Anticipating Flows on the Wiener Space Generated by Vector Fields of Low Regularity
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Path Regularity and Tightness of Capac- Ities
Applications of Compact Superharmonic Func- tions: Path Regularity and Tightness of Capac- ities Lucian Beznea and Michael R¨ockner Abstract. We establish relations between the existence of the L-superharmonic functions that have compact level sets (L being the generator of a right Markov process), the path regularity of the process, and the tightness of the induced capacities. We present several examples in infinite dimensional situations, like the case when L is the Gross-Laplace operator on an abstract Wiener space and a class of measure-valued branching process associated with a nonlinear perturbation of L. Mathematics Subject Classification (2000). Primary 60J45, 31C15; Secondary 47D07, 60J40, 60J35. Keywords. resolvent of kernels, tight capacity, Lyapunov function, right pro- cess, standard process. 1. Introduction A major difficulty in the stochastic analysis in infinite dimensions is the lack of the local compactness of the state space. The developments from the last two decades, in particular the theory of quasi-regular Dirichlet forms (cf. [14]), indicated, as an adequate substitute, the existence of a nest of compact sets, or equivalently, the tightness property of the associated capacity. However, in applications it is sometimes not easy to verify such a property, for example, because the capacity is not expressed directly in terms of the infinitesimal generator L of the theory. It happens that a more convenient property is the existence of an L-superharmonic function having compact level sets; such a function was called compact Lyapunov. The aim of this paper is to give an overview on the relations between the existence of a compact Lyapunov function, the path regularity of the process, and the tightness of the induced capacities. -
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. -
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. -
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. -
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. -
A Divergence Theorem for Hilbert Space
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 164, February 1972 A DIVERGENCE THEOREM FOR HILBERT SPACE BY VICTOR GOODMAN« Abstract. Let B be a real separable Banach space. A suitable linear imbedding of a real separable Hubert space into B with dense range determines a probability measure on B which is known as abstract Wiener measure. In this paper it is shown that certain submanifolds of B carry a surface measure uniquely defined in terms of abstract Wiener measure. In addition, an identity is obtained which relates surface integrals to abstract Wiener integrals of functions associated with vector fields on regions in B. The identity is equivalent to the classical divergence theorem if the Hubert space is finite dimensional. This identity is used to estimate the total measure of certain surfaces, and it is established that in any space B there exist regions whose boundaries have finite surface measure. 1. Introduction. Let B be a real separable Banach space with norm || • ||. Con- sider a linear imbedding i of a real separable Hubert space into B with dense range. Gross [5] has shown that if the function \¿- || is a measurable norm on the Hubert space, then the space B carries a Borel probability measure px uniquely characterized by the identity Px({xeB : iy,xy < r}) = (2Tr\¿*y\2)-1'2 f exp [-(2\¿*y\2Y1s2]ds. J —00 Here, y is any element of the topological dual space of B, ¿* is the adjoint of the imbedding map, and | • | is the norm on the dual Hubert space. The space B, to- gether with the Hubert space and the imbedding, is said to be an abstract Wiener space, andpx is said to be abstract Wiener measure on B. -
Wiener Measures on Certain Banach Spaces Over Non-Archimedean Local fields Compositio Mathematica, Tome 93, No 1 (1994), P
COMPOSITIO MATHEMATICA TAKAKAZU SATOH Wiener measures on certain Banach spaces over non-archimedean local fields Compositio Mathematica, tome 93, no 1 (1994), p. 81-108 <http://www.numdam.org/item?id=CM_1994__93_1_81_0> © Foundation Compositio Mathematica, 1994, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/conditions). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ 81 © 1994 Kluwer Academic Publishers. Printed in the Netherlands. Wiener measures on certain Banach spaces over non-Archimedean local fields TAKAKAZU SATOH Department of Mathematics, Faculty of Science, Saitama University, 255 Shimo-ookubo, Urawu, Saitama, 338, Japan Received 8 June 1993; accepted in final form 29 July 1993 Dedicated to Proffessor Hideo Shimizu on his 60th birthday 1. Introduction The integral representation is a fundamental tool to study analytic proper- ties of the zeta function. For example, the Euler p-factor of the Riemann zeta function is where dx is a Haar measure of Q, and p/( p - 1) is a normalizing constant so that measure of the unit group is 1. What is necessary to generalize such integrals to a completion of a ring finitely generated over Zp? For example, let T be an indeterminate. The two dimensional local field Qp«T» is an infinite dimensional Qp-vector space. -
Mixtures of Gaussian Cylinder Set Measures and Abstract Wiener Spaces As Models for Detection Annales De L’I
ANNALES DE L’I. H. P., SECTION B A. F. GUALTIEROTTI Mixtures of gaussian cylinder set measures and abstract Wiener spaces as models for detection Annales de l’I. H. P., section B, tome 13, no 4 (1977), p. 333-356 <http://www.numdam.org/item?id=AIHPB_1977__13_4_333_0> © Gauthier-Villars, 1977, tous droits réservés. L’accès aux archives de la revue « Annales de l’I. H. P., section B » (http://www.elsevier.com/locate/anihpb) implique l’accord avec les condi- tions générales d’utilisation (http://www.numdam.org/conditions). Toute uti- lisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Ann. Inst. Henri Pincaré, Section B : Vol. XIII, nO 4, 1977, p. 333-356. Calcul des Probabilités et Statistique. Mixtures of Gaussian cylinder set measures and abstract Wiener spaces as models for detection A. F. GUALTIEROTTI Dept. Mathematiques, ~cole Polytechnique Fédérale, 1007 Lausanne, Suisse 0. INTRODUCTION AND ABSTRACT Let H be a real and separable Hilbert space. Consider on H a measurable family of weak covariance operators { Re, 0 > 0 } and, on !?+ : = ]0, oo [, a probability distribution F. With each Re is associated a weak Gaussian distribution Jlo. The weak distribution p is then defined by Jl can be studied, as we show below, within the theory of abstract Wiener spaces. The motivation for considering such measures comes from statistical communication theory. There, signals of finite energy are often modelled as random variables, normally distributed, with values in a L2-space. -
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. -
A Note on the Malliavin-Sobolev Spaces Peter Imkeller, Thibaut Mastrolia, Dylan Possamaï, Anthony Réveillac
A note on the Malliavin-Sobolev spaces Peter Imkeller, Thibaut Mastrolia, Dylan Possamaï, Anthony Réveillac To cite this version: Peter Imkeller, Thibaut Mastrolia, Dylan Possamaï, Anthony Réveillac. A note on the Malliavin-Sobolev spaces. Statistics and Probability Letters, Elsevier, 2016, 109, pp.45-53. 10.1016/j.spl.2015.08.020. hal-01101190 HAL Id: hal-01101190 https://hal.archives-ouvertes.fr/hal-01101190 Submitted on 8 Jan 2015 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. A note on the Malliavin-Sobolev spaces Peter Imkeller∗ Thibaut Mastrolia† Dylan Possamaï‡ Anthony Réveillac§ January 8, 2015 Abstract In this paper, we provide a strong formulation of the stochastic Gâteaux differentiability in order to study the sharpness of a new characterization, introduced in [6], of the Malliavin- Sobolev spaces. We also give a new internal structure of these spaces in the sense of sets inclusion. Key words: Malliavin calculus, Wiener space, functional analysis, stochastic Gâteaux differentiability. AMS 2010 subject classification: Primary: 60H07; Secondary: 46B09. 1 Introduction Consider the classical W 1,2(Rd) Sobolev space on Rd (d 1) consisting of mappings f which are square integrable and which admit a square integra≥ ble weak derivative. -
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. -
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.