduce an a priori estimate for Ily(.)II If a solution z(.) exists on a time interval [0, To] then there exists a sequence { y,,} C C1([O, To]; D(AK)) such that
yn.(t) - y(t), d dtt) - AKyn(t) - F(yn(t) + z(t)) = Sn(t) - 0 as n oo, uniformly on [0,To]. Now for some xt,n E OIIyn(t)II, and t E [0, T],
Chapter 5
86
dt II yn(t)II
0, y E ]0, 1[, A > 0, and C > 0 such that if yo (B(0, R)) > ry then JH
eAIZ12 i4(dz) < C,
and in particular E (I X (t, 0) (2)
0, is irreducible at some t > 0, then all solutions to (6.2.1) are recurrent with respect to all open nonempty sets if and only if all eigenvalues of A, with the exception of at most two, have negative real parts and the remaining eigenvalues either are pure imaginary of multiplicity 1 or are equal to 0, see H. Dym [58], R. Erickson [62] and J. Zabczyk [165]. There is also a simple connection between existence of invariant measures and recurrence. If there exists an invariant measure for an irreducible semigroup Rt, t > 0, corresponding to (6.2.1) then any solution to (6.2.1) is recurrent with respect to all open sets of H. In the infinite dimensional case such a characterization is not known. However, some results in this direction are possible, see J. Zabczyk [164]. In particular we have the following
Proposition 6.2.4 (i) Assume that the transition semigroup Rt, t > 0, corresponding to (6.2.1) is irreducible at some t > 0 and has an
Existence of invariant measures
103
invariant measure. If limt._..,+, S(t)x = 0 then any solution to (6.2.1) is recurrent with respect to all open sets of H.
(ii) There exists an equation of the form (6.2.1) such that the corresponding semigroup Rt, t > 0, is irreducible for all t > 0, and has an invariant measure, but a solution X (t, x), t > 0, to (6.2.1) is not recurrent with respect to a ball. Proof - (i) If µ is an invariant measure then p is unique, Gaussian with its support equal to H. Moreover for arbitrary x E H Pt(x, ) weakly as t -* +oo. Consequently, for any open, nonempty set
rcH lim inf Pt(x, r) = p(r) > 0,
t-+oo and it is enough to apply Proposition 3.4.5.
(ii) Let us consider the transition semigroup Rt, t > 0, corresponding to the linear equation
dZ(t) = AZ(t)dt + BdW(t), Z(0) = x. Fix R > 0. It is not difficult to find i E H such that II S(t)xII > t + R, t > 0. Then
Pt(x, B(0, R)) < Pt (0, B°(0, t)), t > 0. By Chebyshev's inequality Pt(0, B`(0, t)) < t2
(IX(o, t) 12)
,
t > 0.
It easily follows now that +00
J
Pt(i, B(0, R))dt < C f
+- dt
< +oc.
Assume that with probability 1 the set
It > 0 : IX(t,x)I < 2 } is unbounded and define TR = inf{t > 0 : IX(t,x)I < R}.
(6.2.11)
Chapter 6
104
Then inf
]E(rr) = a > 0.
I1sII s. Then dIX..(t)I2
= {2(AX.(t) + F(Xn(t)),Xn(t)) + II B(Xn(t))112}dt
+ 2(Xn(t),B(Xn(t))dW(t)). Consequently, integrating in [s, t], and taking expectations, we have E (IXn(t)12) = Ix12
+IEjt{2(AXn(a) + F(Xn(O1)),Xn(Q)) + II B(Xn(o,))112}do,.
Existence of invariant measures
107
It follows that ate (Ixn(t)12)
= ]E(2(AX.,(t) + F(X.(t)),X.(t)) + -w ]E (I X..(t)12) + 2IF(0)I E(I X..(t)I )
+ JIB(0)112 + 21IB(0)112](IB(X.(t))I2)
< - 2E (Ix.(t)I2) + C1, for a suitable constant C1 > 0. So there exists a constant C2 > 0 such that EI X,.(t)12 < e-w(t-s)l2(I x12 + C2), t > s.
As n tends to infinity we find ]EIX(t)12
s,
(6.3.9)
for some constant C3 > 0.
Step 2 - For b > y > 0 define Z(t) = X(t, -y, x) - X(t, -b, X), t > -y. Using the Ito lemma and proceeding as before we find the estimate ElZ(t)12
-y
from which, recalling (6.3.9),
]EIX(0, -y, x) - X(0, -b, x)12 < C3(1 + IxI2)e-,-,, b > -y. (6.3.10) It follows that the sequence of random variables {X(0, -y, x)}.y>O, +oo, and theresatisfies the Cauchy condition in L2(St; H), as y fore it is convergent to a random variable i E L2(Sl, H), which is independent of x. We claim that the law of q is the invariant mea, sure with the required properties. To see this it is enough to remark
that
L(X (t, 0, x)) = £(X (0, -t, x)) ' £(t7) = µ, weakly as t - +oo,
Chapter 6
108
which is equivalent to
Pt b., -* p weakly as t - +oo. Finally, let cp E Lip (H), then by (6.3.10) Ptw(x) - Pso
Existence of invariant measures
109
We now prove
Theorem 6.3.3 Assume that Hypotheses 5.4, 5.5 and 6.2 hold. Then there exists exactly one invariant measure p for (6.3.11, it is strongly mixing and for arbitrary v E M1(H), Pt *v -> it weakly as t --> +oo. Moreover there exists C > 0 such that, for any bounded Lipschitz continuous function 0 and all x E H I Pt4p(x)
Proof-
- ( -A,
-
a
and $ sup t] EIF(WA,A(s))I, t > -A. EIZA(t)I s Ixi + 1 W
-A,
Taking into account the definition of the process ZA and Hypothesis 6.2-(ii) one arrives at (6.3.13). In a similar way one shows that for all -A E t[
EIX(t, -A, x) - X(t, -7, x)I < e_W(t+-\)]EI X (-A, -,y, x) - xl
0. To see this it is enough to remark that, by (6.3.16), for arbitrary x E H,
A(x, ) = G(X(t, x))
= £(X(0, -t, x)) -> it, weakly as t -+ +oo.
Existence of invariant measures
111
Finally, let cp be a bounded Lipschitz function on H, then, by (6.3.15),
fors>t>0
P' 0,
(iii) there exists an invariant measure for the linear system (6.3.11)
with B = 0. See also Remark 5.5.7. The following theorem is a variation of Theorem 6.3.3. It implies, however, only the existence of an invariant measure.
Theorem 6.3.5 Assume that the operators SK(t), t > 0, generated by AK are compact and that for an w > 0 IISK(t)II < e-wt, t > 0.
If in addition to Hypothesis 5.6 sup iE [IIWA(t)II + a(IIWA(t)II)] < +oo t>o
then there exists an invariant measure for the solution of (5.5.18) on K.
Chapter 6
112
Proof - Arguing as in the proof of Theorem 6.3.3 one obtains that II
Xt
x + JO - WAOtII 0,
an d therefore sup IIX(t+u,x)II o]EIIX(t,x)II < -boo and consequently for arbitrary e > 0 there exists C > 0 such that JF(IIX(t,x)lI > C) < E, for all t > 0.
Since the process has K-continuous trajectories and all the processes have the same distributions as the constant C > 0 can be chosen to give also
P
(
sup IIWA,t(u)II > C < E,
/
uE[0,11
]P
sup a(IIWA,t(u)II) >- C < E. uE[0,11
But then
]P( sup IIX(t+n,x)II >3C) uE[0,1]
+lP
P(IIX(t,x)II >C)
(uE[0,11 sup IIWA,t(u)II ? C +P ( sup a (II WA,t(u)II) > C uE[0,11
3r,
Chapter 6
114
for all t > 0. Taking into account this estimate, the identity (6.3.17), andthe compactness of the operators SK(1) and G°, we easily obtain that the family ,C(X(t + 1, x)), t > 0, is tight, as required. By Theorem 5.5.11 and Theorem 6.3.5 we get the following result.
Theorem 6.3.6 If in addition to the assumptions of Theorem 6.3.5 there exists w E R such that the mapping F - w is dissipative in H then there exists an invariant measure for equation (6.3.11) which in addition is concentrated on K.
6.4
Genuinely dissipative systems
We assume here that Hypotheses 5.4, 5.5 and 6.2 hold and moreover the following.
Hypothesis 6.3 (i) S(.) is of negative type. (ii) There exist r > 1 and C,. > 0 such that (F(x) - F(y), x - y) < -Cr I x - yI i+r. Hypothesis 6.3 implies not only that our system is strongly dissipative, but in addition that the nonlinear part is superlinear. In finite dimensions such systems have been considered by S. Cerrai [18]. Under Hypothesis 6.3 we will be able to prove that Ptcp(x) -+ fH W(x) t(dx) for all W E Lip(H) uniformly on x. Moreover, we show that the convergence of PtW(x) holds for all W E Bb(H), as t --). +oo, under the additional hypothesis following.
Hypothesis 6.4 For any cp E Bb(H) and t > 0 there exists Kt > 0 such that Ptcp is Lipschitz continuous and I Pt9 (x)
-
S Ktlx - yI
for all x, y E H.
Theorem 6.4.1 Assume that Hypotheses 5.4, 5.5, 6.2 and 6.3 hold and let p be the invariant measure for (6.3.11) given by Theorem
Existence of invariant measures
115
6.3.3. Then there exists Cl > 0 such that, for any bounded Lipschitz continuous function co, all t > 0 and all x E H IAW(x) - (sO,A) 1 -A
J
It follows, denoting by xa y t an element from the subdifferential of IZ,,.y(t)I, that dt IZA,y(t)I
-A,
n(o) = Ix - X(-A,-'Y,x)l.
Chapter 6
116
By proceeding as in S. Cerrai [18], one can show that there exists a constant C2(t), independent of x, such that ]GI X (t, -A, x) - X (t, -7, x)I
< e-w(t+a)EIX (-.A, --y, x) - xI
< C3e-w(t+A), t > -A. (6.4.3)
Let now S be the random variable such that (6.3.16) holds, and let cp be a bounded Lipschitz function on H, then, by (6.4.3) we easily find (6.4.1).
Let us prove the last statement of the theorem. Let E B6(H) and let e > 0. Then, setting cp = P' in (6.4.1), we have, by Hypothesis 6.4, and by the invariance of p, IPt+eY'(x)
- (PeY',Ft)I
r. Since IPt'i'(x)
- (',µ)I
0 and all x E E, I Ptc (x) - ( -A, A > 0, dX = (AX + F(X))dt + BdW(t), t > -A, (6.5.2)
X(-A)=x, By Theorem 5.5.13 we know that there exists a unique generalized solution of (6.5.2), which we shall denote by X (t, -A, x), t > -A. We show now that there exists cl > 0 such that
]E(IIX(t,-A,x)II) 5 C+11x4, for all A > 0, for all x E H,for all t > -A. (6.5.3)
We first remark that ZA(t) = X(t, -A, x) - WA,A(t), t > -A, is the mild solution of the problem dt Z = AZ + F(Z + WA,A),
Z(-A)=x, t> -A
l J
It follows, recalling that F is dissipative, that there exists xa t
E
BIIZA(t)II such that dt IIZ),(t)II
= (AZA(t) + F(ZA(t) + WA,a(t)), xa,t) (F(WA,A(t)), xa t) + (F(ZA(t)), xa,t)
0, where
(7.2.5)
is the solution to the integral equation,
t () t= - II I ()II
II
F Ilo
t
sds. s IItt(-)III()
In particular Pt, t > 0, is a strong Feller semigroup.
Proof - Denote by D(Ao) the set of all So such that (i) cp E Cb (H),
(ii) Wx(x) E D(A*) for all x E H and the mapping H - R, x --> (x, A*(px),
(..s) (7.2
Uniqueness of invariant measures
135
belongs to Cb(H). (iii)
W. E Cb(H,Li(H))
Now define a linear operator .Ao on Cb(H) by setting Aoco
= 2 Tr [Q(pxx] + (x, A*(p.),
D(Ao) _
{lp E Cb(H) : lpxx E Li(H), S11prEH IIVxxIILI(H) < +00,
and there exists 0 E CC(H) such that cp(x) = O(A-1x)
and the mapping x --+ (x, 0,,(A-lx)) is in Cb(H)}. It has been shown in S. Cerrai and F. Gozzi [20] that the set D(Ao) is dense in Cb(H) in the following sense: For arbitrary cp E C6 (H) there exists a sequence {cpn} C D(Ao) such that (1) IIPn1Io < 2II 0. Assume first that F is twice differentiable with bounded and continuous derivatives up to the second order and that cp E D(Ao). Then the function v(t,x) = Ptcp(x), t > 0, x E H,
is the strict solution of the Kolmogorov equation:
vt(t, x) = 2 Tr [Qvxx(t, x)] + (x, A*vx(t, x)) (7.2.7)
+ (F(x), vx(t, x)), t > 0, which can be written equivalently as dv
d = Aov(t)
+.Fv(t),
v(0)=yc, t>0.
7.2.8)
J
In (7.2.8) Ao is the operator defined above and
Fb(x) = (F(x), ix), 0 E Cb (H), x E H.
Chapter 7
136
For fixed t > 0 consider the function Rt_3(v(s)), s E [0,T]. Then, for s E [0, T], TS
[Rt-s(v(s))] = -Ao [Rt-3(v(s))] + Rt-s
\ dt (s)/
= -Ao [Rt-s(v(s))] + Rt_3 [Aov(s) +.Fv(s)]. Since the operators Ao and R3 commute we obtain that ds
[Rt-s(v(s))] = Rt_s F, s E [0, t].
(7.2.9)
Integrating (7.2.9) over the interval [0, t] one obtains that t
v(t) = Rtcp + f Rt_3(Fv(s))ds, t > 0.
(7.2.10)
o
The equation (7.2.10) is called a mild Kolmogorov equation. It fol-
lows that, for arbitrary 0 E C6(H) and t > 0, Rt'/ is a function bounded and continuous together with its first derivative and Ilr(t)ll Iloilo.
I
(7.2.11)
Note that Jot
IlFv(s)Ilods
0, is locally bounded. Taking into account (7.2.11) and taking derivatives of both sides of (7.2.10), we arrive at the inequality IID.v(t)Ilo S Ilr(t)II II 0,
valid for all t > 0. Since
is locally bounded by a generalization of Gron-
wall's lemma, II Dxv(t)Il o < fi(t), t > 0,
(7.2.12)
and (7.2.5) follows for all cp E Cb(H). The generalization to arbitrary Borel funtions 0, corresponding to problem (7.1.1) is irreducible.
Proof - We first prove, in the following lemma, that irreducibility of Pt, t > 0, is equivalent to irreducibility of the system
dZ = (AZ + cp(t))dt + B(Z(t))dW(t), t > 0, (7.3.1)
Z(0) = x, where cp is any bounded and adapted process. Then, by choosing cp in a suitable way, we prove that system (7.3.1) is irreducible.
Lemma 7.3.2 Assume, besides the hypotheses on A, B of Theorem 7.3.1, that the process cp is bounded. Then the laws in (H,a(H)) of X(t, x) and Z(t, x) are equivalent.
Proof- Let a(t) = B-1(Z(t))[cp(t)-F(Z(t))]. Then a is bounded and adapted, so that the equation
dM(t) = -(a(t), dW(t))M(t), t > 0, M(0) = 1,
has a unique solution M() and E(M(t)) = 1, t > 0. Now fix T > 0 and set dP*(w) = M(T)dP(w).
Chapter 7
138
Then P* is a probability on (0,Y) and therefore, by the Girsanov theorem (see e.g. G. Da Prato and J. Zabczyk [44]), the process
W*t- Wt
t
is a cylindrical Wiener process in H defined on the probability space (9,.F, I-). Note that P and P* are equivalent. Now ft
Z(t) = S(t)x +
ft
o
0 ft
= S(t)x +
S(t - s)B(Z(s))dW(s)
S(t - s)yp(s)ds + ft
S(t - s)B(Z(s))dW*(s).
S(t - s)F(Z(s))ds +
Consequently Z is the solution of (5.1.1) on the probability space (1, )'*, P*). Since the law of the solution does not depend on the particular choice of the probability space, for every Borel set r C H we have P(X (t, x) E r) = P*(Z(t, x) E r). Since P and P* are equivalent the laws of X (t, x) and Z(t) are also equivalent which is our claim.
We can now proceed to the proof of Theorem 7.3.1. We fix to > 0, r > 0, and x, a E H. We will choose a bounded process cp such that, if Z is the corresponding solution of (7.3.1), we have P(IZ(to) - al < r) > 0,
(7.3.2)
and we will apply Lemma 7.3.2. Let us consider an auxiliary system
dY = AYdt + B(Y)dW(t), Y(0) = x, which clearly has a unique solution Y. Let ti E [0, to[ be a moment to be chosen later. Now we choose V by setting 0 if t E [0, ti [, (P(t) =
f(t,Y(tl))
if t E [tl,to],
Uniqueness of invariant measures
139
where f : [ti, to] x H --r H is bounded, f (t, ) is Lipschitz, and moreover 0
if IyI > 2R, s E [ti, to],
f(s) y) = to i-t,
w. - t i )(a - y)
- A;;
- ,i,o ,
if y I < R s E [t t I
]
where R > 0 and a E D(A) will also be chosen later. We remark that, as is easily checked, the following implication holds: y E H, IyI ]P(Ii=a&II2I 0 if, for arbitrary x, z E H and e > 0, there exists u E L'(0, T; H) such that Iy(T, x; u) - xj < e. Note that Hypothesis 5.1(iii) implies continuity of the solution Z(.) of the linear equation dZ = AZdt + Q112dW(t), (7.4.3)
Z(0) = 0. Theorem 7.4.1 Assume that the operatorA generates aCo -semigroup S(t), t > 0, F is a locally Lipschitz mapping, and that the deterministic system (7.4.2) is approximately controllable in time T > 0. Then the transition semigroup Pt, t > 0, corresponding to problem (7-4. 1) is irreducible in time T.
Proof - We first remark that the law £(Z(.)) of the process Z is a Gaussian measure on L2(0,T;H) with a covariance operator Q given by T
Qcp(t) = f g(t,s)cp(s)ds, P E L2(0,T:H); where
g(t, $)x =
J0
S(t - r)QS*(s - r)sdr. x E H.
JO
Moreover
t
Image Q112 = {v E L2(0,T; H) : v(t) = f S(t - s)Q1/2u(s)ds, 0
t E [0,T], u E L2(0,T;H)}. Since £(Z(.)) is concentrated on Co = {u E C([0, T]; H) : u(0) = 0}, the closure of Image Q1/2 in the Co-norm is the support of £(Z(.)), see G. Da Prato and J. Zabczyk [44, page 41]. Now fix x and z in H, and e > 0. We want to prove that
?(IX(T,x) - zj < e) > 0.
(7.4.4)
Chapter 7
142
We first remark that, by the approximate controllability assumption, there exists u E L2(0, T; H) such that l y(T, x; u) - zI < e.
Take R such that I y(s, x; u) I
R.
(7.4.8)
sE[0,T]
Let
T1 = inf{t > 0 : IX(t,x)I > R}. Then T1 < T and for all t E [0, T1]
IX(t,x)I < R, Iy(t,x; u) I
O
,
qn
and (8.2.2) follows. To prove (8.2.3) we remark that
E
yEr,'Yn>O,ryn-1>0
x
rr jl Htih-1
h¢n
h
P71 qn
Yn(7'n - 1)H-vn-2
n
qn
Chapter 8
152
and, if m 54 n,
1:
Dxn Drm
1
1E r, yn>O, Ym>0 T r
x
rr 11
M. 11.Yn-1 (_) H'ym-1 ( x
Xh
H-r,,_1
qh
.
h$n,m
It follows that
JH Tn IDn
T
2 10
p(dx) =
Y
I2
ryEr,ryn>O,'Yn-1 >0
qn2ryn(7n
and
JH
rnrm7n7m
rnrm I DxnDxmPI2 µ(dx) ^YEr,ryn>O,^Ym>O
gn gm
which yields (8.2.3).
The following proposition was proved in G. Da Prato, P. Malliavin and D. Nualart [40] and independently in S. Peszat [123]
Proposition 8.2.2 For an arbitrary Gaussian measure µ on a Hilbert space H the embedding WR'2(H, p) C L2(H, M)
is compact if and only if rn = +00. n-oo qn 1im
(8.2.4)
In particular W1,2 (H, u) is compactly embedded in L2(H, µ).
Proof - Note that (8.2.4) holds if and only if for any ordering y(k), k E N, of I', limk-.,,o(Qo , y(k)) = +oo. This easily implies the required result.
Densities of invariant measures
153
Remark 8.2.3 Note that the embedding W2,1 (H, M) C W1,2 (H,
µ),
is not compact. To see this define cpm(x) = (x, em), m E N, x E H.
Then
Dxpm(x) = e-m, D2,pm(x) = 0. Therefore II0mIIL2(H,,) = q,,,12
and II(PmIIW1,2(H,,U) = II(mIIW2.2(H,p.) = (1 + gm)1h/2
So the sequence {cpm} is bounded in W2'2(H,µ), converges to 0 in L2(H, p) and does not contain any subs-equence convergent to 0 in W1,2 (H, z).
8.3
Properties of the semigroup Rt, t > 0, on L2(H, µ)
For our study of the transition semigroups Rt, t > 0, and Pt, t > 0 we will need, besides Hypothesis 8.1, the following.
Hypothesis 8.2 (i) Image S(t) C Image Qt 12, t > 0.
(ii) Setting F(t) = Qt 112S(t), the function IIr(t)II, t > 0, is
Laplace transformable, i.e. for sufficiently large A, is finite.
f0+00 e-"tjIr(t)Ildt
Note that if Hypotheses 8.1 and 8.2 hold, then the assumptions of Theorem 7.2.4 are fulfilled, so that Rt, t > 0 is a strong Feller semigroup.
Remark 8.3.1 Note that if Q coincides with the identity operator I, then Hypothesis 8.2 is satisfied, see G. Da Prato and J. Zabczyk [44, Corollary 9.22]. On the other hand if H = Rd, d E N, then Hypotheses 8.1 and 8.2 hold if and only if Q is invertible, see T. Seidman [137].
Chapter 8
154
We introduce, following [26], the linear operator in L2(H, p) : Aow
e
D(Ao) _
1
2
Tr [QV.) + (x, A*wx),
{w(x) = f((x,a,),...,(x,a,a) : f E Cb2(R"),
a1i ..., a. E D(A*), n E N} . It is easy to see that D(Ao) is invariant with respect to the semigroup Rt, t > 0, and is dense in L2(H, p).
The following result is proved in G. Da Prato and J. Zabczyk [47].
Theorem 8.3.2 Assume Hypothesis 8.1. (i) Ao is closable in L2(H, p) and its closure A is the infinitesimal
generator of a Co-semigroup etA,t > 0, in L2(H, p) identical with
Rt, t > 0: etASO(x) = E [ca, (S(t)x + f t S(t - s)dW(s) )J
/
o
Rttp(x),
t>-O, cp E L2(H, p) (8.3.1)
(ii) If moreover Hypothesis 8.2(i) holds, then for all t > 0 and cp E L2(H, p) we have Rtcp E W1'2(H, p), and II DxRtwII L2(H,µ) : IIr(t)II IIPIIL2(H,,L),
(8.3.2)
and for any t > 0, Rt is a compact operator on L2 (H, p). (iii) If, in addition, Hypothesis 8.2(ii) holds, then D(A) C W1'2(H, p)
Proof - (i) If cp E D(Ao) then, for arbitrary x E H Rtw(x) = (x) +
j
Rs(Aow)(x)ds.
Therefore D(Ao) C D(A) where A is the infinitesimal generator of the semigroup Rt, t > 0. From Theorem 1.9 in E. B. Davies [49], it
Densities of invariant measures
155
follows that D(Ao) is dense in the graph norm of D(A) and therefore the closure of AO is A, and (i) follows. Now we prove (ii). Let co E Cb(H). Since IHW(y)N(S(t)x,Qt)(dy),
it follows by the Cameron-Martin formula that, for any h E H, (D=Rt,P(x),h) = J(r(t)h, Qt 112y)(P(y)N(S(t)x,Qt)(dy) Consequently (D.Rtw(x), h) l
JH I
(r(t)h,
Qt
112Y) l
1 v(S(t)x + y)I N(0, Qt)(dy)
and by the Schwarz inequality
I (D.Rtp(x),h)12 < JH I (r(t)h,Qt 1/2y)12H(O,Q,)(dy) X
0, with respect to t Now we are going to show that the semigroup Pt can be extended to a Co-semigroup on L2(H, µ). We will need the following result which is a direct extension of Theorem 3.5 of E. B. Davies [49].
Proposition 8.4.1 Let A be the infinitesimal generator of a Cosemigroup on a Hilbert space 1-l, and let C be a closed operator on 1-l such that IICetAIldt < +oo.
D(C) D U etA(x), t>o
0
Then D(C) D D(A) and for an arbitrary bounded operator F on 7-l the operator
,C = A+ .PC, D(C) = D(A),
(8.4.1)
is the infinitesimal generator of a Co-semigroup etc, t > 0, on 7-l and we have
00
etr_ p
=
Tn(t)4o, t > 0, yO E f,
(8.4.2)
n-o
where etA 0, Tn(t)cp =
t e(t-s)AFCTn_l(s) 0, x E H. Assume, for a moment, that F E Cb (H; H). Then the function u is the unique strict solution of the associated Kolmogorov equation
ut =
1
2
Tr[Qu.,x] + (Ax + F(x), ux), (8.4.4)
u(0, x) = c0(x).
In particular the L2(H, µ)-valued function u(t, ), t > 0, is a solution of the problem u'(t) = Gu(t), t > 0,
u(0) = V,
and therefore u(t, ) =
etf-' for t > 0. If F is only a Lipschitz continuous function one can find a uniformly bounded sequence of mappings
{F7L} C C6 (H) such that limn., F,,(x) = F(x), x E H. It easily follows from (8.4.2) and the definition of Pt that for the corresponding semigroups etc^, t > 0, Pt, t > 0, and cP E D(Ao), we will have et1C
= lira etc°cP = lira Pt cP = Pt(p. n-*oo
Chapter 8
158
So Pt = L2(H,p).
etf- cp, t > 0 for cp E D(Ao) and therefore for all cp E
Property (i) follows directly from the definition of C. To show (ii)
it is enough to remark that 00
E JjD.Tn(t)jj < +oo
n=0
for t > 0, where Tn is defined in (8.4.3), and to use (8.4.2).
We are now in a position to prove the existence of an invariant measure pF for (8.1.1) and that it is absolutely continuous with respect to A.
Theorem 8.4.4 Assume that Hypotheses 8.1 and 8.2 hold. Then there exists 'k E D(,C*),i/> > 0, such that G*V)
0, J O(x) p(dx) = 1.
Moreover 0 is the density of an invariant measure pF for (8.1.1). Finally, if ImageQ is dense in H, the invariant measure is unique.
Proof - Since the operators et'c, t > 0, are compact on L2(H, p), the same is true for their adjoints etc*,t > 0. Since 1 belongs to the spectrum a(etc) of etc, 1 is also an eigenvalue for etr*. Repeating the argument from the proof of Theorem 7.8 in E. B. Davies [49], one obtains that the set
G = {*EL2(H,p):etC*b= Vt>0}
= {'EL2(H,p):G*0=0}, is linear and of finite, positive dimension. It is easy to show that elements of G are densities of possibly signed invariant measures for ett', t > 0. That there are nonnegative and nonzero elements in follows from Lemma 3.4 in I. Shigekawa [139].
To prove uniqueness, it is enough to show that Pt, t > 0 on Bb(H) is a strong Feller and irreducible semigroup; however this is so by Theorems 7.1.1 and 7.2.4.
Densities of invariant measures
8.5
159
Locally Lipschitz nonlinearities
We assume here Hypothesis 8.1(i)(iii)(iv)(v) but, instead of 8.1(ii), we assume that F : H - H is locally Lipschitz continuous and that problem (8.1.1) has a unique mild solution X(t, ), t > 0. We want to extend Theorem 8.4.4 to this more general situation. For this we define the mappings Fn, n E N, Fn(x) =
F(x), Ixi < n, F ' Ix) >
(nn/
(8.5.1)
n,
which are Lipschitz continuous and bounded. Then we consider, besides equation (8.1.1), the following equation corresponding to Fn : dXn = (AXn + Fn(Xn))dt + v/¢dW,
(8.5.2)
Xn(0)=xEH. Let Pt(x, ), Pt (x, ), t > 0, n E N, x E H BE the transition probabilities corresponding to solutions of (8.1.1) and (8.5.2) respectively. We first prove a result on absolute continuity of transition probabilities.
Proposition 8.5.1 Assume that Hypothesis 8.1(i)(iii)(iv)(v) hold, that F is locally Lipschitz continuous and that problem (8.1.1) has a unique mild solution X (t, ), t > 0. If for some t > 0 and all n E N the measures Ptn(x, ) are absolutely continuous with respect to a measure v, then the measure Pt(x, ) is absolutely continuous with respect to v.
Proof - Let F C H be a Borel set such that v(r) = 0. Then Pt(x, r) = ]P(X(t, x) E r) = 1P(X(t, x) E r, r,,i > t)
+ P(X(t, x) E r, rn < t), where
rn = inf{s > 0 : IX(s,x)I > n}.
Chapter 8
160
Since for all s < rn, X(s,x) = X,,,(s,x), we have
Pt(x, r)
t) + P(rn < t)
< Pt (x, r) + P(rn < t).
But Pt (x, r) = 0 and P
lira r,, = +oo) = 1, and consequently
(n-r+00
Pt(x, r) = 0.
//
Now we prove the main result of this section under the following assumptions.
Hypothesis 8.3 (i) Hypotheses 8.1(i)(iii)(iv)(v) and 8.2 hold and Image Q is dense in H. (ii) F is locally Lipschitz continuous and problem (8.1.1) has a unique mild solution.
(iii) Problem (8.1.1) has a unique invariant measure µF.
Theorem 8.5.2 Assume that Hypothesis 8.3 holds. Then the invariant measure µF to problem (8.1.1) is absolutely continuous with respect to the invariant measure y to problem (8.1.2).
Proof - From Theorem 8.4.4 for any n E N problem (8.5.2) has a unique invariant measure µF 0, be the transition semigroup corresponding to the solution Xn(t, x) of the equation
Xn(t) = S(t) +
t
Jo
S(t - s)llnF(llxn(s))ds + Z,,,(t)x, t > 0.
-
It is well known that the theorem is true in finite dimensions. Consequently, if ttn are the projections of p on Hn = llnH and Un(x)
U(ln(x)), x E H, n E N, then setting i4 (dx) = kneunxp(dx), x E E,
one has JE
V(x)Pt ,O(x)eUlnnxl µn(dx) = JE O(x)Pt cp(x)eU(nnx)µn(dx). (8.6.6)
But JE P(x)Pt O(x)eU(nnx) Pn(dx) = JE 4p(Hnx)Pt O(Hnx)eU(nnx)/tn(dx,). (8.6.7) Moreover one can check, see J. Zabczyk [168], that for any sequence {xn} C E convergent to x in E and for any function E Cb(H) one has lim Pt O(xn) = Pt'(x). (8.6.8) n-.+oo
By (8.6.7) and (8.6.8) we can pass to the limit in (8.6.6) provided y0, 0 E Cb(H). Therefore
JE o(x)PtV)(x)eU(x)y(dx) =
JEO(x)PtP(x)eU(x)p(dx), d (o, 0 E Cb(H).
(8.6.9)
By a monotone class argument (8.6.9) follows for arbitrary yo, O E Bb(H).
Densities of invariant measures
8.7
165
Regularity of the density when C is variational
Theorem 8.4.4 implicitly states that the density d is in the domain D(L*). Therefore, one way of obtaining information about the regularity of d"F is to characterize elements of D(L*). In general this seems to be a difficult task. In the present section we restrict ourselves to systems such that the infinitesimal generator of the semigroup Pt, t > 0, is variational. We recall that L is said to be variational if there exists another Hilbert space V continuously and densely embedded in i _: L2(H, µ) and a continuous bilinear form
a: V x V - R, such that (LAP) O)L2(H,µ) = -a(cp, 0), V
E D(L), V 0 E V.
If the operator L is variational then obviously D(L) C V, and since L* is also variational we also have
D(L*) c V. We assume here that
Hypothesis 8.5 (i) ImageQ,, C D(A).
(ii)Q=I. Remark 8.7.1 If S(.) is analytic, and there exists w E JR such that IIS(t)II < ewt for all t > 0, then Hypothesis 8.5 is fulfilled, see G. Da Prato [35, Proposition 2.5].
Remark 8.7.2 More general hypotheses were considered in V. I. Bogachev, M. Rockner and B. Schmuland [12], see also M. Fuhrman [73].
Chapter 8
166
Following [12], we introduce the space E of all exponential functions: E _ {cp E Cb(H) : W(x) = e°(h'x), x E H, h E D(A*)}.
Clearly E is linearly dense in L2(µ, H) and in D(C). Moreover if cp(x) = e'(h°x), x E H, we have Gcp(x) = - 2 Ih12 + i(x, A*h).
(8.7.1)
We now set V = W12(µ, H) and introduce the bilinear form in
VxV, JH(DxcP(x),AQ.DxO(x))µ(dx), W, V) E W1'2(µ,H)
a(V,V) =
(8.7.2)
Under Hypothesis 8.5, a is clearly well defined and continuous. Let us check, following M. Fuhrman [73], that it is coercive.
Proposition 8.7.3 We have a(cP, y') =
-2 I1cPII2y1,2(,,H)
I
V y' E Wl'2(µ,H).
(8.7.3)
Proof - We first remark that for any x E D(A*) the Liapunov equation holds:
-Ixl2
= (Q,, x, A* x) + (QUA*x, x)
By Hypothesis 8.5(i) this implies
AQ,, +
-I.
(8.7.4)
We can now prove the statement. We have a((P, c)
=
2
2
f(D(x), AQ.Dxcp(x))µ(dx)
I
(AQooDxcP(x), DxcP(x))µ(dx)
JH(DxcP(x), (AQ. + (AQ.)*)Dxw(x))µ(dx)
2 1
2IIcPI,W1.2(µ,H),
y' E u'l'2(µ, H).
Densities of invariant measures
167
We finally check that a is the Dirichlet form associated with the infinitesimal generator C of Pt, t > 0.
Proposition 8.7.4 We have D(C) _ {cp E W1'2(µ, H) : a(., c) is continuous in L2(µ, H)}, (8.7.5) and
a(4', ,O) = (ASP, v')
,
d *p E D(C), V 0 E W"' (y, H).
(8.7.6)
Proof - Since £ is linearly dense in D(L) it is enough to prove the identity in (8.7.6) for all c0, 0 E C. For this purpose we notice that (k, x) a(dx) = i(Q 12h, Q 00/2k) e- 21e I2h12, h, k E H. (8.7.7) This follows by (setting ea(h,x)
JH
F(E) = 1 e'(h'x)+e(k'x) y(dx) = e_ , H
E > 0,
and by computing F'(0). Let now W(x) = e'(alx), O(x) = e'(a'x), a, 0 E D(A*). By (8.7.1) and (8.7.7), we have (L 0 Ps is a strong Feller transition semigroup. Then an invariant measure for (9.1.1) exists if and only if there exists M > 0, w > 0 such that IIS(t)II < Me-wt, t > 0.
(9.1.2)
Moreover if p is an invariant measure then it is unique and lim Ptcp(x)
t- +00
(9.1.3)
for every xEH,cpEBb(H). Remark 9.1.2 By Theorem 7.2.1, P8 is a strong Feller semigroup if and only if Hypothesis 7.3 with t = s holds.
9.2
Ornstein-Uhlenbeck processes of wave type
9.2.1
General properties
We start from a general result on stochastic linear systems
dX(t) = AX(t) + BdW(t), (9.2.1)
X(0)=xEH,
Ornstein-Uhlenbeck processes
179
with the operator A generating a Co- group of operators S(t), t E R. Linear stochastic wave equations provide a typical application of this study. As in the preceding section, W(.) stands for the cylindrical white noise on a Hilbert space U, and B E L(U; H).
Theorem 9.2.1 Assume that the operator A generates a Co- group of linear operators S(t), t E R. (i) Equation (9.2.1) has an H-valued solution if and only if B is a Hilbert-Schmidt operator. (ii) If B is a Hilbert-Schmidt operator and dimH = +oo, the transition semigroup corresponding to (9.2.1) is never a strong Feller semigroup.
Proof - (i) If there exists a solution to (9.2.1) then, for any T > 0, the stochastic integral T
J0
S(T - s)BdW(s), t > 0,
should be a well defined Gaussian random variable. In particular its second moment T
2
E f T S(T - s)BdW(s) 0
_
JIs(r)B11Hs ds,
should be finite. From the group assumption it follows that there exist constants cl > 0 and c2 > 0 such that c2IxI < JS(r)xj < cljxj, r E [0,T], X E H.
Let {en} be an orthonormal and complete basis in U. Then II S(r)BII Hs
00 = E IS(r)Ben12,
n=1
and consequently 00
00
C2 > BenI2 < II S(r)BII Hs < ci > Ben12, r E [0,T]. n=1
n=1
Since f o IIS(r)B112 Sds < +oo, part (i) follows.
Chapter 9
180
(ii) It is enough to show that the control system
y' = Ay + Bu, y(O) = x,
(9.2.2)
with B a Hilbert-Schmidt operator is not null controllable in any time T > 0. For this purpose we recall a well known result, that we prove for the reader's convenience.
Proposition 9.2.2 Assume that U and H are separable Hilbert spaces and B is a linear compact operator from U into H. Then the operator G : L2(0,T; U)
rT
--- H, Cu = J S(r)Bu(r)dr, 0
is also compact.
Proof - Since the operator B is a limit, in the operator norm of linear operators with finite dimensional range, one can assume that
U=Band Bu = bu, u E R, where b E H. Then the adjoint operator C* to L has the form
L*x(r) = (S(r)b,x), x E H, r E [0,T]. It is enough to show that C* is compact. Let {xn,} be any bounded sequence in H and let {xnk } be a sub-sequence weakly convergent
to x. Then for arbitrary r E [0,T], (S(r)b, xnk) -+ (S(r)b,x), and therefore, by the Lebesgue dominated convergence theorem, ,C*xnk --+ C*x in L2(0,T; R).
This finishes the proof of the proposition. To complete the proof of (ii) notice that the solution of (9.2.2) at moment T > 0 is zero if and only if
-S(T)x = Lu. Since the range of S(T) is the whole of H, the operator L would be surjective. This is impossible if dim H = +oo since L is compact.
Ornstein-Uhlenbeck processes
181
Assume now that B is a Hilbert-Schmidt operator. Then a necessary condition for existence of an invariant measure for (9.2.1) is
that
f+oo
IIS(r)BII Hs ds < +oo.
0
In particular liminf II S(r)BIIHS = 0. r-.+oo
This means that if B # 0, then the semigroup S(r), r > 0, should have a dissipation property, at least for some x $ 0: liminf IS(r)xI = 0. r-+oo
(9.2.3)
Such systems are considered in the next sub-section.
9.2.2
Second order dissipative systems
Let H be a separable Hilbert space, A a positive definite symmetric
linear operator on H, and C : U -+ H a linear bounded operator. We consider the stochastic equation
dX(t) = Y(t)dt,
dY(t) = (-AX(t) - aY(t))dt + CdW(t),
(9.2.4)
X(0) = x, Y(0) = Y. The system (9.2.4) is of the form (9.2.1) where the Hilbert space H should be replaced by 1-l = D(A1/2) x H, with the norm 2
= IA1/2xI2 + Iy12.
Moreover the infinitesimal generator A can be defined by
D(A) = D(A) x D(A1/2), Al y
l
-(
0 a) (y A
y
) ED(A),
Chapter 9
182
and B is given by
B( y)-(Cx) If a = 0 one can write an explicit formula for the corresponding group of transformations
S°(t)
x
cos t
_
Y
t1(x
A-1/2 sin
_A1/2 sin vrA- t
cosy
t
/f
y
, t > 0.
For the general a write
Cy(t))=S(t)(y ), x
x y
Then by direct calculations we get
ld 2 dt
2
_ -aly(t)12, t > 0.
C y(t) x(t) )
(9.2.5)
So, if a < 0, then
rc
for arbitrary (
ai
x) E 1-l, and (9.2.3) cannot hold. Therefore if C 54 0 Y
and there exists an invariant measure for (9.2.4), then a > 0. But then, see e.g. A. Pritchard and J. Zabczyk [126, Proposition 3.5], IIS(t)117y < Me-"t, t > 0,
for some positive M and w. Taking into account Theorem 9.1.1 we get the following result.
Theorem 9.2.3 Assume that C : U --> H is a non-trivial HilbertSchmidt operator. Then there exists an invariant measure for (9.2.4)
if and only if a > 0. If a > 0 then the invariant measure is unique and Gaussian.
Ornstein-Uhlenbeck processes
183
Example 9.2.4 (Damped wave equation) Formally this can be written in the form a2
ate
X(t, E) = OX(t,x)-c
X(0' )
= x(e),
X (t, )
=
0,
at X (0, e) =
E 0, t >
E 0, t > 0,
where 0 is a bounded closed subset in Rd, and WQ is the Wiener process with covariance Q = BB*. To reduce this equation to the form (9.2.4), we set H = U = L2(0) and define
AY = -Ay, y E D(A) = H2(0) n Ho(0). Then D(A1/2) = H0(O) and the norm 11 JJj is merely the energy norm of the classical theory. The operator Q is usually given in the integral form Qx(e) = 1n q(f, q)x(rl)dq, f E H,
E 0,
with Q being the spatial covariance of the noise process.
9.2.3
Comments on nonlinear equations
Of great interest are nonlinear equations
dX(t) _ (AX(t) + F(X(t)))dt + BdW(t), (9.2.6)
X(0) = 0,
with A generating a Co-group S(t), t E R, on a Hilbert space H. If F is Lipschitz continuous there exists a unique solution X(., x) of (9.2.6) by Theorem 5.3.1. Since the operators S(t) are not compact one cannot use Theorem 6.1.2 for establishing existence of invariant measures. By Theorem 9.2.1 we cannot expect that the solution to (9.2.6) will be a strong Feller one and uniqueness, if it occurs, should be investigated with
Chapter 9
184
different methods from those developed in Chapter 4. In some situations however dissipativity arguments from §6.3 can be adapted and used. Note for instance that by formula (9.2.5) the semigroup S(t), t > 0, is dissipative. However, it can happen that an estimate of some moment of X(.,x) in a norm 11 Ily with Y compactly embedded in H is available. In this case the existence of an invariant measure follows from a tightness argument. See e.g. H. Crauel, A Debussche and F. Flandoli [30] where the existence of an invariant measure, and also of an attractor, for the wave equation is proved.
9.3
Ornstein-Uhlenbeck processes in finance
A slight modification of Theorem 9.1.1 is needed to discuss ergodic properties of a process describing the evolution of a term structure of interest rates as proposed by M. Musiela [121], see also A. Brace and M. Musiela [14]. Let t > 0, _> 0, be the rate at which one could enter a
contract at date t > 0 to borrow or to lend for a period of time at date t + e. The risk related to borrowing and lending is represented by a real, nonnegative volatility function satisfies, see [121], the following equation:
dX(t, e) _
r(e)
X(0,e) = x(e),
Jo
0. The process X
r(,q)dq I dt +
> 0, t > 0, (9.3.1)
where W (t), t > 0, is a one dimensional Wiener process. To cover a large class of functions, we choose as the state space H a weighted
Hilbert space L? = L2(0,+00), 'Y > 0, with weight e-'K,
0.
Thus x E L7 (0, +oc) if and only if +00
J
Ix12
< +00.
(9.3.2)
Let A denote the infinitesimal generator of the (left-shift) semigroup
S(t)x(e) = x(t + ), t > 0,
> 0, x E L,.
Ornstein- Uhlen beck processes
185
Then IIS(t)IIG(L2) < e21, t > 0.
Let moreover B be the one dimensional operator from U = R in L., given by
Bu(d) = ur(e), u E R, e > 0. Then equation (9.3.1) can be written in an abstract way:
dX = (AX + a)dt + BdW(t), (9.3.3)
X(0) = x E H, where
One of the main concerns of the paper [121] was existence and uniqueness of an invariant measure for (9.3.1). The following result is a direct extension of Theorem 6.2.1.
Theorem 9.3.1 (i) There exists an invariant measure for (9.3.3) if and only if
(il) supt>o TrQt < +oo. (i2) There exists an invariant measure for the deterministic system
z'=Az+a.
(9.3.4)
(ii) If conditions (ii), (i2) are satisfied then any invariant measure for (9.3.3) is of the form
y = v * N(0, Q.) S(r)QS*(r)ds, and v is invariant for (9.3.4). (iii) If (il) holds and a E Image A then any invariant measure for (9.3.3) is of the form where
µ= v* 6{b} * A1 (O, Q,,)
where v is invariant for z' = Az and b E D(A) is a vector such that a = Ab.
Chapter 9
186
Proof - The first two parts can be proved in a similar way as Theorem 6.2.1 and therefore the proof will be omitted. To show (iii)
notice that a function z is a solution to z' = Az + a if and only if y(t) = z(t) + b, t > 0, is a solution to y' = Ay and therefore the formula for the invariant measure for (9.3.4) follows. To apply the theorem to equation (9.3.1) notice that the condition (i) is equivalent to +00
(S(t)rl2,dt < +oo.
J0
(9.3.5)
Moreover the condition that a E Image A means that a E L22 and b(E) = fo a(rl)drl, > 0, belongs to L' as well. Starting from these observations T. Vargiolu [152], proved the following result.
Proposition 9.3.2 (i) Assume that -y = 0 and T is a bounded function. Then a necessary and sufficient condition for the existence of an invariant measure for (9.3.3) is
f
+00
ulT(u)ldu < +oo.
0
If this condition is satisfied then there exists exactly one invariant measure for (9.3.3). (ii) If y > 0 then a necessary and sufficient condition for the existence of an invariant measure for (9.3.3) is +00
72(u)du < +oo. JO
If this condition is satisfied then there exists an uncountable number of invariant measures for (9.3.3).
9.4
Ornstein-Uhlenbeck processes in chaotic environment
We assume here that H D U = L2(]Rd ) and that A is the operator
A=A - mI,
Ornstein-Uhlenbeck processes
187
where A is the Laplacian, and m _> 0. In particular, if H = U the corresponding heat semigroup is e-mt
S(t)x(O =
J pt(S (9.4.1)
= e-9"tPt *
x E U, 6 E Rd
where
pt(y) =
(4irt)-d/2e-t, t > 0,
E Rd.
We will consider the equation
dZ = (A - m)Zdt + JdW, Z(O)
(9.4.2)
in two important cases: when Q = I and when Q is a convolution operator,
Qu=q*u, uEU, see Example 5.2.1. Moreover J is the embedding operator of U into H.
9.4.1
Cylindrical noise
If Q = I then equation (9.4.2) has no solution in H = L2(Rd) because T
Tr QT = J e-2mt Tr S(2t)dt = +oo, 0
for arbitrary T > 0. It is therefore of interest to look for solutions of (9.4.2) in larger spaces. As a natural candidate we consider weighted spaces H = L2(Rd), where the weight p is a continuous, positive and integrable function. However, other choices are possible and have been considered by several authors. For instance the space S(Rd) of tempered distributions is taken as the state space for a generalized Ornstein-Uhlenbeck process by R. A. Holley and D. Stroock [88]. This space is natural in some applications to branching diffusions and to statistical mechanics [89], see also D. A. Dawson [50], [51], D. A. Dawson and H. Salehi [53], D. A. Dawson and G. C. Papanicolau [52].
Chapter 9
188
Proposition 9.4.1 Assume that p is a positive, continuous and integrable function on Rd such that, for some w > 0,
pt*p<ewtp,Vt>0.
(9.4.3)
Then, for arbitrary p > 1 and m E R, the formula
S(t)x = e-"(Pt * x), t >_ 0, defines a Co-semigroup on LP(Rd) spaces, and II S(t)II L(LP(j d)) < ep-m)t, t > 0.
Proof - We can assume m = 0. Consider first
(9.4.4)
p > 1. Setting
q = ppl and taking x E LP(Rd) we have
j
p
Rd
(
pt(e - 9)x(77)d?7
f-
(Jf
n)dn )
l (JPt(
ii)I x(ii)I pdri)
p
p /q
- n) d?7)
Consequently
fd I'd pt( - n)I x(n)I pp(e) drl d
I S(t)xI LP(d)
fed I
0,
d) dn
Ornstein-Uhlenbeck processes
189
and therefore the estimate (9.4.4) holds. Continuity of S(t)x, t > 0, is straightforward for functions x which are continuous and with have compact support. The case of general x follows by a standard density argument. Finally, the case p = 1 can be handled in the same way.
Definition 9.4.2 A positive, continuous, integrable weight p is said to be admissible for the operator A - mI if and only if the operators S(t), t > 0, form a Co-semigroup in LP(Rd). Remark 9.4.3 Nonnegative, continuous functions p satisfying (9.4.3) are called w-excessive in probabilistic potential theory, R. M. Blumenthal and R. K. Getoor [10]. They are solutions of the inequality (9.4.5)
AP :5 wP,
understood in the sense of distributions, see L. Schwartz [135]. We will need the following lemma.
Lemma 9.4.4 Assume that p(x) = i(Ixl), x E Rd, where 0 is a C1 function on [0,+oo[, with 0" continuous and integrable on any bounded closed subinterval of ]0, +oo[. Then the estimate (9.4.3) holds if and only if
0"(y) +
d
r
1
'(y) < WOW, y > 0,
and in addition if d = 1 then 0'(0) < 0.
Proof - If d = 1 then, see L. Schwartz [135], Ap(x) = 2L'(0)b{o} + "(Ixl), x E R
and if d > 1 then Op(x) = 0"(I xl) + and the result follows.
dlxl1 b'(Ixl),
x
Chapter 9
190
We will consider two specific families of weights
pr(x) = e-"lVl, x E Rd, where ic > 0 and 1
1 +roll , x E R
p,c,r(X) =
d
where x > 0 and r > d. When the number r is fixed we will write for
short p, We write H" = L'. (Rd) and Hr = Lp,t r(E8d). The latter notation is justified since for fixed r > d the weights pK,r, for all c > 0, give rise to isomorphic spaces. If it is not stated otherwise Hr is equipped with the norm of L2K r(Rd). For the weights introduced we have the following result.
Proposition 9.4.5 For the heat semigroup S(t), t > 0, given. by (9.4.1) we have the following estimates (i) If p = p" and p > 1, then
2
- t > 0. it S(t)II LPP(Xd) < e ()t (ii) If p = p,c,r and p > 1, then e(KP-M)t
IIS(t)IILP Rd
0,
r-d-}-2 (r-1)2(1-*).
r Proof - The proof follows from the previous lemma. We give only basic steps of the calculations.
(i) Here l(y) = e-"y, y > 0. Consequently 1'(0) Moreover Oy)_0(y)1K2-dy1KJ
(y)+dy1 and
0(y)
(K2
if and only if r. 2 < w.
-dy
1
# ) < wO(y), V y > 0,
< 0.
Ornstein-Uhlen beck processes
191
(ii) In this case 1
O(y) = T+ nyr' y > 0, icryr-1
(y) N
(1
+,yr)2, y >
K2r2y2r-2
(y) - 2 (l
0,
/Gr(r - 1)yr-2 (1 + ,cyr)2
+ ,cyr)3
y>0.
Note that 0'(0)=0andforally>0 WOW-,O"(y)-dy101(y) = (1 + 1 r)3
y[w(1
Yvyr)yr-2,r(r
+(1 +
- 2n2r2y2r-2)
+ ryr)2
+ d - 2)].
Therefore
wO(y) - 0ii(y) -
d
1
V), (Y) > 0
Y
for all y > 0 if, for instance, w(1 + kyr)2 > 2n 2r2y2r-2 - ic2r(r + d - 2)y2r-2, y > 0,
or equivalently, if 2r-2 w > l£2r(r + d - 2) (1 +
y > 0.
(9.4.6)
The right hand side of (9.4.6) attains its maximum for y such that yr = r-1. So (9.4.6) holds for all y > 0 if and only if 2
W > K,
as required.
r-d+2 (r - 1) 2(1-T) r
Chapter 9
192
Proposition 9.4.6 (i) Assume that p is an admissible weight for 0 - mI. Then the solution to equation (9.4.2) exists in L2(Rd) if and only if d = 1. (ii) If d = 1, then there exists an invariant measure for equation (9.4.2) in H" if and only if m > 0. (iii) If d = 1, and m > 2 , then the invariant measure for equation (9.4.2) is unique in H (iv) If d = 1, and 0 < m < 4 , then there are many invariant measures for equation (9.4.2) in H'. (v) If d = 1 then there exists exactly one invariant measure for equation (9.4.2) in Hr.
Proof -
(i) Let {ej } be an orthonormal and complete basis in
L2(Rd) and {/3j} independent real-valued Wiener processes. Define, see §5.2, the cylindrical Wiener process W(t) by the formula 00
W(t) = E
(t).
j=1
Assume that WW(t) is L22(Rd)-valued. Then EIN'o(t)I2 = F
P( e)IW(t,e)I2de) (Ld
= J'd P( ) E
I
j
2
S(t - s)ej(e)d(s) j=1
t00
PO d
E I S(O')
d
=1
t 00
_
P(f) J0
j=1
I (ej,p'( - .))I2do, 0 such that "p 6 < m. Then by Proposition 9.4.5(ii) the semigroup S(t), t > 0, is exponentially stable in L2 and Theorem 9.1.1 implies the result.
9.4.2
Chaotic noise
We assume now that the covariance operator Q, of the Wiener process W, is of convolution type:
Qu=q*u, uE U=H=L2(R ),
(9.4.7)
where q is a continuous, integrable, positive definite function on l[8d see Example 5.2.1. In fact we will assume the following:
Hypothesis 9.1 The operator Q is of the form (9.4.7) where the function q is integrable and of the form
q(6) = Jd e1(f>n)g(q)di, 6 E Rd,
(9.4.8)
with g a nonnegative integrable function.
It follows from Hypothesis 9.1 that q and g are continuous and bounded. The representation (9.4.8) is natural in the light of Bochner's theorem.
Theorem 9.4.7 (i) Assume that Hypothesis 9.1 holds and that p is an admissible weight for 0 - m. Then equation (9.4.1) has a weak solution on Lp(Rd).
(ii) If in addition m > 0 then there exists an invariant measure for equation (9.4.1). Moreover if m = 0 then there exists an invariant measure for equation (9.4.1) if and only if f1d g(17)drl < +oo. In the latter case the invariant measure is never unique.
(iii) If p = p" and m > 2 then invariant measures are unique and if m < 2 then they are not unique. (iv) If r > d and m > 0 then there exists exactly one invariant measure for equation (9.4.2) in Hr.
Ornstein-Uhlenbeck processes
195
Proof - Using the same notation as in the proof of Proposition 9.4.6 we have that W (t'
t > 0,
Q112ej
E Rd.
j=1
By Hypothesis 9.1, Q112u = ql * u, u E L2(Rd), where
e2rl)drl,
fLd
E Rd.
So we have, compare the proof of Proposition 9.4.6(i), that E I Wo-m.(t)I2 = E 00
IWo-m.(t, S)I2 = E
it
00
((ps j=1
f (P(e) I Wo-m(t, E)I2 d ,d
F
e-m(t-s)pt-s
110 t
* ql * ej(S)dOj(S)
* qlej(.))i2ds 2
ft
J e-2mslps * g1I22ds,
E
0
Consequently ]E
t
L2
IWo-m(t, )I 2 =
f p(71)dr7 f e-2mslps * gIIL2ds d
0
By Plancherel's theorem ps * q1 IL2 = (27r)d lid e-2sIt
Therefore >E IWO-.(t, t
x J.d
Cfo e
22 = (2.)d fed p(i)dq -2(.+IC12)s
ds g(ri)dri.
2
Chapter 9
196
Since g is an integrable function, lE IWA_m(t,
e)12
p < -boo for arbitrary t > 0.
This shows (i). To prove (ii) it is enough to remark that sup E IWo-m(t, )ILP = t>0
(27r )d fit p(rl)dq d 1
2
g(rl) Rd nz+I7I a dry < +oc,
provided that m > 0, and for m = 0 one has to require that
Jd
(E)
e < +oo.
The proofs of (iii) and (iv) are completely analogous to those of (iii),(iv) and (v) of Proposition 9.4.6 and will be omitted. We will now construct functions q and g such that (9.4.5) holds.
This way we will show that even if m = 0 invariant measures for (9.4.1) may really exist.
Example 9.4.8 For arbitrary y > 1 the even function (1 - Ixx'1) if
11
0 if ICI > 1,
is concave on [0, +oo[ and by a result of Polya, see W. Feller [64, Vol. II], and Plancherel's theorem, it is a characteristic function of an even, integrable, nonnegative continuous function V& : e'1n,0(ii)d?1,
f E R.
We define
g(f) = 2 (h( - 1) + h( + 1)) ,
Ornstein-Uhlenbeck processes
197
Then
f
etEn
q(rl) = J
2
_
1)d +
2ei77
11
(h( - 1) + h(f + 1)) d 2e-in fig ei(£+1)nh( +
2ir cos r1o(i ), rl E R.
It is therefore clear that Hypothesis 9.1 is satisfied. If, in addition, y > 1 then
Jfd< f 12
2-ry
d+ f
5II
so the condition (9.4.5) holds as well.
1
1>1
P
Chapter 10
Stochastic delay systems In this chapter we study invariant measures for stochastic delay equations by the dissipativity method.
10.1
Introduction
Let d, m E N, r > 0, a() be a finite d x d matrix valued measure on [-r, 0], b a d x m matrix, f a mapping from Rd into Rd and W(.) a standard m dimensional Wiener process. In this section we are concerned with a stochastic delay equation of the form
dy(t) = Cf a(d9)y(t + 9) + f(y(t))) dt + bdW(t), 0r
(10.1.1)
Y(O) = xo E Rd,
Y(O) = x1(9), 9 E [-r,0].
For simplicity of the presentation we assume that f is Lipschitz continuous. It is known, see A. Chojnowska-Michalik [24], that if H = Rd x L2(-r,0,Rd) then the H-valued process X,
t>0,
X(t) where
yt(9) = y(t + 9), t > 0, 9 E [-r, 0], 199
Chapter 10
200
is a mild solution of the equation
dX(t) = (AX(t) + F(X(t)))dt + BdW(t)
(10.1.2)
where the operators A, F and B are defined as follows A
P
J
for
=(
1
J
dW
de
(10.1.3)
D(A) _
F(xo X1
c0(0)
E Wl'2(-r,0,Rd) } ,
'(0o) Bu
b0u
X1 ) EH, X1
I, uERm.
(10.1.4) (10.1.5)
The process X will be called an extended or a complete solution to (10.1.1) and denoted by X(t, X) = C y(t; xo, xi)
yt(.;xo,xi)
Stochastic delay equations can be studied and have been studied as equations on Rd, without any reference to the theory of stochastic evolution equations. Basic work in this direction is due to K. Ito and M. Nisio [92]. From more recent publications let us mention M. Scheutzov [131], [132], [133] and S. A. Mohammed and M. Scheutzov [119].
In the present chapter we apply general results developed in the first two parts of the book. We are concerned with linear systems and with applicability of dissipativity method to nonlinear ones.
10.2
Linear case
We start from linear delay equations for which the abstract version is
dX(t) = AX(t)dt + BdW(t), (10.2.1)
X(0)=xEH.
Stochastic delay systems
201
Note that the system (10.2.1) is genuinly infinite dimensional while the noise process W is finite dimensional. To simplify the following presentation we restrict our considerations to the discrete delay case, N
dy(t) _ (aoY(t) +
icl
ay(t + 8)) dt + bdW(t), (10.2.2)
Y(O) = xo, y(8) = x1(8), 0 E [-r,0],
when -r = 01 < 02
0, relevant to the asymptotic behaviour of (10.2.1).
Proposition 10.2.1 (i) The operator A given by (10.1.3) generates a Co-semigroup S(t), t > 0, such that the operators S(t), t > r, are compact.
(ii) The spectrum a(A) of A is of the form N
Q(A)= AEC:
det
AI-ao-Eeao,aj =0
,
(10.2.4)
j=1
and the semigroup S is exponentially stable if and only if sup{Re A : A E Q(A)} < 0.
(10.2.5)
For the proof we refer to J. Hale [82].
Example 10.2.2 Assume that d = 1, and N = 1, then the stability condition (10.2.5) holds (for the corresponding generator A) if and only if
ao < 1, ao < -a1 < y2 + ao, where 0 < y < ir, and y coth 7 = ao, see N. D. Hayes [84].
(10.2.6)
Chapter 10
202
Theorem 10.2.3 Let Pt, t > 0, be the transition semigroup on Bb(H) corresponding to (10.2.2).
(i) Pt, t > 0, is a strong Feller semigroup for all t > r if and only if
N
rank [AI -
e' B' ai, b]
= d,
(10.2.7)
:-1
for all A E C.
(ii) Pt, t > 0, is irreducible for all t > r if and only if N
rank [AI - ao -
i.l
e'e'ai, b] = d, rank [al, b] = n,
(10.2.8)
for all A E C.
Proof - It follows from A. W. Olbrot and L. Pandolfi [122] that (10.2.7) is equivalent to the null controllability of the control system
corresponding to (10.2.2). Then by Theorem 7.3.1 (i) is true. In a similar way the proof of (ii) follows from a characterization of the approximate controllability due to A. Manitius [110] (for an alternative
proof see It. F. Curtain and H. J. Zwart [32]), and the fact that the corresponding control system is approximately controllable in time t if and only if the image of Qt = fo S(t)BB*S*(t)dt is dense in H. Proposition 10.2.4 A sufficient condition for the strong Feller prop-
erty of Pt, t > 0, for all t > r is that rank [b, aob,...,
ao-1 b]
= d, and Image b D Image ai, i = 1, ..., N. (10.2.9)
Proof - Consider the control system N
z'(t) = aoz(t) +
aiz(t + Bi) + bu(t), i=1
(10.2.10)
z(0) = xo, z(9) = x1(9), 9 E [-r, 0],
and introduce new control functions uo and v such that u(t) = uo(t)+
v(t), t > 0, and E"_'1 aiz(t+9i)+bu(t) = bv(t),t > 0. Then (10.2.10)
Stochastic delay systems
203
becomes z'(t) = aoz(t)+bv(t), and, by the rank condition, the system can be steered to zero.
The main result of the present section is formulated in the following theorem.
Theorem 10.2.5 (i) Assume that (
sup
t
N
r
l
1
ReA:det[AI-Eeoai] =0} 0. Some results in this direction can be found in K. Ito and M. Nisio [92]. It is also possible that if the transition semigroup for the linear equation is a strong Feller one, see §4.2, then the semigroup corresponding to the nonlinear equation (10.1.1) is likewise but the proof
Chapter 10
204
of this result is also laking. Consequently, for the proof of the uniqueness of the invariant measure, Doob's theorem cannot be applied at the moment. Results on this topic, based on Girsanov's theorem, are contained in M. Scheutzov [132]. In this section we discuss applicability of the dissipativity method for the equation N
dy(t) =
aoy(t) + E aj y(t + 9j) + f(y(t))
dt + bdW(t),
j=1
y(0) = xo E Rd,
y(9) = x1(9), 0 E [-r,0]. (10.3.1)
We start from the crucial question: under what conditions does exist
a positive number w such that the operator A + wI is dissipative, compare §6.3.2, Unfortunately, with respect to the original norm in H = Rd x L2(-r, 0; Rd), the operator A does not necessarily have this property. If, however, we replace the original norm by an equivalent one of the form 0 In(
20
)III2=Ixol2+ J TT(9)Ixl(9)I2 d9,
xo x1
) E H,
with the function T satisfying 71 < T(0) j=1 r,
f
dx
Bi+1
(T(0),
6i
x(0))
d0.
But
j°e2/39((9),x(9))d9=
e2P6
2 Jeo
= and
(1x(0)12
dB
- e2,111, 1x(01)12) -,310e2119 IX(0)12 dO,
2
e3+1 / ddx (0),x(0)>
JJ
d IX(0)12 dO
de = 2 (Ix(ej+1)12 - I x(0j )12)
Moreover N
Jl < -aolx(0)I2, j2 E IajI Ix(0j)I 1x(0)1 j=1
N.
Stochastic delay systems
207
Therefore x(0) -< A
x
X
x(0)
x
N
< -aplx(0)I2 + E Iajl ix(ej)I Ix(0)I j=1
N +2 E Tj (Ix(ej+1)I2 - I x(Bj)I2) + /
j=1
--0 J
(Ix(0)I2 - e211,
IX(
01)12
2 0e2R°Ix(9)I2dO
< (-aO + TN2
)
Ix(0)I2
N
+2 1:(Tj_1 - Tj)Ix(O )I2 J=2 N
1
2
(71
+'Ye2)3e1)
Ix(81)I2
f
+E IajI I x(Oj)I Ix(0)I - yae2111Ix(0)I2d9. ° j=1
Consequently, for (10.3.7) to hold it is sufficient that
Cap-a-TN2 y )Ix(0)I2 N
+2 j=2
r _i)Ix(Bj)I2 + 2 (71 + ye2001) Ix(91)I2
(10.3.8)
N
E I aj I I x(ej)I Ix(O)I >- 0, - j=1
and ° (,y,3e20e 181
- ar(9)) IX(0)12 dO > 0,
(10.3.9)
Chapter 10
208
for all x E W1'2(-r, 0; Wi). The inequality (10.3.9) holds if and only if _ N e a 200 > 0 E ]01, 9N+1] a
j=1
Tjxloj.Bj+1l'
Taking into account (10.3.5) one easily sees that (10.3.9) is satisfied.
Moreover if Ix(0)I = 0 then the inequality (10.3.8) holds because Tj > Tj_1 for j = 2,...,N and Tl + 7e2Qe1 > 0. If Ix(0)I 0 then dividing both sides of (10.3.8) by Ix(0)I2 we get -Tj-F)-ye2,301)CIx(01)I12
12 EN(7-j N
(Ix(Oi)I) E - aj j=
Ix(o)I J
2
Ix(o)I /l1
j=1
7- a0.
I
(10.3.10)
By the definition (10.3.5), the minimal value of the left hand side of (10.3.10) is _1 4
N E (Tl+7e2P91 + j=2 Tj 2Ia112
1
a
a Ia1I2
27
2Iaj12
e2001 +
T7-1
N .7=2
- e2Qej-1 Iajl2
e2130j
Therefore (10.3.10) holds provided there exists 7 > 0 such that
ao-a> 2 (0-ae 200N + 1 1 1a lal12 +27
a
N
e2/301 + 3 - a =2 j
Iaj12
(10.3.11)
e210j-1
Minimizing the right hand side of (10.3.11) with respect to 7 > 0 one gets the desired result.
Stochastic delay systems
209
Remark 10.3.2 One gets even more explicit conditions implying (10.3.6) if one takes for instance /3 = 2a (or 0 = 3a, /3 = 4a,...). Different choices for the weight r are possible. For instance if N = 1 and T(0) = ye2a0, 0 E [-r, 0],
one finds that the inequality a0 > a + e' lall
implies (10.3.6).
The main result of this section is a direct consequence of Theorem 6.3.3, see Remark 6.3.4, and of Theorem 10.3.1.
Theorem 10.3.3 Assume that condition (10.3.5), holds and that the Lipschitz constant of f is smaller than a. Then, there exists exactly one invariant measure y for the extended solution, of (10.3.1).
Moreover there exists w > 0 such that for all x =
I
xo XI
E H,
cp E Bb(Rd) and t > 0
((y(t;xo,xi)) - f IP(z)µd(dz)/ `- (C + 2lxl)e-"tlIPIILip, d I
where µd is the projection of a on the Rd component of H.
Chapter 11
Reaction-Diffusion equations In this chapter we study invariant measures for stochastic reactiondiffusion equations. First equations on a finite space interval are studied and then equations on the whole of Rd and with the space homogeneous Wiener process are investigated.
11.1
Introduction
Deterministic reaction-diffusion equations are usually written as systems of equations
atk
akAyk(t, e) +
yi(t, c), ..., YK(t, c)), t > 0,
k=1,...,K, (11.1.1)
where fk, k = 1, ..., K, are given real functions from 0 x RK to R, Uk, k = 1, ..., K, are positive constants, and 0 is an open subset of Rd. If 0 # Rd one also adds boundary conditions of Dirichlet, Neumann or mixed types. 211
Chapter 11
212
The stochastic version of (11.1.1) is of the form
dXk(t, ) = (okzXk(t, E) +
X1(t, c), ..., XK(t, ))] dt
+ bk(e, X1(t, ), ..., XK(t, ))dWk(t, ), = xk(e), e E 0 C Rd, k = 1, ..., K, t > 0,
Xk(0,)
(11.1.2)
plus boundary conditions if 0 54 Rd. Moreover W1, , WK are independent Wiener processes and bk, k = 1, ..., K, are given real functions. Setting 0'10
X1(t, E)
X(t,C) =
A
W(t)
Wi(t)
,
XK(t,C)
WK(t)
CJ xd(S))
XI(S)
xl(b),...xd(S))
xd(S)
f1(b,xl(S),..
F(x)(C) _ fd(S, b1 (S, xl
.xd(S ))u1
ul (')
u()
(B(x)u)(C) = bd(C, xl (C), ...xd(C))ud
ud(')
one can write problem (11.1.2) in the standard form
dX(t) _ (AX + F(X))dt + B(X)dW(t), (11.1.3)
X(0) = X. To formalize this procedure one should give precise definitions of the operators A, F and B, of the state space F (or H in the Hilbert space case) and of the space U on which the cylindrical Wiener processes are defined. We will consider only one equation, K = 1, and distinguish two main cases:
Reaction-Diffusion equations
213
(i) d= 1,O=]a,b[, -oo 0, E ]0, 1[. We will require that the Wiener process W is cylindrical on U = L2(0,1). This space will also be our state space H.
In this section it is natural to assume that 2
Ax = d-2 , x E D(A) = Ho'2(0,1) n H2'2(0,1), f(x(E)), Thus problem (11.2.1) is equivalent to problem (11.1.3).
11.2.1
Existence and uniqueness of solutions
We assume that f and b are Lipschitz continuous mappings. Consequently, F is likewise, but B is not from H into L(H). We will check, however, that conditions of Theorem 5.3.1 are satisfied. As the functions
=v sin
n E N,
E [0,1],
form an orthonormal sequence of eigenfunctions of the operator A, corresponding to the eigenvalues -ir2n2, n E N, we have II S(t)(B(x) - B(y))II
Hs
= E I00((B(x) - B(y))en, S(t)em) I2 n,m=1
e-2tl2m2I
(B(x) - B(y))eml2
m=1
But
I(B(x) - B(y))em I2
=2
0 sufficiently small.
Remark 11.2.3 Using the factorization procedure one can prove that solutions of (11.1.3) take values, for t > 0, in E = C'([0, 1]) with a E ]0, 1/2[. Consequently if ti is an invariant measure for (11.1.3) then
µ(E) = JHPt(x,E)() = 1, t > 0, and the measure u is concentrated on E, see It. Sowers [144]. For uniqueness we have the following theorem, see R. Sowers [144], C. Mueller [120], S. Peszat and J. Zabczyk [124]. The present proof and the limit formula for Pt, t > 0, are taken from [124].
Theorem 11.2.4 Assume that f and b are Lipschitz continuous functions and for some constants rc, k > 0
Then there is at most one invariant measure for (11.1.3). If µ is an invariant measure for (11.1.3) then, for arbitrary x E H, I' E ,3(H),
t
iim Pt(x, r) = p(r). 00
Proof - By Theorem 4.2.1 and Proposition 4.1.1 it is enough to show that the transition semigroup corresponding to the solution of (11.1.3) is a strong Feller one and irreducible. The strong Feller property is not a direct consequence of Theorem 7.1.1 because B is not a Lipschitz transformation from H into
Reaction-Diffusion equations
217
L(H). However, as we saw in subsection 11.2.1, B satisfies Hypothesis 5.1(iii), and the proof of Theorem 7.1.1 workes with only one exception. That is we do not know if the mappings B,,,(x), x E H,
are invertible. But in the present particular case their invertibility is a simple consequence of the fact that either b(C) > m for all or -m for all . Irreducibility follows from Theorem 7.3.1.
11.3
Equations with non-Lipschitz coefficients
As in §11.2 we set K = d = 1; however, we no longer assume that the function f is Lipschitz continuous, but to simplify presentation we set the diffusion coefficient to be equal to 1. Thus we will consider now the problem dX (t, £) _
X (t, 0) + f (e, X(t, ))
12
dt + dW(t, e), (11.3.1)
X(t,0) = X(t,1) = 0, X(010 = X(01
for t > 0, E ]0, 1[. We will need the following conditions on
f.
Hypothesis 11.1 (i) The function f is locally Lipschitz continuous and (f ( + i) + l; + )) sgn < a( rl ), for all , q E R, where
a(r) < co + clr", for some co > 0, cl > 0 and p > 1. (ii) For some w E R, the function f - w is decreasing. By K we denote the space Co([0,1]) of all continuous functions on [0, 1] vanishing at 0 and 1 with the usual supremum norm. As in §11.2 we take H = L2(0,1). We define d2
Ax = dC2 - x, x E D(A) = H2(0,1) n Ho(0,1),
Chapter 11
218
and E [0, 1].
Theorem 11.3.1 Under Hypothesis 11.1(i) the solution of problem (11.3.1) on K has an invariant measure concentrated on K. If in addition Hypothesis 11.1(ii) holds then the solution of equation (11.3.1) on H has an invariant measure concentrated on K.
Proof - We will check the hypotheses of Theorem 6.3.5 and Theorem 6.3.6. The semigroup S(t), t > 0, generated by A is of the form
S(t)x = Y, 00 e-«"ten(en,x), x E H,
(11.3.2)
n=1
where an = -a2n2 - 1, n E N, and f sinnir , n E N. The same formula defines the restriction SK(t), t > 0, of S(t), t _> 0, to K. Both semigroups are compact repectively in H and K and I S(t)I
< e-2t, IISK(t)IIK < e-t, t > 0.
It is also clear that the transformation F has the desired properties. Therefore it is enough to check that the process
WA(t) = f S(t - s)dW(s), t > 0, ot
-
has an E-continuous version such that for arbitrary p > 1 supEIIWA(t)Ilfc < +oo. t>o
This follows however from Theorem 5.2.9.
Remark 11.3.2 In I. Gyongy and E. Pardoux [81], C. Donati Martin and E. Pardoux [55], and I. Gyongy [80], equation (11.3.1) is studied under weaker assumptions by using a comparison theorem for SPDEs. For a result about the existence of an invariant measure in this setting see G. Da Prato and E. Pardoux [41].
Reaction-Diffusion equations
11.4
219
Reaction-diffusion equations on d dimensional spaces
In this section we use the notation of §9.2. We will consider equations of the form
dX(t,E) = [(Ax(0, E) =
t > 0, E Rd, t > 0,
(11.4.1)
where A is the Laplace operator, m a nonnegative constant and the covariance operator Q of the Wiener process W is either I or a convolution operator
Qu(a) = q *
E Rd, U E U,
satisfying Hypothesis 9.1.
As far as the function f : R -p IR is concerned we will require that the following is satisfied.
Hypothesis 11.2 The function f is of the form
f=10+fi where fo(b)+ii , e E R, is continuous and decreasing for some rt E R, and for some s > 1 and co > 0 Ifo(E)I 1 and Hypothesis 9.1 holds. Then equation (11.4.1) has a unique generalized solution in H', s; > 0, and H,., r > 0. If in addition x E LP8(Rd) where p = p" or p = Pr, then the generalized solution is mild.
Proof - We will apply Theorem 5.5.8 with H = L2.(Rd), H = and K = LpK(][8d), Is = Lpr(l[8d). By Proposition 9.4.5 the heat semigroup can be extended to semigroups on all those spaces. If A,c,2 i Ar,2 i AK,2a i A,.,2s are the genera-
tors of the semigroups then, again by Proposition 9.4.5, the operators A.,2 + 1), Ar,2 + 17, A,c,2s + 77, Ar,2s +77 are m-dissipative for sufficiently small q7.
To see that the operator F + 77 where
F(x)(e) = f(x(e)), f E D(F)
_
J X E L 2(Rd) :
Rd,
f
E L2(][8d)},
sastisfies Hypothesis 5.4, note that D(F) J K and the domain D(FK) of the part FK of F in K contains LP3(Rd). It is also easy to see that F + 71 and FK + 11 are m-dissipative for small 77 and that F maps bounded sets in K into bounded sets in H. Then Hypothesis 5.4 holds. The following proposition shows that Hypothesis 5.5 is satisfied as well and consequently finishes the proof of Theorem 11.4.1.
Proposition 11.4.2 Let Ap,P, where p = p', K > 0, or p = pr, r > 0, and p > 1, be the generator of the heat semigroup in LP(Rd). Then the Ornstein-Uhlenbeck process WA,,, (t), t > 0, has continuous
trajectories in LP(Rd), provided that condition (i) or condition (ii) of Theorem 11.4.1 holds.
Proof- We will consider only the more difficult case (ii). Without any loss of generality, we can assume that p > 2 and that 00
C'`Q1/2ej(e)Nj(t), t > 0, W(t,S) = 3=11
E Rd.
Reaction-Diffusion equations
221
We will use the factorization formula, Theorem 5.2.5, and investigate first the process t
Ys(t) = f S(t - s)(t - s)-SdW(s), t > 0,
-
o
[. By Gaussianity of Y6(t, e), t E] 2 cl > 0, suitable constant where 5
p,
EIYs(t)I LP(] d) = IE t
lid P(E)E
f
f (t -
E W', for a
_> 0,
)IPdf
d
P
00
s)-s>(S(t
-
d
j=1
Pt -
s)-s E(S(t - s)Q1,2hj)(e)dij(s) j=1 P/2
P(S) t (10
< c1
t U-28 E 00
C
I S(S)Q1/2hj(S)I2ds
j=1
But
S(S)Q112hj(e) = e-asps * ql * hj(e), where
E Rd'
Q1/2u=g1*u, uEU.
Consequently IS(S)Q112hj(S)I2 = e-2as(ps * ql(S
and by Parseval's and Plancherel's identities
E
2 IS(S.)Q112hj(f)
=
-2asjjps * g1IIU
jElld e-2as f, d
AU
Chapter 11
222
Therefore, for a constant c2, P/2
EIYS(t)IPP(Ld) < c2 [g(n) (e-2s(m+InI2)s-2ads)
d(i)J
Since m > 0, for another constant c3,
]EIY6(t)ILP(jd) < [cs f
d (m + g(ij) IrlI2)2s+1
lp/2
+oo
for arbitrary t > 0.
Taking into account, see G. Da Prato and J. Zabczyk [44, p.128],
that WA, (t) =
sin'rbG5Y5(t), ir
t > 0,
where Gb is the operator given by
GsV(t) = f
t S(t-s)(t-s)b-1Cp(s)ds,
t E [0,T], p E LP(O,T;LP(R'1)),
JO
we see that II WAP.P(t)II LP(IlE.d)
` X
sin orb
(ft s4(&-1)IIs(s)IIL(LP(
1
4
d))ds)
7r
(Jot II Ys(s)II iP(nd)ds)1/P
where q = P The function IIS(s)I)L(LP(Ld)), s > 0, is locally bounded and q(b1) > -1. Therefore for arbitrary T > 0 there exists a constant c4 such
that >E
sup tE[0,T]
II WAP,P(t)II LP(]Ld))
< c4 JOT EIIY6(s)IILP(]&d)ds < +oo,
and the required continuity follows.
The next theorem is devoted to the existence and uniqueness of the invariant measures.
Reaction-Diffusion equations
223
Theorem 11.4.3 In addition to the conditions of Theorem 11.4.1 assume that the function fo is decreasing and
m-IIfiIIL,p>LO >0. Then there exists yo > 0 such that the transition semigroup Pt, t > 0, corresponding to the solution of (11.4.1 has a unique invariant measure in both H = L',, (Rd) and H = L2, (Rd), for any ic, r E ] 0, eco[. Moreover there exists c > 0 such that for any bounded Lipschitz function yp on H, all t > 0 and all x E H
IPto(x)
- JH W(x)p(dx) I < (c + 2IxI)e-WtII4oIILip.
Proof - We apply Theorem 6.3.3. Arguing as in the proof of Theorem 11.4.1 one easily shows that condition (i) of Hypothesis 6.2 holds. It remains to check that (ii) holds as well. We will prove that
for arbitrary p > 1 sup ]E
(IiwAP.P(t)IILP(]&d))
< +o0-t>0
Note, compare the proof of Theorem 11.4.1, that for all t > 0 t 00
E II YYAP, (t)II LP(Rd)
=J
d
p(S)
f0 E S(t j=1 t 00
f
< cl
- cl
J d p(e) (10 Ej=1I S(t - s)Q1/2hj(6)I2ds
I
) P!2 g(ij) dp() 1\ 2(m + I7]I2) d d
(
p/2
d6
Chapter 12
Spin systems Invariant measures for classical and quantum spin systems on d dimensional lattices Zd are considered. Using the dissipativity method, conditions for the existence of invariant measures are given. The exponential convergence of the transition probabilities to the invariant measure is established as well.
12.1
Introduction
Let Zd be the d dimensional lattice whose elements will be interpreted as atoms. A configuration is basically any real function x defined on
Zd. The value x(y) of the configuration x at the point -y can be viewed as the state of the atom -y and it is sometime called the spin
of y. Physical properties of a lattice system are determined by a matrix (a.yj),yjeZd ofglobal interactions and a function g : R -> ]R, of local interactions. For an arbitrary finite set I' C Zd the matrix (a.y,j),y jEzd is assumed to be negative definite. The Hamiltonian Hr of the restricted spin system is given by the formula
Hr(x)
a.y,jx.yxj 1',jerd
g(x.y), x E RZ",
(12.1.1)
ryEr
and the corresponding restricted Gibbs measure pr has a density, with respect to the Lebesgue measure on Rr, of the form Cre{-Hr(x)) = Cre'
r,iEZd a"r,ix7xi+E7Er
225
9(X7))'
x E Rr, (12.1.2)
Chapter 12
226
where Cr is a normalizing constant such that
µr(Rr) = 1.
(12.1.3)
For this to be true some conditions on the behaviour of g at infinity have to be imposed. Identifying the space R" with the set of those configurations x E RZd which vanish outside of F, every measure µr can be regarded as a probability measure on ]R . Assume now that {r,,} is any increasing sequence of finite subsets of Zd. All probability measures on the space of configurations that are weak limits of are called Gibbs measures. Of special interest are those Gibbs measures which are concentrated on configurations with polynomial growth. Denote by yr the Gaussian measure on Rr with mean vector zero and covariance matrix Sr =
2
((-a'Y,.7)-Y,.1Er)-1
Then for a possibly different constant Cr
yr(dx) = CreF7er9(x7)vr(dx).
(12.1.4)
In general the sequence of densities
{Crel:,Er s(x,) }f EN
does not converge to a density with respect to the limit of Gaussian measures µr,,. This makes the problem of finding a Gibbs measure rather difficult. Let f be a primitive function of Zg:
d (z) = 2 g(z), z E R.
(12.1.5)
Under general conditions on g the measure µr is the unique invariant measure for the stochastic differential equation
dXry(t) = E a.yjXj(t) + f(Xry(t)) dt + ;Er
7Er, t>0,
(12.1.6)
Spin systems
227
where W.y, y E t, are independent Wiener processes. It is therefore reasonable to expect that Gibbs measures corresponding to ((a1j), g) can be invariant measures for the infinite system of equations dX1(t) _
a-,j XX(t) + f (X1 (t))
dt + dWy(t), (12.1.7)
X,y(0)=x,y, yEZd,
t>0.
J
In the following section we will study system (12.1.7) regarded as a stochastic evolution equation
dX = (AX + F(X))dt + BdW(t), (12.1.8)
X(0)=xEH on a properly chosen Hilbert space H of functions on Zd. We not only prove existence of solutions to (12.1.7) but give conditions under which the corresponding transition semigroup has a unique invariant measure. Generalization to quantum spin systems will be the object of §12.3. We follow G. Da Prato and J. Zabczyk [48].
12.2
Classical spin systems
Let W(t) = {W.y(t)}lEzd, t > 0, be a Wiener process on (1,.F,?) with values in U = 12 (R') and with covariance operator Q E L(U). In particular if Q is the identity operator then the processes W.y, y E Zd, are independent standard real valued Wiener processes. However, Q
can be any nonnegative definite operator on U and for any t > 0 the family {W.y(t)}1Ezd can form a general Gaussian random field including stationary ones if Q is shift invariant. To study system (12.1.7) we apply results on dissipative systems from subsections 5.5.2 and 6.3.2 to (12.1.8). The operators A and F will be given by the formulae A(x.y) _ (>AEzd a9'yx9) , x = (x1) E H, (12.2.1)
F(x.y) = (f (x.y)) , x = (x.y) E H,
J
Chapter 12
228
and H = 1p(Zd) will be a weighted Hilbert space of sequences (x y) with positive summable weight p : Zd R+. B will denote the embedding operator J from U into H. Cb(H) and Bb(H) are respectively the space of all bounded continuous functions in H and the space of all bounded Borel functions in H. The following proposition, which goes back to Schur, see P. R. Halmos [83, Pag. 36], gives sufficient conditions under which the matrix (ayj) defines a bounded operator on 1P(Zd) for arbitrary p E [1, +oo].
Proposition 12.2.1 Assume that (I) sup T, Ia-YjH =a 0 such that for
Chapter 12
234
' E ]0, iE [
(A(x-y)+F(x)-F(y)+(rl-E)(x-y), x-y)t2
(zd)
< 0, x, y E j2"(Zd).
Consequently if tc E 10, r.,[ the operator A + q - e is m-dissipative. Since -IIf1IILip +q - E > w for sufficiently small E > 0 Hypothesis 6.2 is fulfilled.
We will show now that the process WA has bounded p-moments in £P(Zd) spaces for all p > 1: (12.2.7)
+oo.
sup 1EIWA(t)I t>o
Write WA(t) = (Zy(t))dyEz and S(t) = (ary(t)),q-YEZ. Then t
Jsj(u)dWj(u),
Z,y(t) _ jEZd
EIIWA(t)IIeP(zd)
u E(IZ',(t)Ip)-
yEfd
By Gaussianity and assuming, to simplify notation, that Q = I,
E I Z-r(t)I P= cr (E I Z-r(t)l2)pl2
0. (12.3.1)
In the equation (12.3.1), A and F are respectively linear and nonlinear, in general unbounded, operators on a Hilbert space ?-l, (a.yj),y JEzd is a given matrix with real elements and (W.y),yEzd is a family of independent, cylindrical Wiener processes on 1I. Following [4] we will
require that the space f and the operators A and F are of special character although it will be clear that our general schema works in the more abstract setting. We will thus assume that 7.1 = L2(0,1), A =
dZT
(12.3.2)
- a,
D(A) = {x E H2(0,1) : x(0) = x(1), x'(0) = x'(1)},
f(x(e)),
E [0, 1], (12.3.4)
D(F) = {x E L2(0,1) : f (x) E L2(0,1)}, and on f and on the matrix (ayj)'y,.7E Gd we will impose Hypothesis 11.2 and conditions (12.2.3)-(12.2.4). To see that (12.3.1) is of the general form (12.1.8) define x(2d)
H = £ (L2(0,1)) _ {(x y)
P(Y)II xti II 2 < +°O},
E yEZd
K =
£22s(L2s(0, 1))
{(x.y) E (L2s(0,1))(7d)
E,yE2d
p(y)IIx_YIIL2s(o,i)
< +001,
where p = p" or p = pr, K,T > 0, see the previous section.
Spin systems
237
Let A = AO + Al where Al is a bounded linear operator on H given by
Al(x.y) =
a.yjxj
, x E D(A1) = H,
jEZd and
Ao(x.y) _ (Ax.y), x = (x.y) E D(Ao).
D(Ao) _ {(x.y) E H :
P(y)IIAx. II' < +oo}. 'YEZd
Boundedness of Al follows from an obvious generalization of Proposition 12.2.1. Let, in addition,
F(x..) = (Fx.y), x = (x.y) E D(F), D(F) = {(xry) E H :
+oo}.
It is easy to check that the operator AO + 77 on H and its restriction Aop +,q to K are m-dissipative for sufficiently small rt.
Let S(t), So(t), t > 0, be Co-semigroups on H generated by A and AO and write
WA(t) = J S(t - s)dW(s), t > 0, ot
-
WAo(t) = J So(t - s)dW(s), t > 0, ot
where W(t), t > 0, is the Wiener process H.
-
embedded into
Proposition 12.3.1 The processes WA0(t), t _> 0, WA(t), t > 0, have continuous versions with values in fp(LP(0,1)), p > I.
Proof -
Existence of a continuous version of WAo (t), t > 0, in £p(LP(0,1)) can be obtained by factorization, as in the proof of Proposition 11.4.2, and using the diagonal character of the semigroup So(.).
Chapter 12
238
Note that for Z(t) = l4'4(t), t > 0. Z(t) = WA,(t) + JOt So(t - s)A1Z(s)ds, t > 0.
-
(12.3.5)
Since the semigroup go(.) has a Co -continuous restriction to £ ( L ( 0 , 1)), the equation (12.3.1) has a unique solution in
C([O,T);t;(LP(0,1)), for arbitrary T > 0 by an elementary fixed point argument. This 0 proves the result. As a corollary from our discussion and Proposition 12.3.1 we set the following result.
Theorem 12.5.2 Assume that Hypothesis 11.2 and conditions (12.3.2)-(12.3.4) hold. Then for arbitrary x E H the equation (12.3.1)
has a unique generalized solution X(-,x). If x E K the solution is mild.
As in §11.4 and in §12.2 one can derive a result on exponential decay for the transition semigroup Pt, t > 0, corresponding to the solution X(., x) of (12.3.1).
Theorem 12.3.3 In addition to the conditions of Theorem 12.3.2, assume that fo is decreasing and that a - IIfilILip > w > 0. Then there exists ico > 0 such that the semigroup Pt, t _> 0, corresponding to the solution of (12.3.1) has a unique invariant measure p both on H = L2K and on H = L2.. Moreover there exists c > 0 such that for any bounded and Lipschitz function cp on H, all t > 0 and all x E H, IPtV(x)
- JH 4o(y)µ(dy)
< (c + 2IIzjI)e 'tIIPIILip-
Proof - As in the proof of Theorem 11.4.3 it is enough to check the assumptions of Theorem 6.3.3. We will show that
supE(IWA(t)IH) < +00,
(12.3.6)
supE(IF(WA(t))IH) < +0,
(12.3.7)
t>o
t>o
as the remaining conditions hold in an obvious way.
Spin systems
239
To prove (12.3.6) note that for t > 1, t
E I WA(t)I H
= E If S(t - s)dW(s))H
E ( Jot S(s)dW(s)
H
fk+1 S(s)dW(s)
x
k
+
lE
ft] S(s)dW(s) It
[t]-1
E IIS(k)IIHE I f 1 S(s)dw(s)I x+ sup
E 1JU
S(s)dW
k=0
[t]-1
< k= 0
0 such that for the restriction S(t), t > 0, of the semigroup S(t), t > 0, to K, one has c(EIlWA(t)IIK)s.
IIs(t)IIK 0.
It remains now to repeat the proof of (12.3.6).
Remark 12.3.4 Theorem 12.3.3, under slightly stronger conditions, was proved earlier in [4] using specific properties of the one dimensional heat equation. It is, however, clear that the dissipativity method allows us to treat spaces 7-( more general than L2(0,1). Existence of an invariant measure y for (12.3.1) was also obtained in S. Albeverio, Y. G. Kondratiev and T. V. Tsycalenko [4] with more stringent conditions imposed on f, p and (a.yj). Exponential estimates in Theorem 12.3.3 were obtained in G. Da Prato and J. Zabczyk [48].
Chapter 13
Systems perturbed through the boundary In some applications the noise can effect the evolution of a system only through the boundary of a region. For such systems conditions are given for existence and uniqueness of invariant measures. Here once more the dissipativity method is used.
13.1
Introduction
In several applications external forces enter the system through the boundary of a region where the system evolves. The same is true if the forces have a stochastic character. Consider for instance the
nonlinear heat equation in the region G = J0, i[d C Rd, with the noise affecting only a portion I'o
6d=0}, 241
Chapter 13
242
of the boundary F of G. Thus the state equation is of the form
X (t,
_ (0 - m)X(t, ) + f(X (t, e)), t > 0, (t, f)=0, REF\Fo, t>
EG, aax
(t,
aV
(t, f ),
E
I
ro, t > 0, (13.1.1)
where vXis the inner normal vector to the boundary at those points f E F where it is well defined. The process V(t), t > 0, is a Wiener process on U = L2(Fo) with the covariance operator Q not necessarily of trace class. The operator Q could be of integral type, Qu(a) =
fro
u E U,
E ro,
(13.1.2)
with the kernel g being the correlation function of the random field V(1, e), E Fo. Of special interest is the case when the field V (l, ), E Fo, is the restriction of a stationary random field on Rd-1. Then there exists an integrable function qo : Rd-1 -+ R such that rl) = qo( - 77),
, 77
E I'o.
(13.1.3)
Moreover qo is the Fourier transform of the spectral density of the random field. System (13.1.1) is our main example. We will establish existence and uniqueness of Markov solutions using the dissipativity method. We will show also that under additional conditions on f there exists a unique invariant measure for the solution. Asymptotic properties of the corresponding transition semigroup will be established as well.
We start from general information on parabolic systems with non-homogeneous boundary data. We formulate an integral version of a generalization of (13.1.1) and give conditions under which it has a solution. Then we apply the general theory for the specific model (13.1.1). We obtain in this way explicit conditions, in terms of the spectral density of the noise, implying the existence of solutions to (13.1.1) and the existence of a unique strongly mixing invariant measure.
Systems perturbed through the boundary
13.2
243
Equations with non-homogeneous boundary conditions
We start from the linear case. Let G C Rd be a fixed region, r = OG its boundary and let A be a linear second order differential operator and r a linear first order differential operator. A non-homogeneous boundary problem is of the form T (t,) = Ay(t, e), (t, e) E ]0, +oo [ x G, ry(t, e) = u(t, ), (t, e) E 10, +oo [ x r, y(0, e) = x(f ),
(13.2.1)
E I',
where u is a U-valued function and U = L2(r). Under some conditions on the data the solution to problem (13.2.1) is given by the formula t
y(t) = S(t)x + (A - A) J S(t - s)Du(s)ds, t > 0,
(13.2.2)
where \ is a real number, A the infinitesimal generator of a strongly
continuous semigroup S(t), t > 0, on a Hilbert space H = L2(G) and D a properly defined linear bounded operator from U into H. Assume for instance that the linear operator Ax = Ax, (13.2.3)
D(A) = {x E H2'2(G) : r = 0},
generates a strongly continuous semigroup S(t), t > 0. on H. Let A be a real number such that the problem
Aw - Aw = O, rw = u
(13.2.4)
has a unique solution in = Du E H for arbitrary u E U. If is a twice continuously differentiable function with values in U and x - Du(0) E D(A) then there exists a strong solution of the equation
z'(t) = Az(t) + .\Vu(t) - Du'(t), t > 0, (13.2.5)
z(0) = x - DU(O).
Chapter 13
244
In particular r(z(t)) = 0 for all t > 0. Define
y(t) = z(t) + Du(t), t > 0. Then
ry(t) = r(Du(t)) = u(t), t > 0, and
d (y(t) - Vu(t)) = (A - A)(y(t) - Du(t))
Ay(t) - Dot u(t).
Taking into account that (A - A)y(t) = 0, for all t > 0, we see that
at (t) = Ay(t), t > 0, 1
7y(t) = u(t), t > 0,
(13.2.6)
Y(O) = X.
Consequently the function y(t), t > 0, solves the problem (13.2.1) and is given by the formula
y(t) = S(t)(x -Du(0)) j S(t - s)[ADu(s) - Du'(s)]ds+Du(t), t > 0.
Proposition 13.2.1 If is a twice continuously differentiable function and x - Du(0) E D(A) then a solution y(t), t > 0, of the abstract boundary problem (13.2.6) is of the form
y(t) = S(t)x + (A - A)
J0 t S(t - s)Du(s)ds, t > 0.
The proof of the proposition is an immediate consequence of the following well known lemma.
Lemma 13.2.2 Assume that the operator A generates a strongly continuous semigroup S(t), t > 0, on a Banach space E and that p E W1"1(0, T; E) for some T > 0. If x E D(A) and
z(t) = S(t)x + f S(t - s)cp(u(s))ds, t E [0,T], t
(13.2.7)
Systems perturbed through the boundary
245
then z E C1([O,T); E) fl C([0, T]; D(A)). Moreover
f
Az(t) = S(t)x +
t
S(t - s)cp'(u(s))ds + S(t)cp(0) - cp(t), t E [0,T]. (13.2.8)
Proof -
For the completeness of the presentation we sketch a proof. One can assume that x = 0 and that t
cp(t) = c40) + f 0(r)dr, t E [0,T], where 0 E L1(0, T; E). Consequently
z(t) =
ft S(t - s)p(0)ds + f S(t - s)cp(0)ds +
ft
S(t - s) ( fS'(r)dr)
ds
f (jt S(t - s)b(r)ds) dr. t
For arbitrary r < t, f t S(t - s)O(r)ds E D(A) and A It t S(t - s)O(r)ds = (S(t - r) - I)0(r). r
Therefore
Az t = (5(t) - I cp(0) +
- r) - I t f(s(t
' r dr,
which is the required formula (13.2.8). Moreover z E C([0, T]; D(A)) and by direct differentiation t
z'(t) = S(t)y0(0) + f S(t - r),O(r)dr, t E [0,T]. The proof of the lemma is complete. The formula (13.2.2) can be extended by continuity to less regular functions u(.).
In particular assume that W(t), t > 0, is a cylindrical Wiener process defined in a probability space (St, F, IP) adapted to a filtration
Chapter 13
246
.Ft and with values in U. If Q is a nonnegative bounded operator on U such that the process
Z(t) = (A - A) f t S(t - s)DQ1/2dW(s), t > 0,
(13.2.9)
0
is a well defined H-valued process, then it will be accepted as a solution to the linear problem
aZ (t) = AZ(t), t > 0, rZ(t) = Q1/2 OW (t), t > 0,
(13.2.10)
It will be called an Ornstein-Uhlenbeck process in analogy to the classical noise case. A sufficient condition for (13.2.9) to be a well defined process is that t
I IIAS(r)DQ1/2IIL2(U;H)dr < +oo, t > 0,
(13.2.11)
where L2(U; H) stands for the space of Hilbert-Schmidt operators acting from U into H, see G. Da Prato and J. Zabczyk [44] and [45]. Let now F be a nonlinear, usually unbounded, operator on H. Ft-adapted H-valued and H-continuous process X is the mild An solution of the problem
jt-(t) = AX(t) + F(X(t)), t > 0, TX(t) = Q1/2
-W
(t), t > 0,
(13.2.12)
X(0) = x, if it is a solution of the following integral equation: t
X(t) = S(t)x + Z(t) + f S(t - s)F(X(s))ds, t > 0. In particular we have the following result.
(13.2.13)
Systems perturbed through the boundary
247
Proposition 13.2.3 Assume that the operators A and F satisfy Hypotheses 5.4 and 5.5. Then equation (13.2.13) has a unique Markovian generalized solution.
We finish this section by giving sufficient conditions for continuity of
the Ornstein-Uhlenbeck process Z(.) in some important subspaces of H. They will be used in § 13.4. To simplify our general considerations we assume from now on that the operator A is self-adjoint and that there exists a complete orthonormal basis of eigenvectors {gn} of A corresponding to the sequence {-An} 1 -oo of non positive eigenvalues. Let H2a = D((-A)a). If a stronger version of (13.2.13) holds, namely t
3 y> 0 f r--YIIAS(r)DQ1/2IIi2(U;H2a)dr < +oo, t> 0, (13.2.14) then the process Z has a version which is H2a-continuous. In particular we have the following result, see G. Da Prato and J. Zabczyk [45].
Proposition 13.2.4 If for some 'Y > 0 00
(13.2.15)
+00 n=1
then the process Z has an H2a-continuous version.
Proof - Note that
j
rIIAS(r)DQ1/2II(U;H2a)dr t
r--YII((-A)1+a)S(r)DQ1/2II2
L2 (U;H)
= J0
= Jo 'r-III _
Q112D*[((-A)l+a)S(r)]II
i2(U;H2a)dr
00
as+1IIQ1/2D*gn112 /' t
n-1
So the conclusion follows.
0
dr
rrye-2anrdr.
Chapter 13
248
Equations with Neumann boundary conditions
13.3
We will apply the general approach of §13.2 to problem (13.1.1) with
V(t) = Q112W(t), t > 0. For simplicity we set m = 0 and define
H = L2(G), U = L2(ro). Moreover the operator A is the Laplacian and r is the trace in F0 of the normal derivative operator aU and
D(A) = {x E H2(G) : a
0,
E F},
Ax=Ax. It is well known that A is a self-adjoint generator of a semigroup S(t), t > 0. Moreover the eigenvectors of A are of the form E G,
9nl,...,nd =
(13.3.1)
where nl,..., nd E N U {0} and go(d) _
1
2- cos ne,
,
E [0, 7r], n E N U {0},
(13.3.2)
and the corresponding eigenvalues are ^nl,...,nd =
-(ni + ... + nd), n1, ..., nd E N U {0}.
We will now find an explicit formula for the operator D. For this purpose it is convenient to consider elements u E U as functions in L2(F) equal to 0 outside of FO. We will solve (13.2.4) with A = 1. An arbitrary element u E U can be uniquely represented by the expansion with respect to the orthonormal and complete system {9n,,...,nd-1
:
nl, ..., nd_1 E N U {0}},
namely 00
U=
(u, 9n1 i...,nd_1)U 9n1,...,nd_1 nl,...,nd_1 =0
I
Systems perturbed through the boundary
249
and therefore the solution w = Du of (13.2.4) is of the form 00
w=
E
\u, gn1 i...,nd-I )U wn1,...,nd_1
nl ,...,nd_1=0
where wn1,...,nd-1 = Dgnl,...,rid_1, n1i ..., nd_1 E N U
{0}}.
By direct calculation we find that CC
[
[
wnl,...,nd_1 (S1, ..., Sd) = gn1,...,nd_1(S1, ..., bd-1 )hn1 .....nd-1 (Sd),
where hn1,...,nd-1 (ed) _
-
cosh[
+ ni +
+ na_l sink
[
+ n2d_1 ]
1 + ni
(e1, ,'d) E G and n1i...,nd_1 E N U {0}. It is clear that D is a bounded operator from U into H. By the very definition (D*9nl,...,nd)(S1I ..., Sd-1)
= gnl,...,nd-1 (b1, ..., ttbd-1)
[t
CC
10"
C
[
gnd(S )hnl,....nd-1 (S )dS
b(n d )
1 + ni + ... + n2 gnl,...,nd-1
d) E G, (13.3.3)
where
1 ifn=0, 2 if n E N.
Theorem 13.3.1 The Ornstein-Uhlenbeck process Z(t), t > 0, has a continuous version in the space H2a,2(G) provided there exists y > 0 such that
E J Q1/2gnl,...,nd-1 II nl,...,nd_I =0 L>
2
(\1 + n1 + ... + nd-1)
2a-{ ry- 1
< +00. (13.3.4)
,
Chapter 13
250
Proof- We can identify the spaces H",2 (G) with H2, = D((-A)"), a > 0. We will apply Proposition 13.2.4. Note that 1/2D* IIQ
gn1,...,ndll
2
-
b2(nd) 1/2 2 (1 + n2 + ... + n2d)2 IIQ gn1,...,nd_1 II '
nl,..., nd-1 E N U {0}. Consequently the sum 00
D* A2"+'r+l nl,...,nd Q1/2 E nl,...,nd-1=o II
gn1,...,nd
is finite if and only if 00
00
2
IIQ1/2 D*9n1..... ndl)
nl,...,nd-l =o
n=0
(n2 + ... + n2d-1 + n2)2a+'Y+l < +oo. (1 + n1 + ... + n2_1 + n2)2
However, there exist constants Cl > 0 and C2 > 0 such that for arbitrary a > 0 and 2a + < 1 2
00
2)+
C2(1 + a2)2a+'Y-2 < 1 ((1 + a2
)21
< Cl(1 + a2)2«+'Y-2,
n=O
and therefore the result follows.
Remark 13.3.2 If d = 1 then to = {0} and the condition (13.3.4) is automatically satisfied. We finish this section by giving conditions on the space homogeneous
noise implying continuity of the Ornstein-Uhlenbeck process in all LP ((0, ir)d) spaces. Thus we assume that the operator Q is of the form (13.1.2)-(13.1.3) with qo being the Fourier transform of the symmetric spectral density g, see j5.2: J d-1
eit(g(()d(, ( E R.
We will call g the spectral density of the noise process V. We need the following lemma.
Systems perturbed through the boundary
251
Lemma 13.3.3 The function TdT
hd,a(() =
(
11 (j=1
0.0
{(1
+ ni + ... + nd)2a-2
nl,...,nd=0
d
S E Rd
X 1 1+ ((j l+ nj)2 1+ ((jl- nj)211 is finite and continuous if and only if a < 1.
Proof - There exists a constant C > 0 such that d
1
(1+n2+...+nd)2a-2 0 that 00
E
nl,...,nd_1=0
Q II
12
gn1,...,nd-1 II
\I 2a+-y- 2 2((1 + n22 + ... + nd_1l 2
1 G)d-1
W
Ld1 g(() nl,...,nd-1=0 (1
n1 + ... + nd-1)
2
X Ignl,...,nd-1 (() d(
C1
f J
d-1
g(O [
(1 +
L00
n12
n1 ,...,nd-1=0
+ ... + nd-1) 2 2a+-Y-2
d
X
11 j=1
(J
d(. X
1
J
1
g(C)hd-1,a+2 (()d(
C1
Therefore the result follows.
Remark 13.3.5 Knowing the asymptotic property of hd_l,a+ry(C) as ISI -4 +oo one can replace (13.3.5) by more specific conditions. If for istance d = 2 then ((2 + n2)2a+2ry-1
h1,a+'Y C n=0
2
(n+ t) 2
< C(4(a+'Y),
for large ICI.
As a corollary we obtain sufficient conditions for continuity of Z in LP spaces.
Theorem 13.3.6 The Ornstein-Uhlenbeck process Z(t), t _> 0, has a continuous version in LP(G) provided that (13.3.5) and that one of the following conditions hold
W d = 1, 2, 3, p > 1, 4 < ca,
Chapter 13
254
(ii)d>4,1 0, has continuous versions in L2 (G). Consequently all the assumptions of Proposition 13.2.3 are satisfied and the proof of the theorem is complete.
In a similar way we can prove the following theorem on invariant measure.
Systems perturbed through the boundary
255
Theorem 13.4.2 In addition to the conditions of Theorem 13.4.1 assume that the function fo is decreasing and
m-IIf1IILip>W>0, Then the transition semigroup Pt, t > 0 corresponding to the solution of problem (13.1.1) has a unique invariant measure µ in L2(G). Moreover there exists c > 0 such that for any bounded Lipschitz function cp on H, all t > 0 and all x E H we have JH o(y)u(dy) < (c +
Proof - We apply Theorem 6.3.3 and argue in a similar way as in the proof of Theorem 12.3.3.
For other results on invariant measures for stochastic equations with boundary noise see B. Maslowski [118].
Chapter 14
Burgers equation This chapter is devoted to the Burgers equation on a finite interval, perturbed by the cylindrical Wiener process. Existence of a unique solution is established. It is also shown that there exists a unique invariant measure by proving a kind of boundedness in probability of some solutions and by establishing strong Feller and irreducibility properties of the corresponding transition semigroup.
14.1
Introduction
An important role in fluid dynamics is played by the following Burgers equation, see J.M. Burgers [15]. 2
[X2(t,)]
atX(t' ) = v C2X(t,e)+
,
(14.1.1)
2
with t > 0,
E R. It is known, however, that the equation is not a good model for turbulence. It does not display any chaotic phenomena; even when a force is added to the right hand side all solutions converge to a unique stationary solution as time goes to infinity. The situation is different when the force is random. Several authors have indeed suggested using the stochastic Burgers equation as a simple model for turbulence, D. H. Chambers et al. [21], H. Choi et al. [22], Dah-Teng Jeng [33].
The equation has also been proposed in M. Kardar, M. Parisi and J. C. Zhang [94] for studying the growing of a onedimensional 257
Chapter 14
258
interface. Let us recall their model. Let h(t, ) be the height of the interface at the point E R, then we have
a where
q2
02
2 F h(t,
h(t, E) = v a- h(t, e) +
W (t' f ),
+ 0
(14.1.2)
is the space-time white noise. Setting
X (t,) _
h(t,
we arrive, after differentiation with respect to , at the equation
aX(t,E) = v
X(t,6) +
aa
2
W(t, ), (14.1.3)
In § 14.2 we shall prove existence and uniqueness of a solution, following G. Da Prato, A. Debussche and It. Temam [36], of equation (14.1.3) supplemented by Dirichlet boundary conditions
X(t,0) = X(t, 1) = 0, t > 0,
(14.1.4)
and with the initial condition X (0, 6) = x( ),
E [0, 1].
(14.1.5)
Moreover in § 14.3 we prove that the corresponding transition semigroup is a strong Feller one and in §14.4 the existence and uniqueness of an invariant measure. Here we use methods in G. Da Prato and D. Ggtarek [38] (where a more general case with multiplicative noise is treated) and from Chapter 7. However, the proofs we present here are new. For a different approach on existence and uniqueness, based on the Feynman-Kac formula, see L. Bertini, N. Cancrini and G. JonaLasinio [8]. Equation (14.1.2) has also been studied in several space dimensions using the Hopf-Cole transform
h(t, f) = log
(.v(te))
,
which reduces equation (14.1.2) to heat equations with a random force. In this connection we quote H. Holden, T. Lindstrom , B.
Burgers equation
259
Oksendal, J. Uboe, & T. S. Zhang [87], A. Dermoune [54], Yu. Kifer
[97]. We mention finally the papers by L. Bertini and N. Cancrini [7], and G. Tessitore and J. Zabczyk [151], where equation (14.1.2) 82 W (t, ) is studied. with at W (t, ) replaced by In the sequel we will set v = A = 1 for simplicity. Then we write the problem in abstract form. We define
Au = u", V it E D(A) = H2(0,1) n Ho(0,1). A is a negative self-adjoint operator on L2(0, 1) and
D((-A)°) = H2-(0,1) if o E ]0, 1/4[. We denote by S(t), t > 0, the semigroup on L2(0, 1) generated by A, by {ek} the complete orthonormal system on L2(0,1) which diagonalizes A and by {Ak} the corresponding sequence of eigenvalues. We have
ek(e) =
sin
kE
k E N.
Setting X(t) = X(t, ), t > 0, we can write (14.1.3)-(14.1.5) as
dX(t)
AX(t) + 1 Dt(X2(t))J dt + dW(t), (14.1.6)
X(0) = x,
1
where W is defined by
W(t) = E Qheh,
(14.1.7)
h=1
and {Qh} is a sequence of mutually independent real Brownian motions in a fixed probability space (St, F, ]P) adapted to a filtration
Ft, t > 0.
Chapter 14
260
14.2
Existence of solutions
Here we are concerned with the problem
dX(t) _ [AX(t) + 1 De(X2(t))] dt + dW(t), (14.2.1)
X(0) = x, which we write in the mild form t
X(t) = S(t)x + 2 f S(t - s)D,(X2(s))ds + WA(t), where WA(t) is given by t
WA(t) = JO S (t - s)dW(s).
Now, setting Y(t) = X (t) - WA(t), t > 0, we obtain a deterministic equation, a.s. in 1,
Y(t) = S(t)x +
1f
S(t - s)D,[(Y(s) + WA(s))2] ds,
(14.2.2)
which can be considered as the mild form of the evolution equation
Y'(t) = AY(t) +
WA(t))21, t > 0,
Y(0) = X.
In order to give a precise meaning to equation (14.2.3), we introduce the mapping
F°(u)(t) = ft (-A)°S(t -
ds, t E [0,T],
(14.2.4)
0
for any o E [0,1[. If u E C([0, T]; Ho (0,1)) then F, (u) is certainly well defined and belongs to C([0, T]; L2(0,1)).
Lemma 14.2.1 The mapping FF defined by (14.2.4) can be extended to a bounded linear mapping from C([0, T]; L1(0,1)) into C([0, T]; H2o(0,1)), for arbitrary o E [0,14[.
Burgers equation
261
Proof - An essential tool for the proof is the following well known estimate:
I DeS(t)cpjj. < Clt-+
(14.2.5)
II(P112, cp E L2(0,1),
for some constant C1. Let us choose 0 E H2°(0,1) and consider the scalar product /
t
I = ((-A)°b, f S(t - s)Dgu(s)ds) , where u E C([0, T]; Ho (0,1)). Then
I=
=-
f
t
(S(t - s)(-A)°O, Dtu(s)) ds jt
s)(-A)°O), u(s)) ds.
Therefore III
Ilu(s)Illds.
J ot II
But
IIDe[S(r/2)(S(r/2)(-A)°V')]II.
< Cl (j) -3/4lIS(r/2)(-A)°V')II2 Cl
(2)-3/402 (2)-° II01l2
< C3r-(3/4+°)II0II2, r E [0, T],
where C2, C3 are constants. Consequently III < C3 J t(t
- s)-(3/4+°)Ilu(s)IIlds II'II2
(14.2.6)
Chapter 14
262
It follows from (14.2.6) that fo S(t - s)Dgu(s)ds E D((-A)") and I1(-A)O'
J t S(t - s)Dtu(s)dsll2
0, (14.2.10)
Y(0) = X.
1
As is easily seen equation (14.2.10) can be solved by a fixed point argument in C([0, T]; Ho(0,1)), so it has a classical solution. We find now an a priori estimate on the solution Y which will be independent of regularity assumptions on cp and x. Multiplying both sides of equation (14.2.10) by Y and integrating with respect to e in [0, 1] we get lY(t)12de 2 dt
101
f
=I
It follows, integrating by parts, that
2 dt
11 IY(t)I2d +
2
11 y2(t)d
_
-2 j(Y(t) +
Jol Y2(t)Yt(t)d - J ol cp(t)Y(t)YY(t)d -
2 101
Noting that f Y2(t)YY(t)de
3
[Y3(t)] 1
0,
Chapter 14
264
we get 2 dt IIY(t)112 + Joy Yg (t)d
II o(t)II.IIYY(t)II211Y(t)II2 + 2IIp(t)II00IIYe(t)112
0 let z/'R be a C' function on [0, +oo [ taking values in [0, 1] and such that
1 for r E [0, R], 0 for r E [R + 1, +oo[, and define a mollifier
MR(x) = x&(x), x E H.
(14.3.1)
Note that MR E Cy (H) and DXMR(x) = V(IIxII)I +
'OR(II xII )x
® x, x E H.
IIxil
Proposition 14.3.1 If MR is a mollifier given by (14.3.1) then the modified Burgers equation
dX =
!DC (MR(X))2] dt + dW(t), (14.3.2)
X(0)=x,
I
has a unique mild solution on a time interval [0,T]. Moreover the corresponding transition semigroup PR, t > 0 has the strong Feller property on H.
Chapter 14
266
Proof - Step 1-Existence and uniqueness The equation (14.3.2) can be rewritten in the form
Y(t) = S(t)x + 2 F(MR(Y + WA))2 (t), t E [0, T],
(14.3.3)
with X(t) = Y(t) + WA(t). The proof of existence and uniqueness of solutions to (14.3.3) is similar to that of Theorem 14.2.4 provided one can derive an apriori bound of the form (14.2.12). For this purpose we assume that Y satisfies the equation
Y'(t) = AY(t) +
2D (MR(Y + P))2 ,
t E [0, T], (14.3.4)
I with cP E C([O,T]; Ho(0,1)), x E D(A), and proceed as in step 2 of the proof of Theorem 14.2.4. In particular we obtain that 2 dt
IIY(t)IIZ +
-1
R(IIY(t) + sa(t)II) fo 1(Y(t) + V(t))2 YY(t)d 1
11 _ -'OR(IIY(t) + (P(t)II) 11 p(t)Y(t)YY(t)de + 2 P2(t)Yy(t)d6] 0 IIYY(t)II211Y(t)II2 + 2
IIYY(t)112.
Therefore the estimate (14.2.12) also holds in the modified case.
Step 2-Strong Feller property We proceed as in the proof of Theorem 7.1.1. We set
(Xx(t, x), h) = r7h(t, x), t > 0, x E H. It is easy to check that the solution X (t, x) of (14.3.2) is differentiable
with respect to x and that rjh(t, x) is a mild solution of the problem ii' = i?4 + Dg (MR(X)DxMR(X )I) , t E [0,T], (14.3.5)
,q(O) = h, P a.s.
Burgers equation
267
Note that by Holder's inequality, for arbitary x, z E H II MR(x)DSMR(x)zIIl < II MR(x)I12I1 D.MR(x)zII2 _< C1142,
(14.3.6) where C is a constant depending on the choice of the function MGR.
Moreover equation (14.3.5) can be rewritten as
rl(t) = S(t)h + 2F (MR(x)DxMR(x)11) (t), t E [0, T]. Taking into account Lemma 14.2.1 and estimate (14.3.6) we conclude that the solution to (14.3.5) exists in C([0,TJ;H) provided T is sufficiently small. Moreover, for a non-random constant CT, sup II /(t,x)II 5 CTIIh1I2, h,x E H. tE [O,T]
This implies, as in the proof of Theorem 7.1.1, the required strong Feller property on a small time interval, and then, by the semigroup property, on [0, +oo[.
Theorem 14.3.2 The transition semigroup Pt, t _> 0, corresponding to solutions of equation (14.2.1), has the strong Feller property on H.
Proof - For any R > 0 let PtR, t > 0, be the transition semigroup corresponding to solutions XR of equation (14.3.2). Let TR
= inf{t > 0 : IIXR(t,x)II > R}.
It is clear that
X(t,x) = XR(t,x) for all t < TR, x E H. Let now cP E B6(H), then PRV(x)
- Ptw(x)I
=
IE ((P(XR(t, x))
0 and t > 0 it is enough to show that for arbitrary M > 0 and t > 0 lim sup iP(TR 0, on Bb(L2(0,1)) which is a Feller one (because the solutions to (14.4.1) depend continuously on the initial conditions). Therefore, by the general theory, see Chapter 6, to show existence of an invariant measure for (14.4.1) it is enough to prove that the family of measures 1 JT
is tight on L2(0,1). For this we need a lemma.
Lemma 14.4.1 For any e > 0 there exists ME > 0 such that, for all T > 0, T
I
TP
ME) ds < F.
(14.4.2)
Proof - Without any loss of generality we can assume that Y is a strict solution of (14.4.1). Multiplying both sides of this equation by Y(t) and integrating by parts in [0, 1], we find
I
2dt
d+
2
since
d
YWA
0 such that IIYII4 5 C
IIYIIHI/4(0,1)-
It follows, from a well known interpolatory inequality, see e.g. J. L. Lions and E. Magenes [107], that f1
IIYII4 < C1 IIYI12/4 IIYIIH (O,l) =
C1
Y2de]1/s
.
IIYII2/4 [Jo
From (14.4.3) it follows that
+
2 dt IIYII2
I 1/2
+ 2 IIN'iII4 [j1 Y d
]
+ « IIWA-11211Y112.
(14.4.4)
Now recalling the inequality
x'
xy
and setting p = such that
p
y9
1
+ q,
1
p+q
=1, x,y?0,
3q = 5, we see that there exists a constant C2 > 0 5/8 IIYII2/4
C1 IIWA IJ4
1f l y2 d ]
< C211W 114/3IIYII2+
4
101
Y2 d
.
(14.4.5) Moreover
[f1]h/2
2 IIWA112 By substituting
2dtIIYI12 + 2
< 2 IIWAoIJ4 +
(14.4.5))
4
jy2d.
(14.4.6)
and (14.4.6) in (14.4.4) we find
y2
M} dt
By multiplying both sides of (14.4.8) by 1
IIY(t)112 X{Ily(t)II2>M}'
we get 2
2
X{Ily(t)II2>M} < 2C2 2
M IIWA(t)Ii4 + 8 M IIWA(t)II2
Integrating from 0 to t and taking expectation, we find
E(((t) - ((0)) +
2
f
t
p (IlY(S)112 >_
M) ds
t
< 2C2
It [EllW2(s)114/3 + MEIIWA(s)114 +
2E11WA(s)112 ds.
But ((t) - ((0) = ((t) - M > 0, and so, taking into account (14.4.9), t
It
(IlY(s)1122 ? M) ds
f
t>EIIN'A(s)114ds
0, corresponding to solutions of equation (14.2.1).
Proof - We fix a > 0 such that the statement of Lemma 14.4.1 holds. Note that for arbitrary a > 0 the embedding
D((-A)°) C L2(0,1) is compact and therefore to prove the required tightness property it is enough to show that
For any e > 0 there exists M > 0 such that for all T > 1
IT T
P (II(-A)°X(t)112 > M) dt < E.
(14.4.10)
From now on we fix not only a but also a E ]0,1/4[. Repeating the calculations which led to the estimate (14.4.9) we easily deduce that K = supEIt(-A)°WA(t)I12 < +oo. t>1
Therefore by Chebyshev's inequality
IT = T
T
P (11(-A)' W(t)I12 > M) dt
IT TM Jl 1
C
(II(-A)°u'A(t)112) dt
-71) KM
So IT can be made small uniformly in T > 1 provided M is chosen sufficiently large. This way, since X = Y+WA, to show that (14.4.10) holds it is enough to prove it with the process X replaced by Y. This will be done in the same spirit as in the proof of Theorem 6.1.2 exploiting the regularizing effect of (14.1.3).
Chapter 14
274
For the mild solution Y of (14.4.1) and arbitrary t > 0 we have
that
(-A)O'Y(t + 1) = (-A)°S(1)Y(t) f t+1
+ (-A)° J
it
+ a(-A)" J
it
S(t + 1 - r)D (Y(r) + WA(r))2 dr
t+1
S(t + 1 - r)WA(r)dr.
Taking into account Lemma 14.4.1 and Remark 14.2.3 we have (for any p > 1 such that p + or < 1) II (-A)°Y(t + 1)112
< M1IIY(t)112 + M2 sup IIY(t +r)1122 o 6M )
< I21(t) + I22(t)-
In addition 121(t) < P
(1W2e2C2ftt+1
I21(t) S ? (M2e2C2
+p
M/6)+r (IIYtII2
>
ftt+' Ilu'A(s)II413ds > M16) g
(f`+1
[411 u'A(S)114
+
2Z
llw
(S)I12,
ds >
M6)
6)
,
Chapter 14
276
Finally we see that
T f?(IR-AfY(t+ 1)112
+T
IT (e2f"' II'(s)II3ds > P
'T I
/)
>M/6\
I dt
P (IIY(t)112
+T I P
(fI+I
8a2II11'A(s)I121 1411WA(s)I14 +
(It+,
1
+T J T P
dT
> M) dt < T j?(MiIIY(t)II2> M) dt
(M3
i
IIWA(s)IIPds)
P
>
M
ds > M/6 I dt dt.
The first two terms on the right hand side can be made arbitrarily small uniformly in T if M is chosen sufficiently large (by Lemma 14.4.1). The same is true by Chebyshev's inequality with the remaining terms, because, as is easy to check, Sups>0E[IIWA(s)II4/3+ IIWA(s)II4 + IIWA(s) I2 + IIWA(S)II21 < +OC,
compare the derivation of the estimate (14.4.9).
14.4.2
Uniqueness
Finally we prove that the invariant measure for the equation (14.2.1) is unique. According to the general theory it is enough, see Chapter 7, to show that the corresponding transition semigroup is irreducible and has the strong Feller property. Since the latter was proved in Theorem 14.3.2, it remains to show irreducibility.
Proposition 14.4.3 The transition semigroup Pt, t
_>
0, on the
space Bb(L2(0,1)), corresponding to (14.2.1) is irreducible.
Proof-
Let us fix T > 0,a E L2(0,1) and b E Ho (0, 1). We show first that there exists v E L2(0, T; L2(0,1)) such that for the solution
Burgers equation
277
y(t), t E [0, T] of the problem yt(t, e)
= D2Y(t, ) + 2 Dg (y2 (t, e)) + v(t, t > 0,
y(t,0)
),
E [0,1],
(14.4.11)
= y(t,1) = 0, t > 0,
y(0, ) =
a(e), C E [0,1],
one has y(T, e) = b(e),
E [0,1].
Assume in addition that a
E H01(0, 1). Then there exists u E L2(0,T; L2(0,1)) such that for the solution z(t), t E [0,T],
of the linear problem
zt(t, ) = D2z(t, e) + u(t, e), t > 0, z(t, 0)
= z(t,1) = 0, t > 0,
z(0, e)
= a(e), e E [0,1],
E [0,1], (14.4.12)
one has z(T, e) = b(e), e E [0,1],
and one can easily check that z E C([0,T]; H(0,1)). Define
v(t, e) _ - 2 D2 z(t, ) + u(t, ), t > 0,
E [0,1].
Then v E L2(0, T; L2(0,1)) and by direct substitution one sees that z is the solution to problem (14.4.11). Now, from the well known identity for solutions of (14.4.11), 2 dt Ily(t)1122 + II
2
(y, v),
it follows that, if v = 0, then for almost all t E [0,T], y(t) E Ho(0,1). It is therefore enough to define v(s) = 0 in [0, ] where t E ]0, T[ is
Chapter 14
278
a moment such that y(t) E Ho(0,1) and for the remaining interval Ft, T] to use the first part of the proof. To prove the proposition we will estimate the L2 distance between
the solution X to equation (14.2.1), with xo = a, and the function y. Using our previous notation,
X (t) - y(t) = 2
F(X2
- y2)(t) + WA(t) - VA(t),
where
VA(t) = JO S (t - s)v(s)ds, t E [0,T]. t
From the proof of Lemma 14.2.1, see formula (14.2.6), it follows that IIF(X2
- y2)(t)112 < C ft(t - s) 411X2(s) - y2(s)Ilids. 0
But, for arbitrary s E [0,T],
IIX2(S)
- Y2(S)II,
0, represents external force acting on the fluid. 281
Chapter 15
282
In this chapter we first prove the existence of a solution to the stochastic Navier-Stokes equation on an open subset of R2. We use a simple and new fixed point argument instead of the more common Galerkin-Faedo approximation scheme. Then we show the existence of an invariant measure by adopting a method of F. Flandoli [66]. For the proof of uniqueness we refer to F. Flandoli and B. Maslowski [67]. These results can be generalized to the Benard problem, that is a Navier-Stokes equation perturbed by white noise coupled with the heat equation, see B. Ferrario [65]. We shall denote by V the set of all mappings u : D -> R2 of C°° class and having compact support in D, such that
div u(E) = Tr
E D.
0,
For all u = (ul, u2), v = (vl, v2) E V we set
lu(f)12 = ui(e) + u2O,
(u(0, v(0) =
E D,
u2(e)1)2(b), S E D.
For any p > 2, we shall denote by LP the closure of V with respect to the norm HUG = (JD
Iu(S(`)I2det)
1/p
and by V the closure of V with respect to the norm Ilully =
IDH
We
Obviously H is a closed subspace of L2(D) x L2(D) = L2(D;R2),
we shall denote by P the orthogonal projection of L2(D) x L2(D) onto H. We introduce finally the linear operator D(A) = [H2(D) x H2(D)] n v, (15.1.2)
Au = PAu, forall u E D(A). A is a self-adjoint strictly negative operator on H, and V = D((-A)1V2), see e.g. R. Temam [150]. We shall denote by {ek} an orthonormal
Navier-Stokes equations
283
complete system of eigenvectors of A and by {Ak} a sequence of pos-
itive numbers such that Aek = -Akek, k E N. We assume that {Ak} is not decreasing. A central role in our considerations will be played bythe space E = L4(O,T; f,4).
Since L4 is the closure of V with respect to II(xl, x2)114 =
Xl(e)14
(ID
1/4,
+ Ix2(e)I4) de) 1
(I
(x1, x2) E V,
we have 1/4
[IT IIUIIE
r JD
=
(Iul(t,e)I4 +
dd dt
1/4
(fT II u(t)II4dt) We will also need the following inclusion result.
Proposition 15.1.1 For any T > 0 we have L°°(O, T; H) n L2(0, T; V) C E,
and there exists a constant K, independent of T > 0, such that IIIIIIE C K(IIUIILoo(O,T;H) + IIUIIL2(0,T;V)), U E E.
(15.1.3)
Proof - We recall that due to the Sobolev embedding theorem H1/2(D) C L4(D)
and there exists Cl > 0 such that IIVIIL4(D)
C1IIvIIH1/2(D), V E H1/2(D).
(15.1.4)
Chapter 15
284
Define V1/2
= (H1/2(D) x H1/2(D)) n V.
Then V1/2 = D((-A)114). By a well known interpolatory inequality, see J. L. Lions and E. Magenes [107, pag.22], we have for some constant C2 > 0 and all t E [0, T] IIu(t)IIV1,2 s C2IIu(t)IIH2 Ilu(t)IIV2.
It therefore follows from (15.1.3) that
Ilu(t)114 < 2C Ilu(t)II i,2 < 2C C Ilu(t)IIH IIu(t)II i. Now integrating this inequality in [0,T] we easily get (15.1.4).
15.2
Local existence and uniqueness results
By applying the projection operator P to the first equation in (15.1.1)
and setting X = PZ we find the problem
dX(t) = [AX(t) + D4(X(t)) X(t)] dt + dW(t), T > 0, X (O) = x, (15.2.1)
where T > 0 is fixed and W is defined by
W(t) _
Qh/3hgh.
(15.2.2)
h=1
Here {/3h} is a sequence of mutually independent real Brownian mo-
tions in a fixed probability space (St,
F) adapted to a filtration
Moreover {Oh} is a sequence of positive numbers and {9h} is an orthonormal basis in H. We reduce problem (15.2.1) as usual to a deterministic one by setting Y(t) = X(t) - WA(t), t E [0,T], IP' a.s.
Navier-Stokes equations
285
We find
Y'(t) = AY(t) + Dt(Y(t) + WA(t)) (Y(t) + WA(t)) + f(t), t > 0, Y(0) = x, (15.2.3)
where WA(t) is the stochastic convolution
WA(t) = J S(t - s)dW(s), 0t
(15.2.4)
and S(t) = etA, t > 0. We now make the following basic assumption
Hypothesis 15.1 Process
has a version in L4 with trajecto-
ries integrable in the fourth power. Notice that an equivalent formulation of the above hypothesis is that WA(.) has an E-valued version. We will interpret equation (15.2.3) as an integral equation
Y(t) = S(t)x +
J0
t
S(t - s)De(Y(s) + WA(s)) . (Y(s) + WA(s))ds
= S(t)x + F(Y + WA)(t), t E [0, T], where F : E --+ E is a continuous extension of the operator
Fo:CI([0,T];V)-+E given by the formula (Fou)(t) =
J
t
S(t - s)(Gou)(s)ds, t E [0,T],
where
Go : Cl([O,T];V) -+ E, (Gou)(t) = Dtu(t) u(t), t E [0,T]. To show that the operators G and F are well defined it is useful to introduce a trilinear form on V x V x V by setting b(u, v, z)
=
(Dov(e) . u(e), z(e))d JD 2
i9Vh
Uk Zh d h,k-1 ID - k
Chapter 15
286
By the Holder inequality we have Ib(u,v,z)I S 41IvIIvIluII41IzII4,
(15.2.5)
so, by the inclusion (15.1.3), the definition of b is meaningful. We now list some useful properties of b.
Lemma 15.2.1 The following identities hold: b(u, v, z) = -b(u, z, v), V u, v, z E V,
(15.2.6)
b(u, v, v) = 0, V U, v E V.
(15.2.7)
and
Proof of (15.2.6) - We have 2
b(u, v, z)
=
OVh
h,k-1 ID 09G
Uk Zh d
2
auk
h ,k-1
dC 1f V h a-k zh D
2
OA vh uk E ak h,k-1 JD I/r
dC
div u (v, z) d4 - b(u, z, v),
and the conclusion follows since div u = 0.
Navier-Stokes equations
287
Proof of (15.2.7) - We have b(u, v, v)
=
l9yh ukVhd E h,k=1 ID ak
=
1.
2
a
1/
2 h,k=1 ID uk '9 k \vh) 1
2
8uk
2h,k=1 Dak
2
vh
the duality mapping between L2(O,T; V') and L2(O,T;V),
IT >
=
jT
D
(D
u(t, ) . u(t, ), cp(t, )) dt dd
b(u(t), u(t), cp(t))dt = - 1T b(u(t), W(t), u(t))dt.
Chapter 15
288
In addition if u,v E C2([O,T];V) and So E L2(O,T;V) we have >
jTJ(Dut,f =
u(t, ) - Dfv(t,
v(t, e)) 4odt d
T
_ - f [b(u(t), u(t), p(t)) - b(v(t), v(t), p(t))Jdt
Tbut
0
t u(t))
b(v(t),
t v(t))]dt
0T
= J [b(u(t), cp(t), -u(t)) + b(v(t), cp(t), v(t))]dt
_
I
[b(u(t), o(t), v(t) - u(t)) + b(v(t) - u(t), (a(t), v(t))]dt.
It follows from (15.2.5) that > 1< 4 f T [IIu(t) - v(t)II4 IIu(t)II4 IIW(t)IIV
+II u(t) - V(t)114 IIv(t)II4 II (t)IIVI dt T
< 4 (10 Iiu(t) - v(t)II4dt)
(fT
1/4
l1/4
(IT
(Ilu(t)II + Ilv(t)II4dt) I
II (t)II vdt)
< 4IIu - VII E (IIUIIE + II'IIE) IIWIIL2(o,T;V) .
Consequently IIGou - GovII L2(o,T;V) 0, independent of T > 0, such that IIF(u) - F(v)II E < M (IIuIIE + IIvIIE) In - vIIE, u, v E E.
Proof - Using the notation of the previous lemma we define the extension F by the formula
F(u)(t) = y(t,G(u)), t E [0,T], u E E. By Lemma 15.2.2 F E L°°(0,T;H)nL2(0,T;V). Moreover, for u,v E E,
IIF(u) - F(v)IIE
We now continue the proof of the theorem. Define E = E, a = + F(4p),
.1(u) = F(u + p) - F(4o), U E E.
Navier-Stokes equations
291
Then .F(O) = 0 and for u1, u2 E E and a = 44 IIJ'(ul) - F(u2)IIE < 2MaI ul
- U211 0 such that in the space E L4(O,T; L4) II& + F(cp)II E < 11011E + IIF(c)IIE 1/4
0, y is a mild solution of the equation
dt
Ay(t) + Dt(y(t) + co(t)) . (y(t) + cp(t)) + acp(t),
(15.3.8)
Navier-Stokes equations
295
then for arbitrary E > 0 d II y(t) II H + II y(t)II V
- 0. 2
(15.3.9)
Existence of an invariant measure
15.4
We pass now to the existence of an invariant measure for equation (15.2.1) under the condition
Hypothesis 15.2 Process WA(-) has a continuous version in
D((-
A)1
+26
) for some 9 E ]0, 2 [.
We remark that since
D((-A)') C D((-A) 4) C L4, Hypothesis15.2 is stronger than Hypothesis 15.1. The main result of this section is the following theorem. We will adapt, following F. Flandoli, see [66], the technique which proved to be useful for dissipative systems in §6.3. For different ways of investigating asymptotic properties of the stochastic Navier-Stokes equation, we refer to A. B. Cruzeiro [31] and to S. Albeverio and A. B. Cruzeiro [3].
Theorem 15.4.1 Under Hypothesis 15.2 there exists at least one invariant measure for equation (15.2.1).
Let V(t), t > 0, be a Wiener process independent of V but having the same law as W(t), t > 0. We denote by W a Wiener process defined on the whole real line by the formula
W(t), if t > 0,
W(t) =
Chapter 15
296
and by Ft the filtration
Tt=a(W(s), s s, is the unique solution of the stochastic Navier-Stokes equation
dX(t) = AX(t) + D£X(t) X(t) + dW(t), t > s,
}
X(s) = 0.
Let moreover Z(t), t E R, be a solution of the linear equation,
D((A)_ 1 420
(15.4.1)
)-continuous, stationary
dZ(t) = AZ(t) + dW(t), t E R.
Since the semigroup S(t), t > 0, is exponentially stable on H we have
Z(t) =
_It
S(t - s)dW(s), t E R.
00
However, the explicit form of Z will not be needed in the proof.
Remark 15.4.2 Since
D((-A)
i+ze4
) C D((-A)°),
Hypothesis 15.2 implies that the process Z is also D((-A)°)-continuous.
Note that the space D((-A)°) is compactly embedded into H. Consequently if one proves that
the process X(t,0), t > 1, is bounded in probability as a process with values on D((-A)°), (15.4.2)
one gets immediately that the laws L(X(t, 0)), t > 1, are tight on H. This is certainly sufficient for the existence of an invariant measure. The proof of Theorem 15.4.1 will consist of two steps.
Step 1- For any a > 0 denote by Z. the stationary solution of the equation
dZa = (A - a)Zadt + dW(t).
Navier-Stokes equations Since
297
Za(t) = Z(t) +
-a
ft
e(A-a)(t-s)(Za(s)
- Z(s))
e(A-a)(t-o)Z(o)d, t > s,
one can assume, see Remark 15.4.2, that continuous process. Moreover let
is also a D((-A)')-
Ya(t, S) = X(t, S) - Za(t), t > S.
Then
Yc(t) = Ya(t, S), t > S, is the mild solution of the equation
d Y. (t)
= AYa(t) + Dt(Ya(t) + Z. (t)) . (Y. (t) + Za(t))
+ aZa(t), t > s,
= -Za(s) Y(s) The main aim of Step 1 is to show the following proposition.
)
Proposition 15.4.3 There exist a > 0 and a random variable such that P-a.s. IIYa(t,s)II2