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0 and £ e Rd, we have
where m denotes Wiener measure on the space Ct of continuous paths x such that x(0) = 0. The left-hand side of (1.2.5) yields the unique solution, u(t, •) = e~'H(p, at time t > 0 of the heat (or diffusion) equation
with initial condition u(0, •) = p. Here, as before, H = H0 + V, with HO = -1A. Note, however, that we now use the heat semigroup e-tH to represent the solution to the heat equation (1.2.6) whereas we have used earlier the unitary group e~"H to represent the solution to the Schrodinger equation (1.2.2). The "Feynman-Kac formula" (1.2.5) has been extremely useful for a variety of purposes, both in mathematics and in physics (see Section 7.6 for a brief discussion of this along with some references), but it does not by itself resolve the problem of making sense of the Feynman integral since the change from t to —it takes us from quantum theory and the Schrodinger equation to the heat equation. The Feynman-Kac formula does, however, suggest an approach to the Feynman integral. Start with imaginary time and the theoretically powerful Wiener integral and define the Feynman integral by analytically continuing to real time. Indeed, operator-valued analytic continuation in time is another of the approaches to the Feynman integral which will be discussed in detail in this book. These results on the analytic-in-time Feynman integral (at the level of generality found here) are due to Johnson [Jo6] and are the subject of Sections 13.2 and 13.3. We should mention that what is imaginary time from the point of view of quantum theory is real time from the perspective of the heat equation. We shall adopt the latter point of view in Chapter 13 (Sections 13.2, 13.3 and 13.7) and analytically continue from real time to purely imaginary time—going in the process from the Wiener integral to the Feynman integral. We remark that Section 13.7 gives a brief discussion of an extension (see [AIJoMa]) of the analytic-in-time operator-valued Feynman integral which is based on the theory of "additive functionals of Brownian motion" (see [Fuk, FukOT]) and Feynman-Kac formulas in which, for example, the potential V can be replaced by a suitable measure on Rd. The last of the approaches to the Feynman integral which will be treated in detail in this book is operator-valued analytic continuation in mass. Again, one starts with the Wiener integral but this time, the analytic continuation is in a mass parameter (or alternately, in a variance parameter). The connection between Feynman's ideas and the approach to the Feynman integral via operator-valued analytic continuation in mass is discussed in an informal way in Section 7.6, with the approach via the Trotter product formula serving to link the two.
12
INTRODUCTION
The precise discussion of the analytic-in-mass operator-valued Feynman integral is given in Section 13.5. The crucial starting point for this work is Nelson's second approach developed in [Nel]. An earlier paper by Cameron ([Cal], 1960) used scalarvalued analytic continuation in mass; the key contribution of [Cal] was the proof that there is no countably additive "Wiener measure" with a complex variance parameter (see Theorem 4.6.1). This result corrected an error in [GelYag], an interesting and even earlier paper which used analytic continuation. Various extensions of Nelson's results are given in Sections 13.5 and 13.6. Among them, the reader will find hybrids which combine a suitable product formula with analytic continuation in mass. A comparison of the resulting analytic in mass Feynman integrals within their common domain of validity is provided towards the end of Section 13.6. We remind the reader that the analytic-in-mass operator-valued Feynman integral will also be used in Chapters 15-18. Unlike the approaches in Chapter 13 via analytic continuation in mass, this Feynman integral exists for every (rather than Lebesgue almost every) value of the mass parameter. The class of functionals treated in Chapters 15-18 is, in some respects, much larger than in Chapter 13. However, in Chapters 15-18, no attempt is made to deal with potential functions with strong spatial singularities. There are four different versions of the analytic-in-mass Feynman integral discussed in this book, as was alluded to above; in addition, three other approaches to the Feynman integral have already been discussed in this introduction. In the next two paragraphs, we indicate briefly what these are and where they are to be found. The approaches to the Feynman integral that are discussed at any length in this book are all operator-valued. (Recall that we are not taking Chapter 20 into account in our present discussion.) Two of the analytic-in-mass approaches start from the Wiener integral when the mass parameter is real. One of these is discussed in the first part of Section 13.5; the other, which has quite different features, is defined in Section 15.2 and used throughout Chapters 15-18. The last two begin with product formulas for semigroups (in Section 13.5) and resolvents (in Section 13.6) to yield analytic-in-mass versions of the Feynman integral via TPF ([Kat7, BivPi]) and of the modified Feynman integral [BivLa], respectively. The Feynman integral defined via the Trotter product formula for unitary groups is discussed in Section 11.2 and the modified Feynman integral (defined via a product formula for imaginary resolvents established in Section 11.3) is treated in Sections 11.411.6. Finally, the analytic-in-time Feynman integral appears in Sections 13.2 and 13.3, with an extension given in Section 13.7. The role of operator theory As mentioned above, the approaches to the Feynman integral that will be discussed in detail in this book are all operator-valued. Further, there is always at least one unbounded operator involved; much of the time, it is H0 = —1A, the free Hamiltonian, although various physically meaningful substitutes for HO are allowed in Sections 11.4, 11.6, and Sections 13.5-13.6, and more abstract generators are considered in Chapters 11 and 19. In Sections 11.2-11.5, Chapter 12, Sections 13.2-13.4 and 13.7, the theory of (not necessarily bounded) self-adjoint operators and functions of them is sufficient for
RECURRING THEMES
13
our needs. These needs include various forms of the spectral theorem for unbounded self-adjoint operators as well as basic results about unbounded quadratic forms and form sums of operators. This background material is provided in Chapter 10 which is titled "Unbounded self-adjoint operators and quadratic forms". (See also Section 9.6 for introductory material on unbounded self-adjoint operators and the associated semigroups.) The spectral theorem enables us to define the functions e-itH (the unitary group) and, if the spectrum of the self-adjoint operator H is bounded from below, the (self-adjoint) semigroup e-tH. For us, in most applications, H is the Hamiltonian (or energy operator), a suitable self-adjoint extension of H0 + V, where V is the potential. (More specifically, in Section 11.2, H is the unique self-adjoint extension of H0 + V, and, more generally, it is the form sum of HO and V in Sections 11.3-11.5, Chapter 12, Sections 13.2-13.3 and 13.7.) Self-adjoint operators—and the associated unitary groups or self-adjoint semigroups—are not adequate for everything that we will do. Strongly continuous (or (C0)) semigroups of operators will be discussed in Chapter 9 (and in the brief and informal chapter that precedes it). Such semigroups (not necessarily associated with self-adjoint operators) will be used in Sections 11.1, 11.6, 13.5, 13.6, parts of Chapter 14 and throughout Chapter 19. They will also frequently be present in Chapters 15-18 but will be used in a more straightforward way there. Connections between the Feynman-Kac and Trotter product formulas The Feynman-Kac and Trotter product formulas have already been discussed above, but there are additional places in the book where these related formulas or variations of them appear. The Trotter product formula is the main tool in the crucial first step of the proof of the Feynman-Kac formula in Chapter 12. A variation of the FeynmanKac formula, the "Feynman-Kac formula with a Lebesgue-Stieltjes measure", is—along with its connection with Feynman's operational calculus—the topic of Chapter 17, which describes part of the work in [La14-18]. A related product integral, a relative of the Trotter product formula, is discussed in Section 17.6 [Lal8,16]. Example 16.2.7 (in conjunction with Example 15.5.5) looks at the relationship between the Trotter product formula and the Feynman-Kac formula from the point of view of weak (or vague) convergence of measures. This broad perspective is informative even though the results are far less general than those proved in Chapters 11 and 12. A version of the Feynman-Kac formula which substantially extends the one in Chapter 12 is discussed briefly in Section 13.7. There, for example, the potential energy function can be replaced by certain measures (in the space rather than in the time variable, as in Chapter 17) which are singular with respect to Lebesgue measure. Finally, a Feynman-Kac formula for certain complex potentials is contained in the work presented in Sections 13.5 and 13.6. Evolution equations A fundamental concept of quantum mechanics is a quantity called the propagator, and the standard way of finding it (in the non-relativistic case) is by solving the Schrodinger equation. Feynman found another way based on what became known as the Feynman path integral or "the sum over histories" ... Mark Kac, 1984 [Kac5, p. 116]
14
INTRODUCTION
The evolution of physical systems concerns us throughout this book, so it is not surprising that the subject of evolution equations is another recurring theme. Our point of view (following Feynman) is not, however, the usual one. Typically, the evolution equation comes first and is regarded as the model for the physical system. One then looks for a method to solve the evolution equation and the solution gives the evolving state. Our deviation from this point of view is perhaps seen most clearly in Chapter 19. The idea there is: Given the forces involved in the problem, write down and then "disentangle" the exponential of a sum of integrals (from, say, 0 to t) of associated time-ordered operators (see (19.4.8)). The resulting time-ordered perturbation series (see (19.3.14)) should give the evolution of the physical system. Of course, it is of mathematical and physical interest to know if this series solves some related evolution equation. Theorem 19.5.1 shows that this is so under a quite general set of assumptions. As remarked earlier in this introduction, the approach to quantum dynamics provided by "the" Feynman path integral differs in several ways from the standard Hamiltonian approach. The point we wish to make here is that the path integral itself should give the evolving state. No evolution equation is needed ahead of time. Of course, it is of interest to know conditions under which the evolving state given by the Feynman integral satisfies the Schrodinger equation or some variation of it. The different specific approaches to the Feynman integral discussed in this book have differing relationships with the standard Hamiltonian approach to quantum dynamics. Our first comments along these lines pertain to Chapter 17. Recall that in Chapters 15-18, the potentials can be time-dependent and complex-valued but are not allowed to have strong singularities in the space variables. If we take the appropriate Wiener integral involving the usual Feynman-Kac functional e x . p { F 0 ( x ) } , where F0,i is given by (1.1.4) and / is Lebesgue measure on the time interval (0, t), we obtain a function of time and space which describes the evolution of a distribution of heat. By analytically continuing in mass (and making an adjustment in the potential), we arrive at a function giving a quantum evolution. These time evolutions are solutions to the heat and Schrodinger equations, respectively. In Chapter 17, we replace the Feynman-Kac functional exp{ F0,l } by the Feynman-Kac functional exp{F0,n} (where F0,n is given by (1.1.4) and n is a Lebesgue-Stieltjes measure) and follow the procedure above. We show first that the resulting evolutions involve an interesting variety of discrete and continuous phenomena and then also that they are solutions to correspondingly adjusted versions of heat and Schrodinger equations which are quite different from the usual ones (see especially Sections 17.2 and 17.6). Even though Feynman's approach to quantum dynamics does not depend a priori on the usual one, the method of proof for three of the specific approaches discussed in this book, the Feynman integral via the Trotter product formula (Section 11.2), the modified Feynman integral (Sections 11.3 and 11.4), and the operator-valued analyticin-time Feynman integral (Sections 13.3 and 13.7), not only depend heavily on operatortheoretic results but also on the existence of the unitary group as established in the standard Hamiltonian approach. [In the case of the modified Feynman integral with complex (rather than real) potential studied in Section 11.6 ([BivLa]), the Schrodinger
RECURRING THEMES
15
operator must be defined appropriately and the associated time evolution is dissipative but in general, not unitary.] The situation is quite different for the analytic-in-mass operator-valued Feynman integral, whether you begin on the real line with a Wiener integral (Section 13.5) or with product formulas (Section 13.6 and the last part of Section 13.5). Although operator techniques are still heavily involved, they are not the ones based on self-adjointness that are used commonly in quantum mechanics. Moreover, knowledge of the existence of the unitary group from the usual approach to quantum dynamics is not needed in the proof. In fact, for extremely singular potentials (see Examples 13.6.13 and 13.6.18), the analytic-in-mass operator-valued Feynman integral exists but the Hamiltonian approach does not, at least not in an unambiguous way. Functions of noncommuting operators The formation of functions of noncommuting operators is a theme which is implicit in the title of this book and which is of direct concern to us throughout Chapters 1419. Although it is less obvious, the same subject is also involved in Chapters 6-13. For example, if A and B are commuting self-adjoint operators, there is no need for the Trotter product formula (1.2.1); we simply have e - i t ( A + B ) = e - i t A e - i t B . The Trotter product formula has sometimes been referred to as the noncommutative exponential law. (In light of our later work, especially in Chapters 17 and 19, it would be more accurate to describe it as an especially important example but just one of many noncommutative exponential laws.) The spirit of the theory of semigroups of operators is that it is the theory of forming the "exponential function" of operators. In practice for us (and in general), the operator to be "exponentiated" is often of the form A + B, where A and B do not commute. The Feynman-Kac formula expresses the heat semigroup e~'H = e~' 0}, scale-invariant measurability [JoSk7] and the refined equivalence classes of functions that are needed for a careful discussion of the Feynman integral obtained via analytic continuation in mass. This definition of the Feynman integral will concern us in Section 13.5 (the second approach in [Nel]) as well as throughout Chapters 15-18. Measures on subintervals of K (Lebesgue-Stieltjes measures) are used systematically throughout Chapters 14—19 in connection with Feynman's operational calculus. They serve not only to assign weights but also to time-order the integrands which are usually (perhaps after some preliminary steps) products of noncommuting operators. The measures give directions for "disentangling", and a different set of measures can yield very different results. The first few pages of Section 14.2 (through Example 14.2.1) can be read independently of all of the earlier material in this book and will provide the reader with a discussion of Feynman's heuristic "rules" and an extremely simple example of the points made above. 1.3
Relationship with the motivating physical theories: background and quantum-mechanical models What does this book have to say about the physical theories which motivate it? The reader will not find here applications to concrete and detailed physical problems of the mathematical results contained within. However, in certain respects, we do discuss in a number of places related physical theories and especially quantum mechanics. Physical background A discussion of the relevant physical background is provided in key places. Most importantly, Feynman's way of looking at quantum mechanics and his path integral and how this has led to the approaches to the Feynman integral found in this book is the subject of Chapter 7. Chapter 6 contains an extremely brief discussion of some parts of the usual Hamiltonian approach to quantum dynamics; this chapter is included partly for the sake of contrast but also because the two approaches have, of course, some common features. It seems to us that it is difficult to get an appreciation for the mathematics of the Feyman integral without at least some understanding of the physical background. As noted earlier, this book contains a good deal of information about the Wiener integral (see Chapters 2-5, 7 and 12-18). Much of this material, apart from Chapter 3, Section 4.1 and Chapter 5, is not the standard fare but consists of special topics related to the two items in the title of this book. Chapter 2 discusses the character of physical Brownian motion and the way in which that led Norbert Wiener, through the work of Brown, Einstein and Perrin, to what is now known as the Wiener process, the mathematical model of Brownian motion. Chapter 14 is an introduction to Feynman's operational calculus. Some discussion of the physical problems that led Feynman to this calculus can be found there, but much less than one might guess. Why is that?
18
INTRODUCTION
The primary purpose of the paper [Fey8], "An operator calculus having applications in quantum electrodynamics", was to present the ideas and rules which Feynman had developed in connection with [Fey5-7] for forming functions of noncommuting operators. While most of the examples in [Fey8] are from quantum theory, Feynman was well aware that he had found a computational technique with implications beyond that particular setting. [In fact, this point was stressed repeatedly by Richard Feynman himself in a number of conversations with the second-named author (M. L. L.), during the first of which (in about 1981) Feynman mentioned his paper [Fey8] on the subject and urged M. L. L. to develop his operational calculus and to put it on a firm mathematical basis.] Chapter 14 is an exposition of these mathematical (but not mathematically rigorous) ideas of Feynman and how they will be interpreted, extended and developed with mathematical rigor in Chapters 15-19. [The reader may be aware of Feynman's sometimes negative comments about some of the mathematicians' musings (see, for example, [Fey 16,17]). However, he/she may wish to contrast this impression with Feynman's comments in [Fey8, p. 108] regarding the need for mathematical rigor and for further mathematical exploration of his "operator calculus". (See the second quote from [Fey8] at the very beginning of Chapter 14, which is in complete agreement with the second author's conversations with Feynman.) Perhaps it is appropriate at this point to add two more personal recollections. When asked by a physics Ph.D. student how much mathematics he needed to learn, Feynman answered without hesitation: "As much as possible." (This was witnessed by the second author in Los Angeles in 1981.) Finally, and to give a more balanced view, when during his 1983 UCLA public lectures for a general scientifically curious audience (of which his book QED, [Fey15], is an edited version), he was asked what were the relationships between mathematicians and physicists, he began his answer (approximately) as follows: "They are very good friends, but they do not consider the same problems, and they do not have the same point of view. The mathematician looks at a very broad area and is interested in everything related to it. The physicist, on the other hand, who is interested in certain specific questions, can go much further in some particular directions..."] The discussion of physical background and physical interpretation of results goes beyond the introductory chapters mentioned above. It can be found in various places throughout the book. We mention Chapters 11, 13, 15, 16 and especially, Chapters 17 and 19. Highly singular potentials A variety of quantum-mechanical models are discussed in this book. These include in Chapters 11 and 13 highly singular potentials V and the standard Hamiltonian
In (1.3.1), V denotes the operator of multiplication by a time independent, real-valued potential energy function V. [The precise form of H when the mass m and H = (Planck's constant)/2n are not normalized is given in (6.4.1). For the case of an N-particle system where the jth particle has mass mj, j = 1 , . . . , N, see (6.4.2).] The inclusion of highly singular potentials in the approaches to the Feynman integral discussed in Chapters 11
RELATIONSHIP WITH THE MOTIVATING PHYSICAL THEORIES
19
and 13 is a major advantage of those approaches. Some of the most basic potentials of quantum mechanics such as the Coulomb potential are singular in the space variables. (See [FrLdSp] for a detailed account from a physicist's point of view of the role of singular potentials in quantum theory.) A discussion of highly singular central potentials is provided in Example 11.4.7 and pursued in Example 13.6.13. The interesting special case of the inverse-square potential is treated in Example 13.6.18. We give in Example 11.4.12 and in parts of Sections 13.5 and 13.6 a brief discussion of a refined and highly singular Hamiltonian which is obtained by supplementing H in (1.3.1) by a magnetic vector potential. This corresponds to the Schrodinger equation associated with a magnetic as well as an electric field. Further, in Example 11.4.10, we consider the case where a d-dimensional Riemannian manifold replaces Euclidean space Rd. Time-dependent potentials The operator-valued Feynman integral used in Chapters 15-18 is defined via analytic continuation in mass. In those chapters, the emphasis is on Feynman's operational calculus and, in particular, on disentangling via time-ordered perturbation series by using the Wiener and Feynman integrals. The "potentials" allowed there are very general in most respects; they can be time-dependent and complex-valued and no smoothness assumptions are made. However, they are required to be essentially bounded in the space variables; that is, no spatial singularities are permitted. (Hence, for instance, the Coulomb potential is not allowed in this setting since it has a blow-up singularity at the origin.) Potentials which are bounded and may be time-dependent appear in various places in the physics literature. Forces that are under the control of an experimenter provide a natural source of examples of potentials that are both time-dependent and bounded. It is not just the potentials $ that influence the possible physical models in Chapters 15-18, but also the Lebesgue-Stieltjes measure n as in (1.1.4). These measures determine the disentangling (as noted earlier) and, when combined appropriately with an exponential function, determine the evolution of an associated physical system (see Chapter 17). [We refer, in particular, to Section 17.5 for possible physical interpretations of the corresponding results both in the quantum-mechanical (or Feynman) case and in the diffusion (or probabilistic) case.] The fact that such measures may have continuous and/or discrete parts allows us to study both continuous and discrete phenomena and their relationships with one another. This considerably broadens our approach to Feynman's operational calculus via Wiener and Feynman path integrals in Chapters 15-18. Mathematically, it also gives a rich combinatorial structure to the time-ordered perturbation expansions (or generalized Dyson series) and the associated generalized Feynman graphs introduced in Chapter 15 and used throughout the above chapters. A brief discussion is given in Section 13.5 of Haugsby's extension of Nelson's second approach to the Feynman integral. This is the only place in the book where potentials are treated which can be both singular in the space variables and time-dependent. Phenomenological models: complex and nonlocal potentials We are also able to treat certain phenomenological models. By a phenomenological model, we mean one that does not arise from the basic principles of quantum mechanics
20
INTRODUCTION
but has, nevertheless, been found useful in modeling certain quantum systems. We have already mentioned complex potentials above. Such potentials are used in modeling dissipative (or open) quantum systems. An extensive discussion of this topic—including its strengths and weaknesses and its relationship with "the" Feynman integral—can be found in Exner's book [Ex], Open Quantum Systems and the Feynman Integral. Complex potentials are permitted in some of the results in Chapters 11 and 13 (see especially Sections 11.6 and 13.6, as well as the end of Section 13.5) and in nearly all of the results in Chapters 15-18. The setting of Chapter 19 is more general, but operators of multiplication by a potential can be considered, and, when they are, the potentials involved can be both time-dependent and complex-valued. Chapter 19 deals with time-dependent families {B(s) :0 < s < 00} of bounded operators on a Hilbert space. (A strongly continuous semigroup of operators on the Hilbert space and the generator of that semigroup are also involved but are not particularly relevant to the present comments.) Nonlocal potentials are used phenomenologically in many body problems in several areas of quantum physics (see [Tab, ChSa, Mc] and the relevant references therein). The operator is an integral operator whose kernel V(x, y) (or V(s;x, y) if we have time-dependence) is referred to as a "nonlocal potential". It is nonlocal in that this "potential" does not depend on one sharp choice for the space coordinates (see formula (19.7.15)). Such nonlocal potentials are used, for example, in nuclear physics where the kernels used to model various situations are surprisingly simple; they are, in practice, separable kernels of finite (and low) rank (see Example 19.7.5). Finally, we mention that some of the highly singular potentials discussed just above and treated in Section 11.4 and Sections 13.5-13.6 can also be viewed as providing suitable phenomenological models for certain problems occurring in quantum physics or in molecular chemistry. (See, for example, [LL, Nel, FrLdSp].) For instance, the attractive inverse-square potential (Example 13.6.18) and more generally, highly singular attractive or repulsive central potentials (as in Examples 11.4.7 and 13.6.13), can be used to model problems occurring in quantum field theory or in polymer physics. They are often considered as "nonphysical" or only of academic interest because, in particular, they may lead (as in Example 13.6.18) to nonunitary evolutions and thus to Schrodinger operators which are no longer self-adjoint—in contradiction with one of the basic tenets of standard Hamiltonian quantum mechanics. (This unusual aspect is apprehended naturally within the context of the various approaches to analytic-in-mass Feynman integrals discussed in Sections 13.5 and 13.6; see [Nel, Kat7, BivPi, BivLa].) Actually, the situation is somewhat more complicated than that and a suitable dose of pragmatism is needed to decide which model (whether of Feynman type or of Hamiltonian type, say) is most appropriate for a given physical situation; see, for instance, [Case, R, FrLdSp] and Example 13.6.18. In spite of these drawbacks, it can be argued convincingly that such highly singular potentials provide better approximate (or "phenomenological") models of suitable physical systems than their more regular counterparts. (See especially the review article [FrLdSp] as well as, for example, [LL, Nel, PariZi, MarPari] and the relevant references therein.) In closing the main part of this introduction, we briefly return to Chapter 20 which, as was mentioned earlier, is of a very different nature than the rest of this book. We
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21
recall, in particular, that in Section 20.2, we discuss some of the relationships between heuristic Feynman-type integrals (as well as aspects of Feynman's operational calculus) and a variety of subjects from contemporary physics (or mathematics). In addition, in Section 20.2.A, several mathematical or physical models inspired by quantum field theory (specifically, "Chern-Simons gauge theory" defined in terms of a heuristic Feynman-Witten functional integral [Wit 14]), are used to gain insight into (and extend) the celebrated Jones polynomial, along with other topological invariants that are central to modern knot theory and low-dimensional topology. We hope that despite its relative brevity, Section 20.2 will prove helpful to a reader interested in getting a sense of the fascinating interplay between heuristic Feynman path integrals and a number of topics lying at the border of mathematics and theoretical physics. Prerequisites, new material, and organization of the book We end this introduction by making some specific comments about the content and the structure of this book, along the lines suggested in the title of the present italicized subsection. As was mentioned in the preface and further explained earlier on in this chapter, much of the background material needed for the main part of this book is provided here; see especially Chapters 3,4, 6, 8-10, along with Chapter 7. Detailed proofs—based mainly on the background material just referred to—are given for nearly all the theorems which deal with the main topics of this book. Most of the exceptions come in Sections 13.6, 13.7 and in the last part of Section 13.5, as well as in Section 17.6. The reader will see that proofs are provided even for a good portion of the background material itself; see, in particular, Sections 3.1-3.4, 4.1-4.2, 4.5-4.6, and 10.2-10.3. We remark that if the reader is willing to forego the proofs in Sections 11.6,13.5 and 13.6, then the operator-theoretic background needed for the book (especially through Chapter 13) is reduced to the information about self-adjoint operators and quadratic forms found in Section 9.6 and in Chapter 10 plus relatively few basic facts about semigroups of operators provided in Chapters 8 and 9. We should mention that the comments in the preceding paragraphs do not apply to Chapter 20; no attempt there is made to supply proofs. (In the case of Section 20.2, in which much of the material connected with Feynman path integrals is of a heuristic nature, rigorous mathematical proofs are usually not known.) The Lebesgue theory of measure and integration is employed in many places in this book. Precise references are typically given for the results used, but no systematic presentation of measure-theoretic prerequisites is provided. Brief discussions of this subject can be found in the books [BkExH, Appendix A, pp. 531-544], [Ru2, pp. 5-75] and [ReSi1, pp. 12-26]. Fuller treatments are given in many places, for example, in [Roy, Fol2, Coh, Du]. The list of references provided at the end of this book is extensive but certainly not complete. (We note that a significant fraction of the references is connected in some
22
INTRODUCTION
way with Chapter 20, which deals with a broad selection of topics.) When the references are given in the main body of the text, they are typically presented in enough detail so that the relevant material can be easily located. The topics discussed in this book are interrelated in a variety of ways; we try to keep track of these relationships by systematic cross-referencing. A substantial part of the material in this book other than the background material has appeared previously only in the research literature and, in a number of cases, only in the recent research literature. A few results that will play a prominent role come from sources that are not widely available. The primary example of that is the Sherbrooke monograph of the first author [Jo6] which plays a key role in the operator-valued analytic continuation in time results in Chapter 13. There is a significant amount of novelty to the exposition in several places. For example, in Section 4.6, we discuss in detail what is meant by the "nonexistence of Feynman's measure" as well as related issues. Chapter 14 is an introduction to Feynman's operational calculus for noncommuting operators. This subject extends certain aspects of the Feynman integral, a fact that does not seem to be widely understood in the mathematical literature. We explain this in some detail in Chapter 14, and the idea is developed further in Chapters 15-19. It will be clear to the reader of this book that the research interests of the authors have influenced much of the content. However, the influence went the other way as well; the desire to fill in missing pieces of the book directed some of our research in recent years. (For instance, reference [dFJoLa2]—on which Chapter 19 is based—was very much written with our book in mind.) A portion of that work is new here. We wish to call the reader's attention to a few such items. In Section 13.4, we show that under rather general conditions, three of the four approaches that are discussed in this book are closely related. Most of this material is new. The last part of Section 13.5 and nearly all of Section 13.6 deal with product formulas and operator-valued analytic continuation in mass from such formulas (rather than from the Wiener integral). A good portion of this material is new as well, particularly with regard to the explicit connections with the Feynman integral. Further, a comparison of the various analytic-in-mass Feynman integrals is provided, along with related results; see Theorems 13.6.10 and 13.6.11, along with Corollary 13.5.18. In addition, a detailed treatment of highly singular central potentials is given in this context; see Examples 13.6.13 and 13.6.18. In Section 15.7, a "time-reversal map" is introduced and studied for our disentangling algebras in Chapters 15 and 18; see Definition 15.7.5, Theorem 15.7.6 and Corollary 15.7.8. This enables us, in particular, to clarify the connections with the usual "physical ordering" in the context of Feynman's operational calculus. In turn, these changes have repercussions in Chapter 17. We expect that more work will be done along related lines in the future. New examples are provided in several places. We mention one, Example 15.5.3, that may be of particular interest. This example involves the purely continuous but singular measure associated with the Cantor function.
RELATIONSHIP WITH THE MOTIVATING PHYSICAL THEORIES
23
At the end of Chapter 16, we indicate how results from our earlier work (contained in [JoLal j) on "stability" in the measures can be extended. We carry this out in detail in one case (see Proposition 16.2.14) and indicate how to go further in another case. In Remark 11.5.15(d), we point out that the requirements on the negative part of the dominating "potential" in Theorem 11.5.13 (from [Lal2]) can be reduced to membership in the "Kato class" of functions on Rd. (Theorem 11.5.13 is a dominated-type convergence theorem for the modified Feynman integral; it is the subject of Section 11.5 and is also applied in Chapter 13 to other approaches to the Feynman integral.) Additional work on Feynman's operational calculus by Brian Jefferies and the first author ([JeJo]) is discussed briefly in III of Section 14.4. That work provides a nice supplement to the treatment given in Chapters 15-19 of Feynman's operational calculus (and based on [JoLal-4, Lal4-18, dFJoLal,2]). However, some aspects of the new material still need further development and so a fuller discussion of this topic could not be included in this book. Several exercises or problems are proposed throughout the book. They are of varying degrees of difficulty. Typically, the exercises are mainly intended to illustrate a new concept, apply a new technique, or supplement some material in the text. Most of them should be accessible to graduate students. However, in a few instances, some of the proposed problems are extremely difficult and not yet solved in the literature (e.g. Problem 11.3.9). In other cases, they correspond to results already published but the proof of which is not discussed fully in the book (e.g. Problem 17.3.6 or 17.6.28). In addition, a few open-ended problems—the precise interpretation or formulation of which is left to the reader-are provided either formally (e.g. Problem 17.6.31) or in various comments or remarks scattered in the text. When appropriate, we have usually indicated the nature or the difficulty of the problem at hand. The numbering system used in this book is straightforward. For example, Theorem 11.5.13 is the thirteenth numbered item in Section 5 of Chapter 11; a similar comment applies to the numbering of equations. Further, Section 15.4 is the fourth section of Chapter 15. Frequently, unnumbered subsections with italicized headings are used within a given section in order to delineate or highlight certain topics. Indexes for symbols or notation, authors, and subjects are provided just after the bibliography. Along with the detailed list of contents, we hope that they will prove to be a useful guide to the reader throughout this book.
2
THE PHYSICAL PHENOMENON OF BROWNIAN MOTION While examining the form of these particles immersed in water, I observed many of them very evidently in motion; their motion consisting not only of a change of place in the fluid, manifested by alterations of their relative positions, but also not infrequently of a change in form of the particle itself; a contraction or curvature taking place repeatedly about the middle of one side, accompanied by a corresponding swelling or convexity on the opposite side of the particle. In a few instances, the particle was seen to turn on its longer axis. These motions were such as to satisfy me, after frequently repeated observations, that they arose neither from currents in the fluid, nor from its gradual evaporation, but belonged to the particle itself. Robert Brown, 1827 In this paper it will be shown that according to the molecular-kinetic theory of heat, bodies of microscopically-visible size suspended in a liquid will perform movements of such magnitude that they can be easily observed in a microscope, on account of the molecular motions of heat. It is possible that the movements to be discussed here are identical with the so-called "Brownian molecular motion"; however, the information available to me regarding the latter is so lacking in precision that I can form no judgment in the matter. Albert Einstein, 1905 [Ei, p. 1] We will still stay within the realm of experimental reality if, putting our eye on a microscope, we observe the Brownian motion which agitates every small particle suspended in a fluid. In order to obtain a tangent to its trajectory, we should find a limit, at least approximatively, to the direction of the line which joins the positions of this particle at two successive instants very close to each other. But, as long as one can perform the experiment, this direction varies wildly when we let the duration that separates these two instants decrease more and more. So that what is suggested by this study to the observer without prejudice, is again the function without derivative, and not at all the curve with a tangent. Jean Perrin, 1913 [Per, p. 27] 2.1
A brief historical sketch
The distinguished English botanist Robert Brown made the first careful study of "Brownian motion". In 1827, he was investigating the fertilization process in a certain species of flower. While looking at the pollen in water through a microscope, he observed small particles in "rapid oscillatory motion". He examined the pollen of other species with similar results. He first hypothesized that the motion was particular to the male sexual cell of plants and next that the motion involved living matter. His experiments with inorganic material showed that both of these are wrong. What does cause the motion? Brown did an experiment which refutes some explanations that were put forth well after his study; he immersed a drop of water containing one particle in oil and looked at it under the microscope. The motion is observed as before. It is clear then for one thing
A BRIEF HISTORICAL SKETCH
25
that the source of the motion is not a force of attraction between the particles. Brown also noted that when several particles are present, they appear to move independently of one another even when they are very close together. Brown's conjecture, carefully labeled as such, was that matter is composed of small particles, which he calls active molecules, which exhibit a rapid, irregular motion having its origin in the particles themselves and not in the surrounding fluid. Considerably later, it became clear that the surrounding fluid is the source of the motion. A brief discussion of the work between Brown and Einstein is given in [Ne2, pp. 11-13] and by Furth in Note 1 of [Ei, pp. 86-88]. These make interesting reading and illustrate well that progress in science does not always proceed linearly. We restrict our attention to a few positive contributions. Cantoni and Oehl found in 1865 that the movement continued unchanged for a year. S. Exner observed in 1867 that the motion is most rapid with the smallest particles and is increased by light and heat rays. In 1877, Delsaux expressed the now commonly accepted idea that Brownian motion is caused by the impacts of the molecules of the liquid on the particles. Guoy found in 1888 that the motion becomes more lively as the viscosity of the fluid is decreased and that, on the other hand, a strong electromagnetic field has no effect. Like Delsaux, Guoy attributed the motion of the particle to the molecular motions of the fluid. The main points made by observing Brownian motion are as follows: (1) The path followed by the particle is continuous but extremely jagged. (2) The particles move independently of one another. (3) The motion is more active the smaller the particles. (4) The composition and density of the particles have no effect. (5) The motion is more active the less viscous the fluid. (6) The motion is more active the higher the temperature. (7) The motion never ceases. It has been generally accepted for a long while now that the source of the motion is the bombardment of the particle by the molecules of the surrounding fluid. The following remark addresses the extent to which (1)-(7) above are consistent with this explanation. Remark 2.1.1 (a) It is certainly reasonable that a particle under molecular bombardment would follow a continuous path. The extreme jaggedness of the path seems less easy to account for but is plausible given the random and almost continuous bombardment of the particle. (b) Items (2) and (7) above are quite consistent with the molecular-kinetic theory. (c) It is not immediately evident that (3) is consistent. How does one attempt to explain it? Individual molecular hits do not produce observable motion. Such motion occurs when there is a preponderance of hits in one or the other direction. The larger the particle, the less often will the hits be sufficiently unbalanced to produce observable motion. (d) If the motion comes from the bombardment of the particles by the molecules of the fluid, is it reasonable that the density of the particles has no effect as (4) asserts? Not
26
THE PHYSICAL PHENOMENON OF BROWNIAN MOTION
entirely it seems. What probably happens is that within the range of densities, viscosities, etc. where Brownian motion is observed, (4) is approximately true. Relatively high viscosity is present when Brownian motion is observed and so the velocity caused by a bump (or bumps) is quickly damped out. This will also be relevant to our discussion of "independent increments" in the next section. (e) The viscosity of a fluid is a measure of internal friction and so it affects the ease with which a particle moves through the fluid. As such, (5) is not surprising. (f) Since increasing the temperature speeds up molecular action, (6) is certainly consistent with the molecular-kinetic theory. The quantitative theory of Brownian motion began with Albert Einstein. His five papers on the subject between 1905 and 1908 made careful quantitative predictions based on the molecular-kinetic theory of heat. Einstein's first paper in 1905 was a prediction motivated by the molecular-kinetic theory that a phenomenon with properties similar to Brownian motion ought to be observable in nature. He was apparently just becoming aware of the earlier studies on Brownian motion. He says, "It is possible that the movements to be discussed here are identical with the so-called 'Brownian molecular motion'; however, the information available to me regarding the latter is so lacking in precision that I can form no judgment in the matter." On the other hand, his second paper in 1906 is titled "On the theory of the Brownian movement". See [Ei], where his five papers on Brownian motion are reprinted. [In 1905, Einstein formulated the special theory of relativity, studied the photoelectric effect and wrote his first paper on Brownian motion—one of the most productive years in the history of science; see [Srt], where his five 1905 papers are reprinted.] Let p = p(u, t) be the probability density that a Brownian particle starting at the origin at time 0 is at u at time t. Through physical reasoning, Einstein derived the diffusion equation
where D is a positive constant called the coefficient of diffusion. A solution to this equation is given by
Hence, the probability that the particle is in say a cube E at time t
Remark 2.1.2 (a) That Einstein expressed position at time t in probabilistic terms was not surprising since this is the framework of the molecular-kinetic theory. There are too many particles involved to use Newtonian mechanics effectively. (b) In his 1923 paper Differential space ([Wil], reprinted in [Wi3, pp. 455-98]), Wiener obtained the formula (2.1.1) quite differently. He combined relatively elementary
A BRIEF HISTORICAL SKETCH
27
physical considerations with the central limit theorem to obtain (2.1.1). He does not specifically invoke the central limit theorem, but it is clear that he had it in mind. We will give a closely related discussion in the next section. Contained in the formula (2.1.1) is the information that the mean square displacement of a particle in time t is 2Dt. We will return to this fact shortly. Assuming that the particles are spheres of radius a, Einstein also derived the formula
where n is the coefficient of viscosity, T is the temperature and k is Boltzmann's constant. Note how points (3)-(6) earlier are reflected in the formula (2.1.1). [Einstein's papers contain much more, but the key for us will be (2.1.1).] In 1909, the French physicist Jean Perrin observed that the Brownian trajectories appear to have no tangents and mentions the nowhere differentiable curves of Weierstrass. This comment led to a beautiful theorem of Norbert Wiener. Wiener [Wil] quoting Perrin [Per]: "One realizes from such examples how near the mathematicians are to the truth in refusing, by a logical instinct, to admit the pretended geometrical demonstrations, which are regarded as experimental evidence for the existence of a tangent at each point of a curve." Are the paths traced out by the physical Brownian particles actually nondifferentiable? We suppose not. Consider what is likely to happen to the difference quotient h when h is orders of magnitude smaller than the average time between molecular hits. On the other hand, the paths of the Wiener process, the mathematical model of Brownian motion, will turn out to be nondifferentiable. We remark that there is another model for Brownian motion, due to Ornstein and Uhlenbeck, such that the associated paths are differentiable [Ne2]. The Brownian motion studies played an important role in supporting the atomistic theory at a time when this theory was much in doubt. In Einstein's formula (2.1.2), T and n can be calculated and it can be arranged that the particles are all spheres of the same radius a. Further, the mean square displacement 2Dt (see above) at time t = 1 can be estimated statistically. The idea is simple: Observe the paths X j ( . ) , j = 1, . . . ,n, in R3 of n independent Brownian particles starting at the origin at time 0 and record their positions when t = 1. We then have
Hence one arrives at a statistical estimate for Boltzman's constant k.
and so also for Avogadro's number NA = R/k, where R is the universal gas constant. This was done by Perrin (and others) and the value was sufficiently close to the value
28
THE PHYSICAL PHENOMENON OF BROWNIAN MOTION
obtained by another quite different approach to be regarded as a significant confirmation of the atomistic theory. (Recall that Avogadro's number, NA = 6 x 1023, is defined as the number of atoms or molecules in a mole of substance.) Perrin received the Nobel prize in 1926 in recognition of his work on Brownian motion. Remark 2.1.3 Many of the conclusions drawn by Einstein concerning Brownian motion were reached at about the same time by the Polish physicist, M. Smoluchowski [Smo]. Even earlier, L. Bachelier in his 1900 dissertation [Bac] arrived heuristically at some of the same mathematical formulas. Bachelier's study was motivated by a very different problem; he was attempting to analyze the French stock market. It was primarily Einstein's work that influenced Norbert Wiener's later brilliant study of "Brownian motion" ([Wil,2], [Wi3, §IC]). We note that much of the material in this section is adapted from d'Abro [dA, pp. 411-415], Einstein [Ei] (including the notes by Furth on pp. 86-88), and especially from Sections 2 and 3 of Nelson's book Dynamical Theories of Brownian Motion [Ne2]. 2.2 Einstein's probabilistic formula The starting point in the introduction of Wiener measure is Einstein's probabilistic formula (2.1.1) which we now recall: The probability that a particle which starts at the origin at time 0 is in a (Lebesgue measurable) set E at time t
In order to simplify matters, we concern ourselves for now with the motion of the particle in a single direction. Further, we normalize the diffusion coefficient, taking 2D = 1, and we take the set E (E c R now) to be the interval (a, B]. Formula (2.1.1) then becomes: The probability that a particle which starts at the origin at time 0 is in (a, ft] at time t
Letting x = x ( t ) be the location of the particle at time t, we can rewrite (2.2.2) as
In words, the right-hand side of (2.2.3) gives the probability that the path traced out by the particle goes through the gap (a, B] at time t. (See Figure 2.2.1.) We will have to go far beyond the simple sets of paths involved in (2.2.3) to develop a satisfactory mathematical theory. Before doing this, however, we wish to give a largely heuristic discussion of the appropriateness of (2.2.3) as a starting point for a mathematical model of Brownian motion. Some knowledge of introductory probability theory will be needed for what immediately follows but will not be necessary as we continue.
EINSTEIN'S PROBABILISTIC FORMULA
29
FIG. 2.2.1. The Brownian paths passing through the gap (or, B] at time t
, -u2The right-hand side of (2.2.3) involves the density function, -4= e 2r, of an N(0, t) random variable; that is, a random "variable which is normally distributed with mean 0 and variance t. Let us try to get some idea why this density should appear. Relatively high viscosity is assumed so that the effect of a single molecular hit on the motion of a physical Brownian particle is quickly damped out. Hence, it is not unreasonable to assume in the mathematical model of Brownian motion that x(t + s) — x(t) is independent of x(t) for s > 0; i.e., the path x has independent increments. Also, for any integer n, we can write
As long as the physical conditions (for example, the temperature) are unchanged as time goes on, it seems reasonable that the random variables x(jt/n) - x ( ( j — l)t/n), j = 1 , . . .,n, have the same distribution. Thus we see from (2.2.4) that x(t) is written as the sum of n independent, identically distributed random variables. The presence of a normal density function is then not surprising in light of the central limit theorem. Further, for the physical particle, it seems equally likely that x (t) be positive or negative and so a zero mean is reasonable. Now the variance of a random variable with mean 0 is the expected (or average) value of its square. Also the variance of the sum of independent random variables is the sum of the respective variances. Finally, if the physical conditions remain constant, Var[x(s +t) — x(s)] = E[(x(S + t)— x(s))2] should be a function of t alone. With these things in mind, we write
30
THE PHYSICAL PHENOMENON OF BROWNIAN MOTION
and so
Hence doubling t doubles the variance. Similarly, tripling t triples the variance, etc. Thus it seems that Var[x(t)] depends linearly on t . Since Var[x(0)] = 0 , we see that Var[x(t)] = ct. The constant c depends on the diffusion constant and c = 1 is appropriate here because of our earlier normalization. Summary: The physical properties of Brownian motion make the appearance of the N(0, t) density in (2.2.3) (the starting point of the mathematical model) seem quite reasonable. Key ideas in the preceding discussion come from Wiener's 1923 paper [Wi1] or from Lamperti's book [L1,§20].
3
WIENER MEASURE Evolution of Mathematics is, by and large, a continuous process and its growth and progress seldom deviate greatly from the natural historical lines. It is because of this that we tend, in retrospect, to admire most those developments which though born well outside it have grown to join and to enrich the mainstream of our science. It was the great fortune and the great achievement of Norbert Wiener to initiate such a development when, in the early twenties, he introduced a measure, now justly bearing his name, in the space of continuous functions. . . . In retrospect one can have nothing but admiration for the vision which Wiener had shown when, almost half a century ago, he had chosen Brownian motion as a subject of study from the point of view of the theory of integration. To have foreseen, at that time, that an impressive edifice could be erected in such an esoteric corner of mathematics was a feat of intuition not easily equalled now or ever. It was Josiah Willard Gibbs, whom Wiener admired so much, who said that "one of the principal objects of the theoretical research in any department of knowledge is to find the point of view from which the subject appears in its greatest simplicity". Integration in function spaces provided such a point of view over and over again in widely scattered areas of knowledge and it gave us not only a new way of looking at problems but actually a new way of thinking about them. The fate of all great work is to be subsumed; the more attention it attracts the greater the chances of becoming engulfed in a cascade of generalizations and extensions. This is especially true today because of a growing tendency to believe that the latest improvement supersedes all that preceded it and that a generalization constitutes a license to subsume. It is therefore well to repeat that Wiener's contribution to the subject of integration in function spaces will forever be the greatest because he had the idea first; and should anyone try to attribute it to luck let him be reminded that it is the deserving ones who are also lucky. Mark Kac, 1966 [Kac3, pp. 52 and 68] We give in this chapter an introduction to Wiener measure with a strong emphasis on those aspects of the subject which are relevant further on in the book. The perspective is that of an analyst rather than a probabilist. As such, it may serve as a useful introduction to the fascinating Wiener process for those with a minimal background in probability theory. Probabilists should browse through the chapter but will find that they can skip over much of the material. We should emphasize that our treatment of the Wiener process (or Brownian motion) in this chapter and the next is far from complete, We hope that it will stimulate some readers to pursue the subject elsewhere. A few of the many references available are [Dool, Durr, Fre, Hid, L1, StroVa, Wil, Yen].
32
WIENER MEASURE
3.1 There is no reasonable translation invariant measure on Wiener space We begin by introducing some terminology and notation which will be used throughout much of this book. Definition 3.1.1 Given a positive integer d and real numbers a and b with a < b, C([a, b], Rd) will denote the space of continuous functions on [a, b] with values in Rd. Often, in a given discussion, we will have d — 1 or d will be arbitrary but fixed. If it is clear from the context which range is intended, we will usually write C[a,b]or C a,b rather than C([a, b], R.d). In fact, later on we will often take a = 0 and will write Cb in place of C([0, b], Rd). The space of functions x in C([a, b], Rd) such that x(a) = 0 will be denoted C0([a, b], Rd) (respectively, C0[a, b], Coa,b, Cbo). We will often refer to C%'b as Wiener space in light of the role that it plays in connection with Wiener measure. Given x in Ca-b, let
where \\x (t) || denotes the Euclidean norm of x (t) in R rf . It is well known that the function || -|| defined in (3.1.1) is a norm on Ca'b and that (Ca-b, \\ • ||) is a Banach space. Since C^ is a closed subspace of C a,b , it follows that (Ca,b , ||.||) is also a Banach space. By the Weierstrass approximation theorem, the polynomials are dense in C ([a, b], R) and, from this, it follows that the polynomials with rational coefficients are dense in C([a, b], R). By considering components, one sees that C([a, b], Rd) is a separable Banach space. Since every subspace of a separable metric space is separable, Co([a, b], Rd) is separable as well. The paths followed by the Wiener process, the mathematical model of Brownian motion, over the time interval [a, b] will lie in the Banach space CQ'*. Wiener measure will turn out to be a measure defined on a or-algebra containing the "Borel class" of CQ ' . (The basic elements of measure theory used in this chapter—as well as later on in the book—can be found in [Cho3, Coh, Roy, Ru2].) Definition 3.1.2 Given any topological space T, B = B(T), the Borel class of T, is the a-algebra generated by the open subsets of T. It is the Borel class B — B(Ca' b ) of CQ' which presently concerns us. Recall that Lebesgue measure on Rd is translation invariant; that is, Leb.(E + u) = Leb.(E) for every Lebesgue measurable subset E of Rd and for every u in Rd. As we will see later, in formulating the "Feynman integral", Feynman thought in terms of a translation invariant measure on the space of continuous functions. Wiener measure m will play a crucial role in two of the approaches to the Feynman integral which we will treat in some detail. The measure m does have the property of assigning positive measure (or probability) to every ball in the metric space CQ'*, but it is not translation invariant. Since our primary goal is a treatment of the Feynman integral, one might reasonably think that we should be introducing a translation invariant measure on CQ' at this point rather than m. The trouble is that there is no reasonable, nontrivial translation invariant measure on C^'b. The following theorem makes that point sufficiently well
NO REASONABLE TRANSLATION INVARIANT MEASURE ON WIENER SPACE
33
for our purposes and is not hard to prove; however, there are stronger and much more general theorems in the literature. (Theorem 3.1.5 which we will state below is a special case of such a result.) Unless otherwise specified, all measures used here will be positive (and countably additive). Theorem 3.1.3 If a measure u on (€$' , B(€Q' )) is translation invariant and assigns finite measure to some nonempty open set, then u is identically 0. Proof We simplify notation by giving the proof for the case CQ = Co([0, 1], R). Since u is translation invariant and any r-ball Br(y) is the translate of any other r-ball Br(z) (in fact, Br(y) = Br(z) + (y — z ) ) , we see that all r-balls have the same u-measure. (Here, Br (y) denotes the open ball of CQ of center y and radius r.) Let O be a nonempty open set of finite u-measure. Given any x in O, there exists ro > 0 such that Bro(x) c. O. By the monotonicity property of measures, u(Bro (x)) < u ( O ) < oo. Using the translation invariance of u and invoking monotonicity again, we see that if 0 < r < ro, u(B r (y)) < cc for all y in C01. Now we define a sequence of functions {xn}in C01 as follows:
It is easy to see that \\xn|| = ro/2 and that \\xn — xm\\ = ro/2 for n = m. It follows that Bro/4 (xn) n Bro/4(xm) = 0 if n = m and that B ro / 4 (x n ) c Bro(0) for every n. Hence Bro (0) contains an infinite disjoint sequence of sets {Br0/4 (*n)} all having the same measure and such that 0 < u(B r o /4(x n )) < oo. Since u(Bro(0)) < oo, we see that u(B ro /4(x n )) = 0 for every n. Now C10 is separable and so there exists a countable dense set {yj}^.] • It is easily checked that CQ = Uj=1, B ro /4(y j ) and so
It follows that n is the identically 0 measure, as claimed.
D
Reflections on the proof of Theorem 3.1.3 lead one to suspect that there are related results of a much more general nature. One such result can be found in the book of Gelfand and Vilenkin [GelVi, Theorem 4, p. 359]. We will state a special case of this result. Definition 3.1.4 Let X be a Banach space and let u be a measure on (X, B(X)). We say that u, is quasi-invariant provided that A in B(X) with n(A) = 0 implies that u(A + x) — O for every x in X, Further, we say that u is a -finite if X =U00n=1Bn,with Bn in B(X) and u (Bn) < oo. Theorem 3.1.5 Let X be a separable infinite dimensional Banach space and let u be a measure on (X, B(X)) which is a-finite and quasi-invariant. Then u is identically 0.
34
WIENER MEASURE
Corollary 3.1.6 If a measure u is translation invariant and a-finite on (€$' ,B(C^' )), then u is identically 0.
3.2 Construction of Wiener measure We will now outline the construction of one-dimensional Wiener measure. Except for notational complication, the construction of cf-dimensional Wiener measure is much the same, as will be discussed in more detail at the end of this chapter. Hence, unless otherwise specified, we will assume in the following that d — 1. We begin by stating another variation of Einstein's probabilistic formula (2.2.3) when d = 1. Temporarily fix t so that a < t < b and suppose that —oo < a < B < +00. The normalized and one-dimensional version of Einstein's formula for a particle which starts at MO at time a is as follows (see Figure 3.2.1):
where
The function p(u, uo, t — a) is the N(U0, t —a) density. The mean MO and variance t — a seem entirely appropriate here since the particle starts out at MO and the elapsed time is t -a. Based on the consideration above, Wiener's problem was to demonstrate the existence of a countably additive probability measure m = ma,b on C0a,b such that if a = to < t1 < • • • < tn < b and if cj, Bj are extended real numbers such that
FIG. 3.2.1. Paths starting at MO at time a and going through the gap (a, B] at time t
CONSTRUCTION OF WIENER MEASURE
35
FiG. 3.2.2. Wiener paths starting at 0 at time a and going through the gap (a,-, Bj] at time tj, for j = 1,2,3
Figure 3.2.2 illustrates the set of paths involved in (3.2.3) in the case n — 3. The first factor in the integrand in (3.2.3) represents, as before, the probability density for being at u1 at time t\ having started at 0 at time a. The second factor gives the probability density for moving from u1 at time t1 to u2 at time t2. The appropriateness of this factor comes in part from our assumption in the mathematical model for Brownian motion that x(t2) — x(t\) is independent of x(t\). We will almost always write the right-hand side of (3.2.3) using more abbreviated notation such as one of the following:
36
WIENER MEASURE
Subsets 7 of Coa,b as in the first expression in (3.2.4) will be called intervals and we will write
The collection of all such intervals (also called "cylinder sets") will be denoted J. Wiener's idea was to develop, using (3.2.3) (or (3.2.4)) as a starting point, a full fledged Lebesgue type integral over the (infinite dimensional) space CQ' , thus making available the powerful results of the Lebesgue theory. Wiener succeeded in this endeavor; his achievement was a major advance which has had an amazing number of repercussions. You will appreciate Wiener's contributions ([Wil,2], [Wi3, §IC]) all the more if you keep in mind that his work was done when the Lebesgue theory was not exactly ancient history and about a decade before the appearance of the book of Kolmogorov [Kol], which is regarded as the beginning of probability theory as a separate discipline. We will proceed to show that the collection T of intervals is a semi-algebra and m is well-defined and additive on 1. In fact, although we will not carry out the proof, m is countably additive on J. Thus the Caratheodory extension process ([Roy, Theorem 12.8, p. 295] or [Con, Theorem 1.3.4, p. 18]) can be applied to obtain a countably additive measure on a(I), the a-algebra generated by 1. Further, we will show that a (I) = B(C^'b). Finally, the measure space (C^, B(C^'b), m) can be completed producing the cr-algebra S1 of Wiener measurable sets and the complete measure space (C^' , S1, m). There are several ways of introducing Wiener measure (see, for example, [Dool, Durr, Hid, Kall, Nel, Wil, Yeh]). We choose the approach which we have just outlined since almost all mathematicians and mathematical physicists are familiar with the Caratheodory extension process, at least on M. If one wishes to include the proof of countable additivity, there are more efficient ways to proceed, and which you prefer is somewhat a matter of background and taste. We now turn to carrying out the steps outlined above. We begin by reviewing the definition of a "semi-algebra". Definition 3.2.1 A collection R of subsets of a set X is called a semi-algebra if and only if (a) 0 and X are in H, (b) A and B in K implies that A n B is in R, and (c) the complement of any set in H is a finite disjoint union of sets in R. Proposition 3.2.2 The collection T of intervals is a semi-algebra of subsets of C^' . Proof (a) 0 = [x in C%'b : 1 < x(b) < 1} and C$b = {x in C%'b : -oo < x(b) < +00} and so 0 and CQ' are in I. (b) Let 7, J be in J. The set of restriction points for 7 n J is the union of the restriction points for 7 and the restriction points for J. The restriction interval for any tj which is
CONSTRUCTION OF WIENER MEASURE
37
common to both I and J is the intersection of the restriction intervals. The restriction interval for a tj which is a restriction point for one of the intervals, say /, but not for the other, say J, is just the appropriate restriction interval associated with /. We illustrate the situation in (b) with a simple example. Suppose that a < t\ < t2 < t3, 0. Then
38
WIENER MEASURE
Proof We begin by considering the exponent of the integrand in (3.2.8), but without the minus sign. This is a quadratic function of v and, by completing the square, we obtain
where KI and K^ have their obviously intended meaning. Using (3.2.9) and the translation invariance of the Lebesgue integral, we see that the left-hand side of (3.2.8) equals
where the last equality in (3.2.10) follows from the well-known formula
(See (4.7.1) below.) Finally, some routine algebraic calculations show that
and so (3.2.8) is established.
D
The formula (3.2.3) (or (3.2.4)) for m(7) is based on the representation (3.2.6) for / in terms of the numbers tj, a; and Bj. Such representations are not unique as we will presently discuss and so there is some question as to whether m(7) is well-defined. If there is a fixed set of ts involved, say Tn ~ [t\, ...,tn] with a = IQ < t\ < • • • < tn < b, then it is clear that the numbers oij and f}j uniquely determine the interval / and conversely. One can however always add t-values without changing /. Suppose, for example, that s is in (a, b]\Tn. Then 7 can alternatively be written as
But if one adds further ts, the new "restrictions" must always be — oo and +00. Also one can remove tj whenever ay = —oo and Bj = +00 but not otherwise.
CONSTRUCTION OF WIENER MEASURE
39
In spite of the ambiguities in the representation of /, we have the following: Proposition 3.2.4 m is well-defined on I. Proof There is a minimal representation for / such that for each j, at least one of cj, Bj is finite. Of course, there is a corresponding expression for m(7). Any alternate representation for / must involve additional points with the artificial restrictions — oo and +00. We will show that, in the case of one additional point, the corresponding formula for m(7) agrees with the formula associated with the minimal representation for I. The case of N additional points with artificial restrictions can be done by applying the procedure below N times. Suppose that the minimal representation of / is given by (3.2.6) and that the extra point, say s, satisfies tk < s < tk+1. (The argument takes a slightly different and easier form if s is in (a, t) or (tn, b).) Then
and the corresponding formula for m(7) is
Now the integrand in (3.2.12) is positive and so we may apply Tonelli's theorem and integrate out with respect to v. By the Chapman-Kolmogorov equation, the result is the right-hand side of (3.2.3) as desired. d Exercise 3.2.5 Show that m(C0([a, b], R)) = 1. Exercise 3.2.6 Define F : C a,b x [a, b] -> R by F(x, t) = x(t). Show that F is a continuous function on the product space Ca'b x [a, b]. Proposition 3.2.7 (i) If is in I and I=0, then m(7) > 0. (ii) m is finitely additive on I. Proof (i) Let 7 be given by (3.2.6). If 7 = 0, then cj < Bj for j = 1, . . . , n. Hence in formula (3.2.4) for m(7), a positive function is being integrated over a set of positive Lebesgue measure. Thus m(7) > 0. (ii) Let 7, J and 7 U J be in I with 7 n J = 0. By adding points with artificial restrictions, we can always arrange to have 7 and J based on the same set of restriction
40
WIENER MEASURE
points. Accordingly, using the notation of (3.2.6), let / = If((a\, /3\] x • • • x (c n , B n ] ) and J = It((xi, 81 x • • • x (Yn, Y be a function from X to Y. Define A := {B c Y : f - l ( B ) e £}. Show that A is a a-algebra of subsets of Y. (Clearly, f is £ — A measurable because of the way A is defined.) (b) Let X and Y be topological spaces and let f be a continuous function from X to Y. Show that f is Borel measurable; i.e., f-1 (B) e B(X)for every B & B(Y). Theorem 3.2.11 a (I) = B(C^b). Proof Let a < t < b and define P, : C^b ->• R by
It is easy to see that Pt is continuous. (This also follows from Exercise 3.2.6.) Hence, by Exercise 3.2.10(b), P, is Borel measurable. Now for an arbitrary interval / e I given by (3.2.6), we can write
It follows from (3.2.15) and the Borel measurability of each Ptj that / is in B(C^b). Hence I is a subclass of B(C%'b) and so a(T)C. B(CQ~b) as desired. It remains to show that H(CQ' & ) c a (I), and, for this, it suffices to show that every open subset of CQ' is in a(T). Now CQ' is a separable metric space and such spaces have the property that every open subset of them is a countable union of open balls. Thus it suffices to show that every open ball is in a(T). But every open ball Br(xo) is the countable union of closed balls; specifically,
Thus it suffices to show that every closed ball, say Bs(yo), is in a (I). Now pick a countable set of t-values, [t1, t2, . . .}, which is dense in [a, b]. For each
Each KN belongs to a (I) since
42
WIENER MEASURE
Hence to finish the proof it suffices to show that
It is clear that B& (yo) c KN for every N and so the proof will be finished if we show that
Suppose that y £ Bs(yo). Then there exists .y e (a, b] and 8\ > 0 such that \y(s) — yo(s)\ = S + Si. Let {tjk} be a sequence from the dense set [t\,t2,...} such that tjk -*• s. Then y(tjk) -> y ( s ) and yo(tjk) -*• yo(s). Hence there exists k such that \y(tjk) — yo(tjk)\ > 8 + (Si/2). Thus y £ KN for any N such that tjk e (t\, . . ., tN}. Therefore
The Caratheodory process actually carries us beyond the a -algebra generated by I to its completion with respect to m; i.e., to the R be Lebesgue measurable. Then, with Wn(7, U) given by (3.2.5), we have
where the equality is in the strong sense that if either side of (3.3.9) is defined (in the sense of the Lebesgue theory), whether finite or infinite, then so is the other side and they agree.
WIENER'S INTEGRATION FORMULA AND APPLICATIONS
45
Proof Using the change of variable theorem (Theorem 3.3.2), we can write
where the last equality comes from Corollary 3.3.4 which tells us that the image measure m o Pr~ tn is the same as the measure given by the density Wn(t, U) and Lebesgue measure on R". D Remark 3.3.6 (a) The right-hand side of (3.3.9), and hence also the left-hand side, is finite for many functions f which grow rapidly at oo. On the other hand, if f has a singularity at a point (u\,..., ~u~^) e W which is not LI with respect to Lebesgue measure, the function f(-)Wn(t, •) has a non-L1 singularity at the same point. (b) Formula (3.3.9) reduces an integral over the infinite dimensional path space CQ' to an ordinary Lebesgue integral over the finite-dimensional space K". Such a reduction is not possible in general; it works here because the function x H>• / (x(t\),..., x(tn)) depends only on the value of the path x at n points. In spite of its finite-dimensional nature, (3.3.9) is the starting point for various Wiener integration formulas, for example, the Cameron-Martin translation theorem [CaMal], which are truly infinite dimensional in character. One begins by applying (3.3.9) where larger and larger finite subsets of a dense set of f-values are involved. Applications Next we turn to some examples illustrating the use of the formula given in Theorem 3.3.5. Proposition 3.3.7 Let t satisfying a < t < b be fixed. Then
Proof The function /(MI) := u\ is Lebesgue measurable and u\W\(t, u\) is Lebesgue integrable. It follows from Theorem 3.3.5 that Pt(x) is S\-measurable and Wiener integrable and that
where the last equality in (3.3.12) holds since the integrand in the second expression in (3.3.12) is an odd function. n Remark 3.3.8 (a) We know from Corollary 3.3.4 that P, is Si-measurable. We mentioned it in the proof above to emphasize that Wiener measurability is a part of what can be concluded from Theorem 3.3.5.
46
WIENER MEASURE
(b) In the notation of expected (or average) values, (3.3.11) becomes
Formula (3.3.13) says, in the language of stochastic processes, that the mean function of the Wiener process is 0. Our next application of Theorem 3.3.5 is a calculation of the "covariance function", Cov(x(t\,), x ( t 2 ) ) , of the Wiener process. We will carry out this computation in some detail but will often abbreviate similar arguments as we continue. Proposition 3.3.9 Let t1, ti belong to (a, b] with t\ = t2. Then F(x) := x(t1) • x(t2) is a Wiener integrable function on C^b and
Proof Suppose for definiteness that a < t\ < t1 < b. It suffices to show that
Let f(u1, u2) = u1 • u2- Since / is Lebesgue measurable and f(-)W2(f, integrable over R2, we have that F is Wiener integrable and
•) is Lebesgue
Applying these facts to the right-hand side of (3.3.16) and then using the Fubini theorem, we obtain
WIENER'S INTEGRATION FORMULA AND APPLICATIONS
47
where the last equality in (3.3.17) results from the fact that V2 K-»- V2e~v* is an odd function. Now using the earlier integration formula (3.2.11) as well as the formula (established in (4.7.2) of Appendix 4.7)
we can finally write
as claimed. A simpler calculation than the above establishes the following: Exercise 3.3.10 Let t be such that a < t R be a random variable (i.e. a measurable function) on £2. Recall that the distribution function FX of X is defined for every /3 e R by
If X, : £2 -> R, y = 1 , . . . , n, are n random variables, then the joint distribution function FXl *„ of X i , . . . , Xn is defined for all (A,..., $,) € R" by Fx1,...,xn(B1, . . . . ., Bn) := P({a> e S2 : X1(o>) < B , . . . , X«(o>) < #,}).
(3.3.26)
Our last two examples (for the present) will deal with the calculation of certain special distribution functions. These examples are related to one another and to a discussion which will soon follow. Proposition 3.3.17 Let a < t\ ) , . . . , Xn(a>)). Note that the distribution function is expressed simply in terms of the distribution measure: Fx(fi) = (PoX~l)(—oo, ft]. Also, since the intervals of the form (—00, /J], ft G K, are a generating class for B(R), the distribution function FX completely determines the distribution P o X~'. Similar remarks apply to the joint distribution; in particular,
The condition (3.3.30) for the independence of the random variables X\, . . ., Xn, is equivalent to the assertion that the distribution measures satisfy the relationship
(Equation (3.3.33) corresponds closely to the intuitive idea of the independence of X 1 , . . . , Xn, as the reader familiar with elementary probability theory can verify.) Axiomatic description of the Wiener process In books on probability theory or stochastic processes, for example [KalKar, L2, Wil], a "stochastic Wiener process" is often defined in terms of axiomatic properties. One choice of the list of properties is as follows: a (standard) Wiener process on a probability space (£2, A, P) is a family of random variables {xt : a < t < b] satisfying:
Our set-up provides one particular realization of these properties. For us, (£2, A, P) = (C^'b, S1, m) and x t (w) is replaced by x(t) where x plays a dual role as a path (or function on [a, b]) and as an element of £2. Properties (i) and (iv) in (3.3.34) are satisfied in our setting for every a> since our space £2 = CQ' consists entirely of continuous functions which are 0 at a. Propositions 3.3.18 and 3.3.17 assure us that properties (ii) and (iii) hold. Remark 3.3.20 (a) The time interval involved above need not be finite. The intervals [0, +00) and [0, b] are the most commonly used. (b) When alternatives to the list of axioms in (3.3.34) are given, they often include the covariance function. Proposition 3.3.9 and Remark 3.3.11 identified this function for us. 3.4 Nondifferentiability of Wiener paths Wiener paths are, with probability 1 (that is, a.s.), everywhere continuous but nowhere differentiable. In light of the remarks of Perrin and the earlier discussion of our mathematical model of physical Brownian motion, the nowhere differentiable nature of the
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WIENER MEASURE
Wiener paths is not a complete surprise. Recall, however, that it was long believed that a continuous function must be differentiable except possibly on some "small" subset of the interval in question. Of course, the famous example of Weierstrass [Fal, §11.1] showed that this is not true, but the result for Wiener paths shows that everywhere continuous, nowhere differentiable functions exist in great abundance. To put it somewhat facetiously, if instead of constructing his function, Weierstrass had selected at random a function from the "urn" C^'b of continuous functions, he would have been incredibly unlucky, in the sense of Wiener measure, to pick a function that was differentiable at even a single point. The result that we will actually prove in this section is that Wiener paths are, with probability 1, differentiable at most on a set of Lebesgue measure 0. This is easier to establish and makes the point sufficiently well for our purposes. An important corollary is that Wiener paths are almost surely not of bounded variation on any subinterval. We will see later that the nondifferentiability (a.s.) of the Wiener paths has implications for the theory of the Feynman integral. We begin by introducing some sets that will be useful to us. Given h > 0, 0 0 as m -> oo. Since xm e CYh(t,t'),xm(t) - xm(t') e [-h\t -t'y,h\t - t'\Y] for m = 1 , 2 , . . . . Hence x(t)-x(t') = lim [jc m (0--tm(«')]isalsointheclosedinterval[-/i|r-r / |> / ,ft|r-f'| )/ ]. m—»oo
Thus CYh (t, t') is a closed subset of CQ' & . Since the intersection of an arbitrary family of closed sets is again closed, it follows that CYh (t) and Cvh are also closed. D Lemma 3.4.2 For t ^ t', we have the inequality
NONDIFFERENTIABILITY OF WIENER PATHS
53
Proof We may assume without loss of generality that t' > t. We will also assume that a < t. The case a = t is easier as the reader can check. From (3.3.8) of Corollary 3.3.4 and (3.4.1) above, we see that
where
Hence (3.4.4) holds.
D
Lemma 3.4.3 If y > 3, then m(C^(t)) = 0 and so, of course, m(C£) = 0. Proof Let {tk} be a sequence of points in [a, b] which are distinct from / and are such that tk —*• t as k —> oo. By (3.4.2) and Lemma 3.4.2, we have
for all k. Since the last expression in (3.4.7) goes to 0 as A: -» oo, we see that m(Cyh(t))=0. D Although it is not the main point that we are driving towards, we can easily prove a corollary of Lemma 3.4.3 which is of some interest. Let 0 < y < 1. Recall that a function x : [a, b] -» R is said to be Holder continuous of order y if and only if there exists a positive constant h such that
for all t, t' in [a, b].
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Corollary 3.4.4 Let | < y b. Now each /„ is continuous as a function of x and t and so is certainly measurable. Let
G* is measurable since it is the set where a sequence of measurable functions has a finite limit. Since G c G*, to show that (m x Leb.)(G) = 0, it suffices to show that (m x Leb.)(G*> = 0. But, by the Fubini theorem,
where G*t = {x e C%b : (x, t) e G*} is the f-section of G*. We will show that m(G*) = 0 for all a < t < b, and from (3.4.14), it will then follow that (m x Leb.)(G*) = 0 as desired. Let a < t < b. Of course,
For every positive integer h, let
Clearly G* C (J°i, Kh(t). Hence to show that tn(Gf) = 0, it suffices to show that m(Kh(t)) = 0 for h = 1,2, ....But
Using (3.4.15) and Lemma 3.4.2, we see that for every n ,
Since the last expression in (3.4.16) converges to 0 as « —»• oo, it follows that m(Kh(t)) = 0 as desired.
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Now that we have the measurability of the function F given by (3.4.10), we can apply Fubini's theorem and write
But, in the notation of Lemma 3.4.6, F(x, t) — XD (•*)• Hence from (3.4.17) and Lemma 3.4.6 we obtain
From (3.4.18) it follows that for m-almost every x, fa F(x, t)dt = 0. Hence for malmost every x, F(x, 0=0 for Leb.-almost every t. It follows that for m-almost every x, the derivative x'(t) exists at most on a set of Lebesgue measure 0 as claimed in the theorem. n We can now easily prove the following corollary. Corollary 3.4.8 Wiener paths are, with probability 1, of unbounded variation on every subinterval of [a, b}. Proof If a function x is of bounded variation on any interval, then it is differentiable Lebesgue almost everywhere on that interval [Roy, Corollary 5.6, p. 104]. Hence
But the right-hand side of (3.4.19) has Wiener measure 0 and so the left-hand side does as well. D Remark 3.4.9 (a) The paths of physical Brownian motion are wildly varying, but it seems most unlikely that they have infinite variation on every time interval, no matter how small. (b) A good deal is known about the properties possessed, with probability 1, by Wiener paths. (See [Durr, Fre, Tayl3], for example.) Many of these results are mathematically appealing and definitely nontrivial. We will not need much more along these lines, however, and will discuss what we do need as the occasion arises. We just state (for d = 1) two further sample path results below. These theorems emphasize the gap between a typical Wiener path and the paths (or functions) which we ask students to deal with in elementary mathematics. Theorem 3.4.10 With probability 1, the local maxima of a Wiener pathform a countable dense set.
APPENDIX: CONVERSE MEASURABILITY RESULTS
57
Theorem 3.4.11 With probability \, the zero set, Z(jc) := {t € [a, b] : x(t) = 0}, of a Wiener path x is perfect (i.e. closed and without isolated points) and so is uncountable, but has Lebesgue measure 0. The exercise which we are about to state asks you to prove the last assertion in Theorem 3.4.11. Exercise 3.4.12 Show that, with probability 1, the zero set of a Wiener path has Lebesgue measure 0. Finally, we mention that the "fractal" properties of Wiener paths have been analyzed in great detail; see, e.g., ([benA], [Fal, §16.1], [leGa], [Mand, Chapter 25], [Tayll-3]). d-dimensional Wiener measure and Wiener process For notational simplicity, we have outlined in this chapter the construction of onedimensional Wiener measure m and studied, in particular, some of the sample path properties of the associated one-dimensional Wiener process. When d > 2, the construction of d-dimensional Wiener measure (still denoted by m)—and hence of the associated d-dimensional Wiener process, the mathematical model of Brownian motion in W1—is entirely analogous. Alternatively, the d-dimensional Wiener measure can be defined as the product of d copies of one-dimensional Wiener measure; of course, m (or its completion) is now viewed as a probability measure on (C^' , B(C^' )) (or on (C^'b, SO), where C^ = Co([a, b], Rd). Further, the d-dimensional Wiener process can be viewed as the product of d independent copies of the one-dimensional Wiener process. All the theorems of this chapter (with the exception of Theorems 3.4.10 and 3.4.11)— as well as of Chapters 4 and 5 below, where we mostly work in dimension d = 1 for notational convenience—remain valid when d > 2, with the obvious adjustments. In the rest of this book, the dimension should be clear from the context. 3.5 Appendix: Converse measurability results In this appendix, we will establish converses to measurability results given in Section 3.3. Fix ti, ...,tn such that a = to < t\ < • • • < tn < b and let Pr, ,n be given by (3.3.2). We saw in Proposition 3.3.1 that if B e B(E"), then P,"1 tn(B) e B(C^'b). In fact, the converse holds as well. Proposition 3.5.1 P^...,tn (B) e B(CQtb) if and only if B e B(E"). Proof We need only show that P^1 ..,,„(#) 6 B(C°'b) implies that B € B(W). Define //,,,...,,„ : M" -» C^b as the polygonal path in C^b with vertices at (a,0), (t\, MI), ..., (tn,un) and (b, un); that is,
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WIENER MEASURE
where HO = 0. Since t\,..., tn will be fixed throughout, we simplify notation and write P and H in place of />,, ,n and //»,,...,/„, respectively. The function H is easily seen to be continuous and so, by Exercise 3.2.10(b) and the present assumption, it follows that H~l (P-1 (B)) belongs to H(K"). Thus to finish the proof it suffices to show that
Let (MI, . . . , un) belong to the left-hand side of (3.5.2). Then //(MI, . . . , « „ ) 6 P~ 1 (fi)andso
However, (u\,...,un) — (H(u\,..., un)(t\),..., H(u\,..., «„)(*„)) and therefore, (MI , . . . , un) e B as we wished to show. Conversely, let (MI , . . . , un) e B. We need to show that H (MI , . . . , un) € P~l(B); that is,
This is so, however, under the present assumption since (u\,...,un) (//(MI, . ..,un)(t\),..., //(MI, .. .,Un)(tn)). Hence the result is established.
= D
The converse measurability results in Proposition 3.5.1 (as well as Theorem 3.5.2 below) will be used, in particular, in Proposition 4.2.12. We know by Corollary 3.3.4 that Pt~,l..,ta (E) is Wiener measurable provided that E is a Lebesgue measurable subset of R". Is the converse valid? This question was posed by Robert Cameron in his lectures at the University of Minnesota and was solved but never published by Fulton Koehler, a colleague who was attending Cameron's course. Koehler's argument was quite involved. The simpler proof below was given by Siegfried Graf after the first author mentioned Koehler's theorem during some lectures at the University of Erlangen in 1980. The key to Graf's proof is the fact that m, as a finite measure on the complete separable metric space C^'b, is regular (see, e.g., [Con, Proposition 8.1.10, p. 258]). Theorem 3.5.2 Let E c W. Then P,~^tn(E) is Wiener measurable if and only if E is Lebesgue measurable. Proof We need only prove one direction. Accordingly, assume that P,~l holds since K c P~l(E) implies that P(/T)C EmdsoP-l(P(K)) c ?-'(£). The sets P(AT) appearing on the left-hand side of the third equality in (3.5.3) are compact subsets of E since K is compact and P is continuous; the inequality < follows immediately. The inequality > follows from the fact that P~l(L) C P~~l(E) and from the first two equalities in (3.5.3). As we saw in the proof of Proposition 3.3.3, the image probability measure m o P"1 equals the probability measure v defined on B(R") by
Further, since v is mutually absolutely continuous with respect to Lebesgue measure on R", the a -algebra associated with the completion of v is precisely Leb.(Rn). We can now summarize the results of (3.5.3) by writing In order to complete the proof, it will be helpful to recall some facts about the inner measure v+ associated with v. By definition [Coh, p. 39], v* is given by for any subset A of R". Since v is regular, for F e Leb.(R") we have Putting (3.5.6) and (3.5.7) together, we obtain Combining (3.5.5) and (3.5.8), we see that for any subset E of R" for which P"1 (E) e <Si, we can write Now if E is such that P~l(E) e S}, then P~l(Ec) = P~l(E)c e Si also. (Here, Ec denotes the complement of E in R".) Hence, by (3.5.8), From (3.5.9) and (3.5.10) and the fact that m(Co'fo) = 1, we obtain
Further, every subset A of R" satisfies [Coh, Problem 7, p. 42] It follows from (3.5.11) and (3.5.12) (with A = E) that v*(E c ) = v*(Ec). But this implies [Coh, Proposition 1.5.5, p. 39] that Ec, and hence also E, is v-measurable. Since
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the v-measurable sets coincide with Leb.(R") as noted earlier, we see that E e Leb.(M") as desired. D We now easily obtain the following two corollaries. Corollary 3.5.3 There exist subsets ofC^b which are Wiener measurable but not Borel measurable. Proof Let a < t < b and take E e Leb.(R)\6(R). By Theorem 3.5.2 and Proposition 3.5.1, Pt~l(E) € Si\B(C%'b). n Corollary 3.5.4 There exist subsets ofC^'b which are not Wiener measurable. Proof Let a < t < b and take a subset E of K which is not Lebesgue measurable. Then, by Theorem 3.5.2, P~l (E) i S\. Q Remark 3.5.5 As far as we know, the first discussion of Theorem 3.5.2m the literature appeared in a paper of David Skoug [Sk] where Yeh-Wiener (or two-parameter Wiener) space was the primary object of interest. A quite general result along the lines of Theorem 3.5.2 has been established by Chang and Ryu [ChanRyl]. 3.6 Appendix: B(X x Y) = B(X) ® B(Y) Let X and Y be topological spaces. Throughout this discussion we assume that X x Y has the product topology. Proposition 3.6.1 Let X and Y be topological spaces. Then
Proof Let n\ and n2 be the projection maps from X x Y onto X and Y respectively. Clearly, it\ and n2 are continuous. (Indeed, the product topology on X x Y can be described as the weakest topology on X x Y making n\ and n2 continuous.) Hence n\ and 7T2 are Borel measurable. SinceB(X)®B(Y) := a([BixB2 : BI e S(X)andB 2 e B(y)}),(3.6.1)willfollow if we show that BI x B2 e B(X x Y) for every B\ e B(X) and B2 e B(Y). Accordingly, let BI e B(X) and let B2 e B(Y). Then BI x Y = n^(Bi) and X x B2 = n~l(B2) are both in B(XxY). Therefore BI x B2 = (Bi xY)C\(XxB2) =srf 1 (Bi)nn- 2 ~ 1 (fl 2 ) e B(X x Y), as desired. D Proposition 3.6.2 If X and Y are second countable topological spaces, then
Proof By Proposition 3.6.1, it suffices to show that
Since X and Y are second countable, there are countable bases [U\, U2,...} and {Vj, V2,...} for X and Y respectively. It is well known then (and easy to prove) that
APPENDIX: B(X x Y) = B(X) ® B(K)
61
{£/y x Vk : j = 1, 2 , . . . , k = 1, 2 , . . . } is a basis for the product topology on X x F. Now we write
where only the first containment in (3.6.4) needs commenting on. Since {Uj x Vk} is a base for the product topology on X x Y, every open subset of X x Y can be written as a countable union of sets of the form Uj x Vk.lt follows that
and so the proof is complete. Corollary 3.6.3 IfX and Y are separable metric spaces, then
Proof It is well known that every separable metric space is second countable and so the claim made here follows immediately from Proposition 3.6.2. D Remark 3.6.4 The equality B(X) B(F) = B(X x F) is in fact true for Souslin spaces, a much larger class of topological spaces (see, e.g., [Coh] or [Schwl]). However, it is not true for arbitrary topological spaces X and Y. Some further information related to the topic of this appendix can be found in [Bau, Problem 2, p. 384].
4 SCALING IN WIENER SPACE AND THE ANALYTIC FEYNMAN INTEGRAL Notre imagination se lassera plutot de concevoir que la nature de fournir. [Our imagination will rather grow weary of conceiving than nature of providing.] Paul Levy, citing Leibniz, and discussing the complexity of Brownian paths. (French text quoted in [benA, p. 8].) Mathematicians, however, have well understood the lack of rigor of such considerations, said to be geometric, and how, for example, it is childish to want to prove, by simply drawing a curve, that every continuous function admits a derivative. If the differentiable functions are the simplest, the easiest to deal with, they are nevertheless the exception; or if one prefers a geometric language, the curves which do not have a tangent are the rule, and the nicely regular curves, such as the circle, are quite interesting, but very particular cases. Jean Perrin, 1913 [Per, p. 25] There are very interesting mathematical problems involved in the attempt to avoid the subdivision and limiting processes. Some sort of complex measure (italics added) is being associated with the space of functions x(t). Finite results can be obtained under unexpected circumstances because the measure is not positive everywhere, but the contributions from most of the paths largely cancel out. Richard P. Feynman, 1948 [Fey2, p. 372]
Let CT > 0. The simple transformation of scale change, x i-»- ax, in CQ'* has some surprisingly pathological features with respect to Wiener measure m. When a is taken as fixed, as it is in most situations, these features do not cause any particular concerns. However, one of the approaches to the Feynman integral which will most interest us involves analytic continuation in the scaling parameter; in that setting, scalings by all positive numbers cr will be involved and we will need to exercise caution. The basis of the results in Sections 4.2-4.5 is Levy's famous quadratic variation theorem, established in Section 4.1. This theorem is well known to probabilists but the content of Sections 4.2^.5—which discuss consequences of this result for scaling—is less well known. In Section 4.5, we define the scalar-valued analytic Feynman integral and explain the relevance of the earlier material for this topic. In particular, we will see in Example 4.5.3 that the usual equivalence relation for functions on Wiener space (i.e., equality m-a.e.) is not adequate for our purposes. A more refined equivalence, introduced in Definition 4.5.5, is shown to have the appropriate properties in Theorem 4.5.7. We discuss in Section 4.6 the nonexistence of Feynman's "measure "; specifically, we show in Theorem 4.6.1 that the natural analogue of Wiener measure with complex variance parameter a (a = X ~ 2) is not a countably additive (complex) measure. Section 4.6
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63
fits well with the rest of this chapter but is also directly relevant to Chapter 7, where the Feynman integral will be introduced heuristically. For notational simplicity, we will assume that d = 1 throughout this chapter although all the results presented here are valid in any dimension d > 1.
4.1 Quadratic variation of Wiener paths The key to the results of this section is Levy's 1940 theorem on the quadratic variation of Wiener paths ([Levl], [Lev2, Chapter VI)). We w^ll prove the special case which was discovered later (1947) but independently by Cameron and Martin [CaMa3] in connection with their investigation of change of variable formulas in Wiener space. Our proof, an "analyst's proof", will essentially follow [CaMa3]. For a "probabilist's proof" of the general result, see, for example, [Tuc, p. 243]. The quadratic variation result is of independent interest and we will discuss some aspects of it which have no particular relevance to the Feynman integral. Given a partition n of [a,b],a = to(x) = 0 since
as n —> oo. Theorem 4.1.2 Wiener paths x e C^b satisfy formula (4.1.2) almost surely.
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Proof We restrict attention to the case [a, b] = [0,1]. This will simplify the notation without making any essential difference in the proof. We will begin by showing that
from which it will immediately follow that
Since [a, b] = [0, 1], given any positive integer k, we have tj = {, j = 0, 1 , . . . , k and
Let
We know from Proposition 3.3.17 that x(tj)-x(tj-\) It follows from this and (4.1.5) that
~ N(0, tj-tj-i), j = 1 , . . . , k.
To evaluate Ik it remains to calculate /c«,* S^(x)dm(x). Using Theorem 3.3.5 and the fact that tj — tj-\ = l/k, j = 1,..., k, we can write
QUADRATIC VARIATION OF WIENER PATHS
where MQ '•— 0- The change of variables vj —
J
65
/TJT[ , j = 1, • • •, k, has Jacobian
(l/k)k/2, and so we obtain
where the next to last equality comes from formulas (4.7.1)-(4.7.3) of Appendix 4.7 to this chapter. Combining (4.1.6), (4.1.7) and (4.1.9), we now see that
as claimed in (4.1.3). It immediately follows from (4.1.3) that Sk — l —>• 0 in the L2-norm on Wiener space, but this does not yet give us the conclusion we seek since L2-convergence does not in general imply almost sure convergence. However, "fast enough" L2-convergence does imply convergence almost surely; the rest of the proof will consist of showing this in our present setting. (A general result of this type will be stated in Exercise 4.1.3 below.) Formula (4.1.3) yields (4.1.4) immediately as observed earlier. Now let
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We must have
since, if not,
which contradicts (4.1.4). Let
Then, by (4.1.12),
where K is twice the sum of the convergent geometric series £]tlo ,3 Lt • Now for x e C2'*\Fn = n£ln ££, where Eck denotes the complement of Ek in CQ'*, it follows from (4.1.11) that | S2* Cx) - 11 < 2~/:/3 for it = n, n + 1,.... Hence, if there is an n such that x £ Fn, then limjt_>oc ^W = !• Thus limt_>.oo ^(jc) = 1 except possibly on n£L, F&. Thus it suffices to show that tnCn^lj F*) = 0. But, for every n, we have, by (4.1.13),
The result follows from (4.1.14) since (£/2 n / 3 ) ->• 0 as n -> oo. Exercise 4.1.3 Let (n, A, P) be a probability space and let X, Xn, n = 1, 2, . . . , be (R-valued) random variables on £2 such that
Show that X n ((w) ->• X (w) P-a.s. Can you get the same conclusion if P is just required to be a finite measure? What if the number 2 in (4.1.15) is replaced by some number r >0? Remark 4.1.4 (a) If we had given Exercise 4.1.3 before the proof of Theorem 4.1.2, we would have been done with the proof of Theorem 4.1.2 as soon as (4.1.4) was
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established. Note, however, that Exercise 4.1.3 does not allow us to conclude from (4.1.3) that Sk -> 1 a.s. (b) In the notation of probability theory, (4.1.15) is written more compactly as (c) The results in Exercise 4.1.3, in our opinion, deserve to be better known than they seem to be. Exercise 4.1.5 Use Theorem 4.1.2 to show that the set of functions in C^'b which are of bounded variation on [a, b] is a set of m-measure 0. (Corollary 3.4.8 already gave us a stronger conclusion than is provided by this exercise, but it is somewhat informative to see that Theorem 4.1.2 is related to these facts.) Next we state without proof the stronger quadratic variation result which we mentioned earlier and which is due to Levy. (See [Levl], [Lev2, Chapter VI].) Theorem 4.1.6 Let {FI^} be any nested sequence of partitions of [a, b] whose norm approaches 0. Then Wiener paths x almost surely satisfy
where Snk is given by (4.1.1). The nature of the sums in (4.1.1) makes the term "quadratic variation" seem reasonable. However, a more straightforward extension of the usual definition of total variation to the quadratic case would be
where x is a function from [a, b] to R and where the supremum is taken over all partitions n of [a, b], a = to < t\ < • • • < tk — b. In light of Theorem 4.1.6, it may seem surprising that Wiener paths x almost surely satisfy Quad.Var.(jc) = +00. This follows from the work in [Fre, p. 48]. 4.2 Scale change in Wiener space Much of the next four sections of this chapter is adapted from the 1979 paper of Johnson and Skoug [JoSk7] which discusses and expands on results of Cameron and Martin [CaMa3] and Cameron [Ca2], as well as relates them to various results in the more recent literature. We turn now to our main concern in this chapter, the implications of Theorem 4.1.2 for scale change transformations in Wiener space. Given a > 0, let
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where
We will use m \ rather than m, at least in the remainder of this chapter, to denote (standard) Wiener measure on C^'b. Let ma (a > 0) be the image measure
(By this, we mean that ma is the image measure of m1 by the continuous map <pa : CQ'& -> CQ' given by (pa(x) = ax.) The measure ma is certainly defined on B(C^' ); in fact, ma(B) — mi(ff~ 1 fl), for B e B(C^'b). (Before long, we will consider its completion as well.) In probabilistic terms, the space (Co' 6 ,nv) is a realization of the Wiener process with variance parameter a2; that is, axioms (i)-(iv) of (3.3.34) are unchanged except that (iii) becomes
The properties (i), (ii), (iii)CT and (iv) are readily verified for the space (CQ'*, ma); in fact, (i) and (iv) are immediate simply because C^'b is our probability space. We will show in Appendix 4.8 to this chapter that (iii)CT holds. We go next to some elementary but rather informative propositions. The first of these tells us that the sets £2ff are disjoint from one another but simply related and that ma is concentrated on £2CT. Proposition 4.2.1 (i) The set Q,j is Borel measurable for every a > 0. (ii) Given any a\, ai> 0, we have
and, in particular, for every a > 0,
(iii) ma(Qa) = 1 for every a > 0. (iv) If a\ ^ 02 (a\, 02 > 0), then J2ffl D Qa2 — 0 so that the measures mai and nv2 are mutually orthogonal. Proof (i) $2" is continuous and hence Borel measurable for every n. It follows from (4.2.1) that f2CT is Borel measurable for every a > 0. (ii) This is an immediate consequence of (4.2.1) and (4.2.2). (iii) By Theorem 4.1.2, m1(£2i) = 1. Using this fact as well as (4.2.5) and (4.2.3), we can write
(iv) This follows immediately from (4.2.1) and (iii).
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Remark 4.2.2 In light of (iii), we see that £la is a set of full ma-measure. We will say that nv is concentrated on £la. However, we will not refer to &a as the "support" of trie, since there is a rather standard definition of support [Schwl, p. 40] to which the present situation does not correspond. (In fact, Supp(ma) = C^'b for every a > 0.) The following definitions will be important to us as we continue this chapter and also later in connection with the analytic Feynman integral. Definition 4.2.3 A subset A of CQ ' is said to be scale-invariant measurable provided a A € Si for all a > 0. A scale-invariant measurable set N is said to be scale-invariant null provided that m\ (aN) = Ofor all a > 0. A property which holds for all x e CQ'& except for a scale-invariant null set is said to hold scale-invariant almost everywhere (briefly, s-a.e.). The classes of scale-invariant measurable and scale-invariant null sets will be denoted S and J\T, respectively. Remark 4.2.4 The definition just given may well seem most peculiar to many readers. If A e S\, one would expect it to follow that, for example, 2A € 5i. We will see below, however, that this does notfollow; multiplication by 2 is a nonmeasurable transformation on Wiener space (Cg' , Si, mj)Definition 4.2.5 We let Sa be the a -algebra obtained by completing the measure space (Cp1*, B(CO'*), nv) and we letMa denote the class of ma-null sets. We show in our next proposition that Sa = a S1 and that nv(,E) = m1(a~ l E) for every E e Sa. Proposition 4.2.6 (i) N e No if and only if a~lN € N1; equivalently, Na = oN1. (ii) E & Sa if and only if a~lE 6 S1; equivalently, Sa = a S1. (iii) We have ma(E) = mi (a~l E) for every E e Sa. Proof (i) We first show that N € Na implies that a ~l N € A/i. Let N € Ma. Then there exists a Borel set M such that N c M and mCT (M) = 0. By definition of nv. mo (M) = m\(a~lM). Thus a~lM is a Borel set which is mi-null. But a~lN c &~1M and so a ~l N e .A/i as claimed. A similar argument, which we leave to the reader, shows that if o--lN eM.thenW eMa. Next we show that Ma C a MI. Let N e Ma. By the above, a~lN e A/i- Thus N = a(a~lN) e aj\f\. Similarly, one can easily show that aN1 c A/"CT. (ii) We will carry out the proof that a~lE e S1 implies that E e Sa and leave the rest to the reader. Accordingly, let a ~l E e S1. Then there exists a Borel set B and N e M such that a~lE = BUN.By (i), crN € 0. Proposition 4.2.7 (i) S is a a-algebra; in fact, S = C\a>o^"-
di)M=r}a>0M 0. Thus, by (ii) of Proposition 4.2.6, A = a(cr~l A) e crS\ = Sa for everyCT> 0. Hence 5 c no SaConversely, suppose that A e f\>o ^ • Then A e Sa = crSi for every a > 0. Thus a"1 A € .Si for every a > 0. But then A e 5. Hence n^o^V £ 5. It now follows that 5 = Ho-s-o'Sff an 0. (ii) N € M if and only if Nrt£la e Afa for every a > 0. Proof (i) Suppose that E e S and letCT> 0 be given. By Proposition 4.2.7, E € Sa and, by Proposition 4.2.1, £2a is a Borel set. Therefore, E n £2CT € «Scr. Conversely, suppose that E r\Sla e Sa for every cr > 0. To show that E e 5 it suffices, by Proposition 4.2.7, to show that E £ Sa for every a > 0. But, by (iii) of Proposition 4.2.1, Co'fe\atf is nv-null and so E *= (£nS2 < T )U(£n(Co'*\n ( r )) e £„. We leave the proof of (ii) to the reader. D Our next result shows rather well what scale-invariant measurable sets and scaleinvariant null sets are like and how they compare to Wiener measurable sets and Wiener null sets, respectively. Theorem 4.2.10 (i) £ e 5 if and only if E has the form
where each Ea is an ma-measurable subset of £2CT and L is an arbitrary subset of c b o' \ Ua>o ®a- Further, for E written as in (4.2.6), we have
for all a > 0. (ii) N e A/* if and only ifN has the form
where each Nf, is an m^-null subset of &„ and L is an arbitrary subset of C*\U>o^-
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Proof (i) Suppose that E is in 5. Let Ea :— E n £2CT for every a > 0 and let £ := E n (CQ'*\ Uo ^2 0. Also it is clear that L is a subset of Cg'fc\ Ucr>o ^- The decomposition (4.2.6) now follows since, by (iv) of Proposition 4.2.1, the union
is a painvise disjoint union. Conversely, suppose that E has the form (4.2.6). To show that E e S it suffices by Proposition 4.2.9(i) to show that E D Sla e Sa for every CT > 0. But, from (4.2.6), E n fiff = Ea and Ea 6 Sa. Hence E is in <Sff as desired. The formula nta(£') = m^ (Ea) follows from the fact, established in Proposition 4.2.1, that mCT is concentrated on Qa. (ii) Suppose that N e A/". Then N e S and so can be written in the form N = (Uo Na)\JL, where each Na is in 5a and L c Cg' fe \ U(7>o *V We only need to show that mCT (Na) = 0. But N e N implies, by Proposition 4.2.7, that N e Ma. Hence, by (4.2.7), 0 = ma(N) = m.a(Na) as desired. The converse of (ii) follows from the representation (4.2.8) and from the formula (4.2.7). D Remark 4.2.11 (a) It is noted in the proof of Theorem 4.2.10 that the union on the right-hand side of (4.2.9) is a painvise disjoint union. It follows that the union in (4.2.6) is a painvise disjoint union. (b) Theorem 4.2.10 shows strikingly that there are many more Wiener measurable sets (i. e. sets in S\) than there are scale-invariant measurable sets. It is quite clear that a set E is Wiener measurable if and only if it has the form E\ UL, where E\ is an mi-measurable subset of £l\ and L is an arbitrary subset o/Uo-so^i ^CT U (^o \Ucr>o^< 7 )- The reader should compare this with (4.2.6). The set E is Wiener null if and only ifE] in the decomposition E = E\ U L is an mi-null subset of£l\. Our next proposition compares the a-algebras B(C^' ), S and Sag for anyCTQ> 0. Proposition 4.2.12 For every a0 > 0, we have B(£^b) CSC
Sao.
Proof The containments are clear since B(C^'b) c Sa for every er > 0 and since, by Proposition 4.2.7, S = (~}a>0Sa. Let a < t < b and let G e Leb.(R)\S(R). By Proposition 3.5.1, Pt~l(G) = (x 6 C0'fe : x(t) € G} i B(Cg'b). However, for every a > 0, multiplication by a is a Leb.(R) - Leb.(R) measurable map from K to E. Hence aG € Leb.(E) for every a > 0. Since aP~l(G) = Pt~l(aG), it follows from Theorem 3.5.2 that aP,~l(G) e Si for every a > 0. Thus P,~l(G) 6 S, and so now we know that P,~l(G) 6 <S\B(C0'*). Now let G be a subset of K which is not Lebesgue measurable and take a\ > 0 such that a\ ± cr0. By Theorem 3.5.2, Pt~l(G) £ Si. Therefore,
