DOCSLIB.ORG

Functions in the Fresnel Class of an Abstract Wiener Space

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Functions in the Fresnel Class of an Abstract Wiener Space J. Korean Math. Soc. 28(1991), No. 2, pp. 245-265 FUNCTIONS IN THE FRESNEL CLASS OF AN ABSTRACT WIENER SPACE J.M. AHN*, G.W. JOHNSON AND D.L. SKOUG o. Introduction Abstract Wiener spaces have been of interest since the work of Gross in the early 1960's (see [18]) and are currently being used as a frame­ work for the Malliavin calculus (see the first half of [20]) and in the study of the Fresnel and Feynman integrals. It is this last topic which concerns us in this paper. The Feynman integral arose in nonrelativistic quantum mechanics and has been studied, with a variety of different definitions, by math­ ematicians and, in recent years, an increasing number of theoretical physicists. The Fresnel integral has been defined in Hilbert space [1], classical Wiener space [2], and abstract Wiener space [13] settings and used as an approach to the Feynman integral. Such approaches have their shortcomings; for example, the class offunctions that can be dealt with is somewhat limited. However, these approaches also have some distinct advantages including the fact, seen most clearly in [14], that several different definitions of the Feynman integral exist and agree. A key hypothesis in most of the theorems about the Fresnel and Feynman integrals is that the functions involved are in the 'Fresnel class' of the Hilbert space being used. Theorem 2 below is a general result insuring membership in :F(B), the Fresnel class of an abstract Wiener space B. :F(B) consists of all 'stochastic Fourier transforms' of certain Borel measures of finite variation. (Precise definitions will be given in Section 1 below.) A surprising number ofconsequences follow from Theorem 2; indeed, the rest of this paper after Theorem 2 consists of eleven corollaries. Corollaries 3 through 7 are in the setting of classical Wiener space and show, in conjunction with Corollary 1, that various functions of Received August 23, 1990. *This research was supported by the Ministry of Education, Korea (1989). 245 246 J.M. Ahn, G.W. Johnson and D.L. Skoug interest in quantum mechanics, are in the Fresnel class of the classical abstract Wiener space. We should mention that all of the corollaries of [6] could have been included in this paper although they are not. Corollaries 9, 10, and 11 in Section 4 illustrate how Theorem 2 can be applied to abstract Wiener spaces and associated Gaussian stochastic processes other than the classical Wiener process. The processes in­ volved are, respectively, the n parameter Wiener process, the Brownian bridge, and a family of processes described in [20, pp.7-9]. We include, without proof, the necessary background information concerning these processes. Corollary 8, the first corollary of Section 4, is in the set­ ting of the classical abstract Wiener space but is very different from the other corollaries in this paper and from all of the corollaries in the earlier work [4;6]. We close this introduction by remarking that the results of [4] which are in the Hilbert space setting of [1] are less directly comparable to the present paper than are the results of [6]. The Hilbert space and Wiener space settings are closely related however as was shown in [10] and [13] in the classical and abstract Wiener space settings, respectively. 1. Definitions and preliminaries Let H be a real separable infinite dimensional Hilbert space with inner product h,} and norm 1·1. Let p be the cylinder set measure on H defined by p(A) = (21r)-n/2 Lexp {_1~2} dx, where A = P-1(E), E is a Borel set in the image of an n-dimensional projection P in H and dx is Lebesgue measure in P(H). A norm 1/·1/0 on H is called measurable if for every € > 0 there exists a finite dimensional projection Po such that p({x EH: IIpxllo > €}) < € whenever P is a finite dimensional projection orthogonal to Po. It is known (see [18]) that H is not complete with respect to .11 . 110' Let B denote the completion of H with respect to 11·110' Let i denote the natural injection from H into B. The adjoint operator i* is one-to-one and maps B* continuously onto a dense subset of H*. By identifying H* with H and B* with i*B*, we have a triple B* c H* == H C B and (x,y) = (x,y) for all x in H any y in B*, where (".) denotes the FUnctions in the Fresnel class of an abstract Wiener space 247 natural dual pairing between B and B*. By a well-known result of Gross, poi-I has a unique countably additive extension v to the Bore! O'-algebra B(B) of B. The triple (B,H,v) is called an abstract Wiener space, and the Hilbert space H is called the generator of (B,H,v). Given an arbitrary real separable Banach space B with norm 11·110, and a countably additive Gaussian measure von (B, B(B», it is known that there exists a Hilbert space H such that H ~ B, B is the completion of H under 11 ·110' the norm 11·110 is a {neasurable norm on H with respect to the canonical normal distribution p and v = po i-I. For more details, see Kuo [18]. Let {ej} denote a complete orthonormal system on H such that the e;'s are in B*. For each h E H and x E B, define a stochastic inner product (.,.)- between B and H as follows: if the limit exists otherwise. It is well-known that for every h E H, (x, h) exists for v-almost all x in B and is a Borel measurable function on B having a Gaussian 2 distribution with mean zero and variance Ih1 • Furthermore, it is easy to show that (x, hf equals (x, h) almost everywhere on B if h E B*. Note that if both h and x are in H, then (x, h) = {x, h}. Let M(H) denote the space of complex-valued countably additive measures defined on B(H), the Borel class of H. A complex-valued countably additive measure p necessarily has finite total variation IIpll [19, p.119]. Under the norm 11·11 and with convolution as multiplication, M(H) is a commutative Banach algebra with identity [1, p.17]. Given two complex-valued functions F and G on B, we say that F = G s-almost sw-ely (F ~ G) iffor ~h a > 0, v( {x E B IP(ax) 1= G(ax)}) = 0, where v is an abstract Wiener measure. For a function F on B, we will denote by [F] the equivalence class offunetions G which are equal to F s-a.e., i.e., [F] = {G IG = F s-a.s.} = {GI G ~ F}. We will now introduce the Fresnel class :F(B) offunctions on B. The space :F(B) is defined as the space of all stochastic Fourier transforms of elements of M(H), i.e., :F(B) = ([F] IF(x) = fH exp(i(x, h))dp(h), x E B, p E M(H)}. As is customary, we will often identify a nmc­ tion with its s-equivalence class and think of :F(B) as a collection of functions on B rather than as a collection of equivalence classes. 248 J.M. Ahn, G.W. Johnson and D.L. Skoug It is well-known (see [13, 14]) that :F(B) is a Banach algebra with the norm IIFII = IIplI and the mapping p 1-+ F is a Banach algebra isomorphism, where p is related to F by F(x) = Lexp{i(x, hf}dp(h), x E B, pE M(H). ~ Although it will not be a direct concern of this paper, we note that for each F in :F(B), the analytic Feynman integral, the sequential Feynman integral and the Fresnel integral of F all exist [14J and are given by fHexp{-ilhI2}dp(h). 2. Functions in :F(B): a general theorem We begin by quoting an Unsymmetric Fubini Theorem from [4, pp.310-311] which we use to prove our main theorem. THEOREM 1. Let (t'; Y, 77) be a measure space, where 77 is either a nonnegative, u-finite measure or a complex-valued measure, and let (Z,Z) be a measurable space. For 177I-a.e. y, let v y be a complex­ valued measure on (Z,Z). Suppose that for any E in Z, vy(E) is Y-measurable function of y. Then . A. for any E in Y x Z, vy(E(Y) is Y-measurable where E(y) is the y-section ofE, and B. for any bounded, complex-valued, Y x Z-measurable function 8 on Y X Z, fz 8(y, z)dvy(z) is a Y-measurable functions ofy. Ifwe add the assumption that IIVyll ::; hey), where h is in Ll(y,Y, 1771), and define p on Y x Z by the formula (2.1) then c. P is a complex-valued, countably additive measure on Y X Z with IIplI ::; IIhlh, and D. H8(y, z) is bounded andYxZ-measurable, then fz 8(y, z)dvy(z) is in L 1 (y, Y, \771), and we have Functions in the Fresnel class of an abstract Wiener space 249 (2.2) i [l O(y,Z)dVy(Z)] dTJ(Y) = ixz O(y,z)dp(y,z). We are now ready to establish the general theorem discussed in the introduction. This theorem shows (as we will see in the next section) that :F( B) contains many functions ofinterest in connection with Feyn­ man integration theory and quantum mechanics.
Load more

Recommended publications
  • A. the Bochner Integral
    A. The Bochner Integral This chapter is a slight modification of Chap. A in [FK01]. Let X, be a Banach space, B(X) the Borel σ-field of X and (Ω, F,µ) a measure space with finite measure µ. A.1. Definition of the Bochner integral Step 1: As first step we want to define the integral for simple functions which are defined as follows. Set n E → ∈ ∈F ∈ N := f :Ω X f = xk1Ak ,xk X, Ak , 1 k n, n k=1 and define a semi-norm E on the vector space E by f E := f dµ, f ∈E. To get that E, E is a normed vector space we consider equivalence classes with respect to E . For simplicity we will not change the notations. ∈E n For f , f = k=1 xk1Ak , Ak’s pairwise disjoint (such a representation n is called normal and always exists, because f = k=1 xk1Ak , where f(Ω) = {x1,...,xk}, xi = xj,andAk := {f = xk}) and we now define the Bochner integral to be n f dµ := xkµ(Ak). k=1 (Exercise: This definition is independent of representations, and hence linear.) In this way we get a mapping E → int : , E X, f → f dµ which is linear and uniformly continuous since f dµ f dµ for all f ∈E. Therefore we can extend the mapping int to the abstract completion of E with respect to E which we denote by E. 105 106 A. The Bochner Integral Step 2: We give an explicit representation of E. Definition A.1.1.
    [Show full text]
  • Patterns in Random Walks and Brownian Motion
    Patterns in Random Walks and Brownian Motion Jim Pitman and Wenpin Tang Abstract We ask if it is possible to find some particular continuous paths of unit length in linear Brownian motion. Beginning with a discrete version of the problem, we derive the asymptotics of the expected waiting time for several interesting patterns. These suggest corresponding results on the existence/non-existence of continuous paths embedded in Brownian motion. With further effort we are able to prove some of these existence and non-existence results by various stochastic analysis arguments. A list of open problems is presented. AMS 2010 Mathematics Subject Classification: 60C05, 60G17, 60J65. 1 Introduction and Main Results We are interested in the question of embedding some continuous-time stochastic processes .Zu;0Ä u Ä 1/ into a Brownian path .BtI t 0/, without time-change or scaling, just by a random translation of origin in spacetime. More precisely, we ask the following: Question 1 Given some distribution of a process Z with continuous paths, does there exist a random time T such that .BTCu BT I 0 Ä u Ä 1/ has the same distribution as .Zu;0Ä u Ä 1/? The question of whether external randomization is allowed to construct such a random time T, is of no importance here. In fact, we can simply ignore Brownian J. Pitman ()•W.Tang Department of Statistics, University of California, 367 Evans Hall, Berkeley, CA 94720-3860, USA e-mail: [email protected]; [email protected] © Springer International Publishing Switzerland 2015 49 C. Donati-Martin et al.
    [Show full text]
  • Arxiv:Math/0409479V1
    EXCEPTIONAL TIMES AND INVARIANCE FOR DYNAMICAL RANDOM WALKS DAVAR KHOSHNEVISAN, DAVID A. LEVIN, AND PEDRO J. MENDEZ-HERN´ ANDEZ´ n ABSTRACT. Consider a sequence Xi(0) of i.i.d. random variables. As- { }i=1 sociate to each Xi(0) an independent mean-one Poisson clock. Every time a clock rings replace that X-variable by an independent copy and restart the clock. In this way, we obtain i.i.d. stationary processes Xi(t) t 0 { } ≥ (i = 1, 2, ) whose invariant distribution is the law ν of X1(0). ··· Benjamini et al. (2003) introduced the dynamical walk Sn(t) = X1(t)+ + Xn(t), and proved among other things that the LIL holds for n ··· 7→ Sn(t) for all t. In other words, the LIL is dynamically stable. Subse- quently (2004b), we showed that in the case that the Xi(0)’s are standard normal, the classical integral test is not dynamically stable. Presently, we study the set of times t when n Sn(t) exceeds a given 7→ envelope infinitely often. Our analysis is made possible thanks to a con- nection to the Kolmogorov ε-entropy. When used in conjunction with the invariance principle of this paper, this connection has other interesting by-products some of which we relate. We prove also that the infinite-dimensional process t S n (t)/√n 7→ ⌊ •⌋ converges weakly in D(D([0, 1])) to the Ornstein–Uhlenbeck process in C ([0, 1]). For this we assume only that the increments have mean zero and variance one. In addition, we extend a result of Benjamini et al.
    [Show full text]
  • The Gaussian Radon Transform and Machine Learning
    THE GAUSSIAN RADON TRANSFORM AND MACHINE LEARNING IRINA HOLMES AND AMBAR N. SENGUPTA Abstract. There has been growing recent interest in probabilistic interpretations of kernel- based methods as well as learning in Banach spaces. The absence of a useful Lebesgue measure on an infinite-dimensional reproducing kernel Hilbert space is a serious obstacle for such stochastic models. We propose an estimation model for the ridge regression problem within the framework of abstract Wiener spaces and show how the support vector machine solution to such problems can be interpreted in terms of the Gaussian Radon transform. 1. Introduction A central task in machine learning is the prediction of unknown values based on `learning' from a given set of outcomes. More precisely, suppose X is a non-empty set, the input space, and D = f(p1; y1); (p2; y2);:::; (pn; yn)g ⊂ X × R (1.1) is a finite collection of input values pj together with their corresponding real outputs yj. The goal is to predict the value y corresponding to a yet unobserved input value p 2 X . The fundamental assumption of kernel-based methods such as support vector machines (SVM) is that the predicted value is given by f^(p), where the decision or prediction function f^ belongs to a reproducing kernel Hilbert space (RKHS) H over X , corresponding to a positive definite function K : X × X ! R (see Section 2.1.1). In particular, in ridge regression, the predictor ^ function fλ is the solution to the minimization problem: n ! ^ X 2 2 fλ = arg min (yj − f(pj)) + λkfk ; (1.2) f2H j=1 where λ > 0 is a regularization parameter and kfk denotes the norm of f in H.
    [Show full text]
  • Change of Variables in Infinite-Dimensional Space
    Change of variables in infinite-dimensional space Denis Bell Denis Bell Change of variables in infinite-dimensional space Change of variables formula in R Let φ be a continuously differentiable function on [a; b] and f an integrable function. Then Z φ(b) Z b f (y)dy = f (φ(x))φ0(x)dx: φ(a) a Denis Bell Change of variables in infinite-dimensional space Higher dimensional version Let φ : B 7! φ(B) be a diffeomorphism between open subsets of Rn. Then the change of variables theorem reads Z Z f (y)dy = (f ◦ φ)(x)Jφ(x)dx: φ(B) B where @φj Jφ = det @xi is the Jacobian of the transformation φ. Denis Bell Change of variables in infinite-dimensional space Set f ≡ 1 in the change of variables formula. Then Z Z dy = Jφ(x)dx φ(B) B i.e. Z λ(φ(B)) = Jφ(x)dx: B where λ denotes the Lebesgue measure (volume). Denis Bell Change of variables in infinite-dimensional space Version for Gaussian measure in Rn − 1 jjxjj2 Take f (x) = e 2 in the change of variables formula and write φ(x) = x + K(x). Then we obtain Z Z − 1 jjyjj2 <K(x);x>− 1 jK(x)j2 − 1 jjxjj2 e 2 dy = e 2 Jφ(x)e 2 dx: φ(B) B − 1 jxj2 So, denoting by γ the Gaussian measure dγ = e 2 dx, we have Z <K(x);x>− 1 jjKx)jj2 γ(φ(B)) = e 2 Jφ(x)dγ: B Denis Bell Change of variables in infinite-dimensional space Infinite-dimesnsions There is no analogue of the Lebesgue measure in an infinite-dimensional vector space.
    [Show full text]
  • Graduate Research Statement
    Research Statement Irina Holmes October 2013 My current research interests lie in infinite-dimensional analysis and geometry, probability and statistics, and machine learning. The main focus of my graduate studies has been the development of the Gaussian Radon transform for Banach spaces, an infinite-dimensional generalization of the classical Radon transform. This transform and some of its properties are discussed in Sections 2 and 3.1 below. Most recently, I have been studying applications of the Gaussian Radon transform to machine learning, an aspect discussed in Section 4. 1 Background The Radon transform was first developed by Johann Radon in 1917. For a function f : Rn ! R the Radon transform is the function Rf defined on the set of all hyperplanes P in n given by: R Z Rf(P ) def= f dx; P where, for every P , integration is with respect to Lebesgue measure on P . If we think of the hyperplane P as a \ray" shooting through the support of f, the in- tegral of f over P can be viewed as a way to measure the changes in the \density" of P f as the ray passes through it. In other words, Rf may be used to reconstruct the f(x, y) density of an n-dimensional object from its (n − 1)-dimensional cross-sections in differ- ent directions. Through this line of think- ing, the Radon transform became the math- ematical background for medical CT scans, tomography and other image reconstruction applications. Besides the intrinsic mathematical and Figure 1: The Radon Transform theoretical value, a practical motivation for our work is the ability to obtain information about a function defined on an infinite-dimensional space from its conditional expectations.
    [Show full text]
  • Methods for Analysis of Functionals on Gaussian Self Similar Processes
    METHODS FOR ANALYSIS OF FUNCTIONALS ON GAUSSIAN SELF SIMILAR PROCESSES A Thesis submitted by Hirdyesh Bhatia For the award of Doctor of Philosophy 2018 Abstract Many engineering and scientific applications necessitate the estimation of statistics of various functionals on stochastic processes. In Chapter 2, Norros et al’s Girsanov theorem for fBm is reviewed and extended to allow for non-unit volatility. We then prove that using method of images to solve the Fokker-Plank/Kolmogorov equation with a Dirac delta initial condition and a Dirichlet boundary condition to evaluate the first passage density, does not work in the case of fBm. Chapter3 provides generalisation of both the theorem of Ramer which finds a for- mula for the Radon-Nikodym derivative of a transformed Gaussian measure and of the Girsanov theorem. A P -measurable derivative of a P -measurable function is defined and then shown to coincide with the stochastic derivative, under certain assumptions, which in turn coincides with the Malliavin derivative when both are defined. In Chap- ter4 consistent quasi-invariant stochastic flows are defined. When such a flow trans- forms a certain functional consistently a simple formula exists for the density of that functional. This is then used to derive the last exit distribution of Brownian motion. In Chapter5 a link between the probability density function of an approximation of the supremum of fBm with drift and the Generalised Gamma distribution is established. Finally the self-similarity induced on the distributions of the sup and the first passage functionals on fBm with linear drift are shown to imply the existence of transport equations on the family of these densities as the drift varies.
    [Show full text]
  • A Limiting Process to Invert the Gauss-Radon Transform
    Communications on Stochastic Analysis Vol. 13, No. 1-2, (2019) A LIMITING PROCESS TO INVERT THE GAUSS-RADON TRANSFORM JEREMY J. BECNEL* Abstract. In this work we extend the finite dimensional Radon transform [23] to the Gaussian measure. We develop an inversion formula for this Gauss- Radon transform by way of Fourier inversion formula. We then proceed to extend these results to the infinite dimensional setting. 1. Introduction The Radon transform was invented by Johann Radon in 1917 [23]. The Radon n transform of a suitable function f : R R is defined as a function f on the set → R Pn of hyperplanes in Rn as follows (αv + v⊥)= f(x) dx, (1.1) Rf ∫αv+v⊥ where dx is the Lebesgue measure on the hyperplane given by αv + v⊥. The Radon transform remains a useful and important tool even today because it has applications to many fields, included tomography and medicine [10]. Some of the primary results related to the Radon transform involve the Support Theorem and the various inversion formulas. Using the Laplacian operator or the Fourier transform one can actually recover a function f from the Radon transform f [14]. The is one of the primary reasons the Radon transform has proved so usefulR in many applications. This transform does not generalize directly to infinite dimensions because there is no useful notion of Lebesgue measure in infinite dimensions. However, there is a well-developed theory of Gaussian measures in infinite dimensions and so it is natural to extend the Radon transform to infinite dimensions using Gaussian measure: Gf (P )= f dµP , (1.2) ∫ where µP is the Gaussian measure on any infinite dimensional hyperplane P in a Hilbert space H0.
    [Show full text]
  • Macroscopic Limits for Stochastic Partial Differential Equations of Mckean–Vlasov Type
    Probab. Theory Relat. Fields (2010) 146:189–222 DOI 10.1007/s00440-008-0188-0 Macroscopic limits for stochastic partial differential equations of McKean–Vlasov type Peter M. Kotelenez · Thomas G. Kurtz Received: 23 August 2006 / Revised: 22 October 2008 / Published online: 12 December 2008 © Springer-Verlag 2008 Abstract A class of quasilinear stochastic partial differential equations (SPDEs), driven by spatially correlated Brownian noise, is shown to become macroscopic (i.e., deterministic), as the length of the correlations tends to 0. The limit is the solution of a quasilinear partial differential equation. The quasilinear SPDEs are obtained as a continuum limit from the empirical distribution of a large number of stochastic ordinary differential equations (SODEs), coupled though a mean-field interaction and driven by correlated Brownian noise. The limit theorems are obtained by application of a general result on the convergence of exchangeable systems of processes. We also compare our approach to SODEs with the one introduced by Kunita. Keywords Stochastic partial differential equations · Partial differential equations · Macroscopic limit · Particle systems · Stochastic ordinary differential equations · Exchangeable sequences Mathematics Subject Classification (2000) 60H15 · 60F17 · 60J60 · 60G09 · 60K40 This research was partially supported by NSF grant DMS 05-03983. P. M. Kotelenez (B) Department of Mathematics, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, OH 44106, USA e-mail: [email protected] T. G. Kurtz Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison, WI 53706-1388, USA e-mail: [email protected] 123 190 P. M. Kotelenez, T. G. Kurtz 1 Introduction Let N point particles be distributed over Rd , d ∈ N.
    [Show full text]
  • Convexities and Optimal Transport Problems on the Wiener Space Vincent Nolot
    Convexities and optimal transport problems on the Wiener space Vincent Nolot To cite this version: Vincent Nolot. Convexities and optimal transport problems on the Wiener space. General Mathe- matics [math.GM]. Université de Bourgogne, 2013. English. NNT : 2013DIJOS016. tel-00932092 HAL Id: tel-00932092 https://tel.archives-ouvertes.fr/tel-00932092 Submitted on 16 Jan 2014 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. UNIVERSITE DE BOURGOGNE UFR Sciences et Techniques Institut de Math´ematiquesde Bourgogne THESE pour obtenir le grade de Docteur de l'Universit´ede Bourgogne Discipline : MATHEMATIQUES par Vincent Nolot Convexit´eset probl`emesde transport optimal sur l'espace de Wiener. Soutenue publiquement le 27 Juin 2013 devant le Jury compos´ede Bernard BONNARD Universit´ede Bourgogne (examinateur) Guillaume CARLIER Universit´eParis Dauphine (examinateur) Luigi DE PASCALE Universit´ede Pise (rapporteur) Shizan FANG Universit´ede Bourgogne (directeur de th`ese) Ivan GENTIL Universit´ede Lyon (examinateur) Nicolas PRIVAULT Universit´ede Singapour (rapporteur) 2 R´esum´een Fran¸cais L'objet de cette th`eseest d'´etudierla th´eoriedu transport optimal sur un espace de Wiener abstrait. Les r´esultatsqui se trouvent dans quatre principales parties, portent • Sur la convexit´ede l'entropie relative.
    [Show full text]
  • GEOMETRIC STOCHASTIC ANALYSIS on PATH SPACES 3 Where P : H → L2(E; R) Is the Paley-Wiener Map
    GEOMETRIC STOCHASTIC ANALYSIS ON PATH SPACES K. D. ELWORTHY AND XUE-MEI LI Abstract. An approach to analysis on path spaces of Riemann- ian manifolds is described. The spaces are furnished with ‘Brown- ian motion’ measure which lies on continuous paths, though differ- entiation is restricted to directions given by tangent paths of finite energy. An introduction describes the background for paths on Rm and Malliavin calculus. For manifold valued paths the approach is to use ‘Itˆo’ maps of suitable stochastic differential equations as charts . ‘Suitability’ involves the connection determined by the stochastic differential equation. Some fundamental open problems concerning the calculus and the resulting ‘Laplacian’ are described. A theory for more general diffusion measures is also briefly indi- cated. The same method is applied as an approach to getting over the fundamental difficulty of defining exterior differentiation as a closed operator, with success for one & two forms leading to a Hodge -Kodaira operator and decomposition for such forms. Fi- nally there is a brief description of some related results for loop spaces. Classification. Primary 58B10; 58J65; Secondary 58A14; 60H07; 60H10; 53C17; 58D20; 58B15. Keywords. Path space, Hodge-Kodaira theory, infinite dimensions, connection, de Rham cohomology, stochastic differential equations, Malliavin calculus, Sobolev spaces, abstract Wiener spaces, differen- tial forms. arXiv:1911.09764v1 [math.PR] 21 Nov 2019 1. Introduction 1.1. Analysis with Gaussian measures. Classical differential and geometric analysis is based on Lebesgue measure. The non-existence of an analogue of Lebesgue measure in infinite dimensions is demonstrated by the following theorem: Theorem 1.1. If µ is a locally finite Borel measure on a separable Banach space E such that translations by every element of E preserve sets of measure zero, then either µ =0 or E is finite dimensional.
    [Show full text]
  • On Bounded Pseudodifferential Operators in Wiener Spaces Laurent Amour, Lisette Jager, Jean Nourrigat
    On bounded pseudodifferential operators in Wiener spaces Laurent Amour, Lisette Jager, Jean Nourrigat To cite this version: Laurent Amour, Lisette Jager, Jean Nourrigat. On bounded pseudodifferential operators in Wiener spaces. Journal of Functional Analysis, Elsevier, 2015, 269 (9), pp.2747 - 2812. 10.1016/j.jfa.2015.08.004. hal-01881667 HAL Id: hal-01881667 https://hal.archives-ouvertes.fr/hal-01881667 Submitted on 26 Sep 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. On bounded pseudodifferential operators in Wiener spaces Laurent Amour, Lisette Jager and Jean Nourrigat Universit´ede Reims Dedicated to the memory of Bernard Lascar Abstract We aim at extending the definition of the Weyl calculus to an infinite dimensional setting, by replacing 2n 2 the phase space R by B , where (i; H; B) is an abstract Wiener space. A first approach is to generalize the integral definition using the Wigner function. The symbol is then a function defined on B2 and belonging to a L1 space for a gaussian measure, the Weyl operator is defined as a quadratic form on a dense subspace of L2(B). For example, the symbol can be the stochastic extension on B2, in the sense of L.
    [Show full text]