DOCSLIB.ORG

Change of Variables in Infinite-Dimensional Space

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read 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. However, there is an analogue of Gaussian measure. Denis Bell Change of variables in infinite-dimensional space Brownian motion x w(t) T Denis Bell Change of variables in infinite-dimensional space Figure : a2 + b2 = c2 The Wiener process The Wiener processes is the mathematical model for Brownian motion. It is a stochastic process w : [0; T ) × Ω 7! R (a random function) with the properties (i) w0 = 0: (ii) The function t 7! wt is (a.s.) continuous. n (iii) For a measurable set B ⊂ R and times 0 < t1 < ··· < tn ≤ T P(w(t1);:::; w(tn) 2 B) = 1 × n=2p (2π) t1(t2 − t1) ::: (tn − tn−1) 2 2 2 Z 1 x1 (x2−x1) (xn−xn−1) − 2 t + (t −t ) +::: (t −tn ) e 1 2 1 n 1 dx1 ::: dxn B Denis Bell Change of variables in infinite-dimensional space Underlying geometric structure: the Cameron-Martin space Consider the space H of continuous, piecewise C 1 functions R T 0 2 h : [0; T ] 7! R with h0 = 0 and energy 0 ht dt < 1: Equip H with the inner product Z T 0 0 < h; k >≡ ht kt dt; h; k 2 H: 0 Then P(w(t1);:::; w(tn) 2 B) = Z 1 − 1 jjh(x)jj2 n p e 2 dx (2π) 2 t1(t2 − t1) ::: (tn − tn−1) B where x = (x1 ::: xn) and h(x) is the polygonal path between the points (0; 0); (t1; x1);::: (tn; xn); (T ; xn). Denis Bell Change of variables in infinite-dimensional space The Wiener measure Denote by C0 the Banach space of continuous paths fσ : [0; T ] 7! R; σ0 = 0g: We define the Wiener measure γ on C0 to be the law of Brownian motion w. i.e. γ(S) = P(w 2 S) on Borel sets S ⊂ C0. Denis Bell Change of variables in infinite-dimensional space Change of variables for Wiener measure: the Girsanov theorem Theorem Let h : [0; T ] 7! R be a continuous adapted path (for any t, ht = f (ws =s ≤ t)). Define φ : C0 7! C0 by Z t φ(w)t = wt + hs ds: 0 Then for any Borel set B ⊂ C0, Z R T h dw− 1 R T h2ds γ(φ(B)) = e 0 s 2 0 s dγ: B Compare with finite-dimensional formula. Note that there is no requirement that the map w 7! h (and hence φ) be differentiable. Denis Bell Change of variables in infinite-dimensional space The Gaussian cylinder set measure on a Hilbert space Let H be an (infinite-dimensional) Hilbert space. Suppose P 2 L(H; Rn) be surjective. Then P induces an inner n n product < :; : >P on R : for x; y 2 R ; < x; y >P ≡< x~; y~ >H , where P(~x) = x; P(~y) = y; x~; y~ 2 (ker P)?. For any Borel set B ⊂ Rn we define Z jjxjj2 −1 1 − P µ~(P (B)) ≡ e 2 dx n=2 (2π) B where x = (x1;::: xn) and dx is the Lebesgue measure corresponding to < :; : >P . Denis Bell Change of variables in infinite-dimensional space The following consistency condition can be shown: If P−1(B) = Q−1(C), where P2 L(H; Rn); B ⊂ Rn; Q 2 L(H; Rm); C ⊂ Rm, then µ~(P−1(B)) =µ ~(Q−1(C)). Thusµ ~ defines a "measure" on the ring of cylinder sets fP−1(B)g in H. µ~ is finitely but not countably additive (in the case dim H = 1). Denis Bell Change of variables in infinite-dimensional space Let i : H 7! E denote an embedding of H into a separable Banach space E. Then there is an induced cylinder-set measureγ ~ on E defined by γ~(P−1(B)) ≡ µ~((P ◦ i)−1(B)); P 2 L(E; Rn); B ⊂ Rn: Theorem If the norm on H is sufficiently stronger than the norm on E, then γ~ is countably additive on the cylinder sets of E and hence extends to a measure γ on E. Denis Bell Change of variables in infinite-dimensional space Abstract Wiener space The previous theorem is due to L. Gross (1967). It generalizes Wiener's original theorem. Gross called this construction an abstract Wiener space. The result is a measure on an arbitrary Banach space E whose finite-dimensional images are Gaussian measures. It is convenient to think of E ∗ as a subset of E via the maps i ∗ i E ∗ 7! H∗ =∼ H 7! E: Denis Bell Change of variables in infinite-dimensional space Change of variables formula in abstract Wiener space The following result is due to H-H.Kuo. Theorem Let B be an open subset of E and φ : B 7! φ(B) ⊂ E be a diffeomorphism of the form φ = I + K; where K is a C 1 map from B to E ∗. Then Z (K(x);x)− 1 jjK(x)jj2 γ(φ(B)) = e 2 H detH Dφ(x)dγ: B Denis Bell Change of variables in infinite-dimensional space Divergence operators Definition Let E be a normed vector space equipped with a measure µ.A linear operator L with domain D (a subset of the set of vector fields on E) is a divergence operator if L satisfies the integration-by-parts formula Z Z DZ fdµ = f · (LZ)dµ E E for a dense class of C 1 real-valued functions f on E, and Z 2 D. Denis Bell Change of variables in infinite-dimensional space Examples of divergence operators In euclidean space: Let E = Rn and µ the Lebesgue measure. Then L is the classical n n divergence operator: for Z = (Z1;:::; Zn): R 7! R , n X @Zi LZ = − @xi i=1 defined on C 1 vector fields. Denis Bell Change of variables in infinite-dimensional space Divergence Theorems Theorem There is a divergence operator L for the Wiener space, with domain D = Z T n 0 2 o h : [0; T ] 7! R adapted, absolutely continuous; hs ds < 1 : 0 Z T 0 Lh = ht dw; h 2 D: 0 (Gaveau &Trauber) Denis Bell Change of variables in infinite-dimensional space Theorem Let µ be a Gaussian measure on E arising from the abstract Wiener space i : H 7! E. There is the divergence operator (LK)(x) = (K(x); x) − traceH DK(x) with domain . D = fK K : E 7! E ∗ is C 1g (L. Gross) Denis Bell Change of variables in infinite-dimensional space Change of measure under the flow of a vector field Let K be a vector field on E. For x 2 E, consider the ODE dα = K(α(t)): dt α(0) = x: The map φt : x 7! α(t) is called the flow of K. Denis Bell Change of variables in infinite-dimensional space Theorem Suppose a measure µ on E admits a divergence operator (L; D). Let K 2 D and let φt denote the associated flow. Then (under appropriate conditions) there is the change of variables formula Z R t − (LK)(φs (x))ds µ(φt (B)) = e 0 dµ. B (B., Comptes Rendus, Paris, 2006) Denis Bell Change of variables in infinite-dimensional space An example In general, it is not possible to compute the flow φt explicitly, but there is a case when it is trivial, i.e. when K is a constant vector in E. Suppose this is the case. Then the differential equation dφ t = K(φ ); φ = x dt t 0 has the solution φt (x) = x + Kt: Denis Bell Change of variables in infinite-dimensional space Suppose we are in the setting of the classical Wiener space. Let K be a Cameron-Martin path. We compute the change of measure formula in this case. Recall that the divergence of K is given by the It^ointegral Z T 0 (LK)(w) = Kudw: 0 Denis Bell Change of variables in infinite-dimensional space Thus Z T 0 (LK)(φs (w)) = Kud(w + sK) 0 Z T Z T 0 0 2 = Kudw + s Ku du: 0 0 Hence we have Z − R t (LK)(φ (w))ds γ(B + tK) = e 0 s dγ B 2 Z − t R T K 0 dw+ t R T K 0 2du = e 0 u 2 0 u dγ: B This formula is known as the Cameron-Martin theorem.
Load more

Recommended publications
  • 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.
    [Show full text]
  • 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]
  • Gaussian Measures in Traditional and Not So Traditional Settings
    BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 33, Number 2, April 1996 GAUSSIAN MEASURES IN TRADITIONAL AND NOT SO TRADITIONAL SETTINGS DANIEL W. STROOCK Abstract. This article is intended to provide non-specialists with an intro- duction to integration theory on pathspace. 0: Introduction § Let Q1 denote the set of rational numbers q [0, 1], and, for a Q1, define the ∈ ∈ translation map τa on Q1 by addition modulo 1: a + b if a + b 1 τab = ≤ for b Q1. a + b 1ifa+b>1 ∈ − Is there a probability measure λQ1 on Q1 which is translation invariant in the sense that 1 λQ1 (Γ) = λQ1 τa− Γ for every a Q1? To answer∈ this question, it is necessary to be precise about what it is that one is seeking. Indeed, the answer will depend on the level of ambition at which the question is asked. For example, if denotes the algebra of subsets Γ Q1 generated by intervals A ⊆ [a, b] 1 q Q1 : a q b Q ≡ ∈ ≤ ≤ for a b from Q1, then it is easy to check that there is a unique, finitely additive way to≤ define λ on .Infact, Q1 A λ 1 [a, b] 1 = b a, for a b from Q1. Q Q − ≤ On the other hand, if one is more ambitious and asks for a countably additive λQ1 on the σ-algebra generated by (equivalently, all subsets of Q1), then one is asking A for too much. In fact, since λQ1 ( q ) = 0 for every q Q1, countable additivity forces { } ∈ λ 1 (Q1)= λ 1 q =0, Q Q { } q 1 X∈Q which is obviously irreconcilable with λQ1 (Q1) = 1.
    [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]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Infinite Dimension
    Tel Aviv University, 2006 Gaussian random vectors 10 2 Basic notions: infinite dimension 2a Random elements of measurable spaces: formu- lations ......................... 10 2b Random elements of measurable spaces: proofs, andmorefacts .................... 14 2c Randomvectors ................... 17 2d Gaussian random vectors . 20 2e Gaussian conditioning . 24 foreword Up to isomorphism, we have for each n only one n-dimensional Gaussian measure (like a Euclidean space). What about only one infinite-dimensional Gaussian measure (like a Hilbert space)? Probability in infinite-dimensional spaces is usually overloaded with topo- logical technicalities; spaces are required to be Hilbert or Banach or locally convex (or not), separable (or not), complete (or not) etc. A fuss! We need measurability, not continuity; a σ-field, not a topology. This is the way to a single infinite-dimensional Gaussian measure (up to isomorphism). Naturally, the infinite-dimensional case is more technical than finite- dimensional. The crucial technique is compactness. Back to topology? Not quite so. The art is, to borrow compactness without paying dearly to topol- ogy. Look, how to do it. 2a Random elements of measurable spaces: formulations Before turning to Gaussian measures on infinite-dimensional spaces we need some more general theory. First, recall that ∗ a measurable space (in other words, Borel space) is a set S endowed with a σ-field B; ∗ B ⊂ 2S is a σ-field (in other words, σ-algebra, or tribe) if it is closed under finite and countable operations (union, intersection and comple- ment); ∗ a collection closed under finite operations is called a (Boolean) algebra (of sets); Tel Aviv University, 2006 Gaussian random vectors 11 ∗ the σ-field σ(C) generated by a given set C ⊂ 2S is the least σ-field that contains C; the algebra α(C) generated by C is defined similarly; ∗ a measurable map from (S1, B1) to (S2, B2) is f : S1 → S2 such that −1 f (B) ∈ B1 for all B ∈ B2.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Probability Measures on Infinite-Dimensional Stiefel Manifolds
    JOURNAL OF GEOMETRIC MECHANICS doi:10.3934/jgm.2017012 ©American Institute of Mathematical Sciences Volume 9, Number 3, September 2017 pp. 291–316 PROBABILITY MEASURES ON INFINITE-DIMENSIONAL STIEFEL MANIFOLDS Eleonora Bardelli‡ Microsoft Deutschland GmbH, Germany Andrea Carlo Giuseppe Mennucci∗ Scuola Normale Superiore Piazza dei Cavalieri 7, 56126, Pisa, Italia Abstract. An interest in infinite-dimensional manifolds has recently appeared in Shape Theory. An example is the Stiefel manifold, that has been proposed as a model for the space of immersed curves in the plane. It may be useful to define probabilities on such manifolds. Suppose that H is an infinite-dimensional separable Hilbert space. Let S ⊂ H be the sphere, p ∈ S. Let µ be the push forward of a Gaussian measure γ from TpS onto S using the exponential map. Let v ∈ TpS be a Cameron–Martin vector for γ; let R be a rotation of S in the direction v, and ν = R#µ be the rotated measure. Then µ, ν are mutually singular. This is counterintuitive, since the translation of a Gaussian measure in a Cameron– Martin direction produces equivalent measures. Let γ be a Gaussian measure on H; then there exists a smooth closed manifold M ⊂ H such that the projection of H to the nearest point on M is not well defined for points in a set of positive γ measure. Instead it is possible to project a Gaussian measure to a Stiefel manifold to define a probability. 1. Introduction. Probability theory has been widely studied for almost four cen- turies. Large corpuses have been written on the theoretical aspects.
    [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]