Differential and stochastic equations in abstract Wiener space

This paper studies the differential equation ( ∂ ∂t) u(t, x) = 1 2trace A(t, x) D2u(t, x) A*(t, x) + 〈σ(t, x), Du(t, x)〉 - V(x) u(t, x), u(0, x) = φ(x) in infinite dimensional space. A Kac\u27s type representation of solution in terms of function space integral is proved. Kac\u27s method is modified to work nicely regardless of the dimensionality. © 1973

On Fourier transform of generalized Brownian functionals

Let (L2) B ̇- and (L2) b ̇- be the spaces of generalized Brownian functionals of the white noises Ḃ and ḃ, respectively. A Fourier transform from (L2) B ̇- into (L2) b ̇- is defined by φ{symbol}̂(ḃ) = ∫S*: exp[-i ∫Rḃ(t) Ḃ(t) dt]: b ̇φ{symbol}( B ̇) dμ( B ̇), where : : b ̇ denotes the renormalization with respect to ḃ and μ is the standard Gaussian measure on the space S* of tempered distributions. It is proved that the Fourier transform carries Ḃ(t)-differentiation into multiplication by iḃ(t). The integral representation and the action ofφ{symbol}̂ as a generalized Brownian functional are obtained. Some examples of Fourier transform are given. © 1982

Stochastic integrals in abstract wiener space

Let W(t, ω) be the Wiener process on an abstract Wiener space (i, H, B) corresponding to the canonical normal distributions on H. Stochastic integrals (formula presented) and (formula presented) are defined for non-anticipating transformations f with values in B(B, B) such that (x(t, ω) — I)(B) B* and C with values in H. Suppose (formula presented), where u is a non-anticipating transformation with values in H. Let fit, x be a continuous function on R x B, continuously twice differentiable in the indirections with D2 f(t, x) e H) for the x variable and once differentiable for the t variable. Then (formula presented) where \u3c, \u3e is the inner product of H. Under certain assumptions on A and a it is shown that the stochastic integral equation (formula presented) has a unique solution. This solution is a homogeneous strong Markov process. © 1972 Pacific Journal of Mathematics

Potential theory associated with Uhlenbeck-Ornstein process

Some parallel results of Gross\u27 paper (Potential theory on Hilbert space, J. Functional Analysis 1 (1967), 123-181) are obtained for Uhlenbeck-Ornstein process U(t) in an abstract Wiener space (H, B, i). Generalized number operator N is defined by Nf(x) = -lim∈←0{E[f(U(τ∈ξ))] - f(x)}/E[τ∈ξ, where τxε{lunate} is the first exit time of U(t) starting at x from the ball of radius ε{lunate} with center x. It is shown that Nf(x) = -trace D2f(x)+〈Df(x),x〉 for a large class of functions f. Let rt(x, dy) be the transition probabilities of U(t). The λ-potential Gλf, λ \u3e 0, and normalized potential Rf of f are defined by Gλf(X) = ∫0∞ e-λtrtf(x) dt and Rf(x) = ∫0∞ [rtf(x) - rtf(0)] dt. It is shown that if f is a bounded Lip-1 function then trace D2Gλf(x) - 〈DGλf(x), x〉 = -f(x) + λGλf(x) and trace D2Rf(x) - 〈DRf(x), x〉 = -f(x) + ∫Bf(y)p1(dy), where p1 is the Wiener measure in B with parameter 1. Some approximation theorems are also proved. © 1976

The fourier transform in white noise calculus

Let S* be the space of termpered distributions with standard Gaussian measure μ. Let (S) ⊂ L2(μ) ⊂ (S)* be a Gel\u27fand triple over the white noise space (S*, μ). The S-transform (Sφ{symbol})(ζ) = ∫S* φ{symbol}(x + ζ) dμ(x), ζ ∈ S, on L2(μ) extends to a U-functional U[φ{symbol}](ζ) = «exp(·, ζ), φ{symbol} a ̊ exp( -∥ζ∥2 2), ζ ∈ S, on (S)*. Let D consist of φ{symbol} in (S)* such that U[φ{symbol}](-iζ1T) exp[-2-1 ∫Tζ(t)2 dt], ζ ∈ S, is a U-functional. The Fourier transform of φ{symbol} in D is defined as the generalized Brownian functional φ{symbol}̌ in (S)* such that U[φ{symbol}̌](ζ) = U[φ{symbol}](-iζ1T) exp[-2-1 ∫Tζ(t)2 dt], ζ ∈ S. Relations between the Fourier transform and the white noise differentiation ∂t and its adjoint ∂t* are proved. Results concerning the Fourier transform and the Gross Laplacian ΔG, the number operator N, and the Volterra Laplacian ΔV are obtained. In particular, (ΔG*φ{symbol})^ = -ΔG*φ{symbol}̌ and [(ΔV + N)φ{symbol}]^ = -(ΔV + N)φ{symbol}̌. Many examples of the Fourier transform are given. © 1989

On gross differentiation on Banach spaces

Let Pt (x, ·) denote the Wiener measure in an abstract Wiener space (H, B) with variance parameter t \u3e 0 and mean x in B. It is shown that if [FORMULA PRESENTED]; 0 and x are fixed, then the function ptf defined by [FORMULA PRESENTED] for h in H is infinitely Gross differentiable at x. The first two derivatives are given by [FORMULA PRESENTED], where h and k are in H. Moreover, [FORMULA PRESENTED] is a Hilbert-Schmidt operator and [FORMULA PRESENTED]. An application to Uhlenbeck- Ornstein process is also given. © 1975 Pacific Journal of Mathematics

Stochastic integrals in abstruct wiener space II: Regularity properties

This paper continues the study of stochastic integrals in abstract Wiener space previously given in [14]. We will present, among other things, the detailed discussion and proofs of the results announced in [16]. Let H ⊂ B be an abstract Wiener space.</jats:p

Roles of Log-concavity, log-convexity, and growth order in white noise analysis

In this paper we will develop a systematic method to answer the questions $(Q1)(Q2)(Q3)(Q4)$ (stated in Section 1) with complete generality. As a result, we can solve the difficulties $(D1)(D2)$ (discussed in Section 1) without uncertainty. For these purposes we will introduce certain classes of growth functions $u$ and apply the Legendre transform to obtain a sequence which leads to the weight sequence \{\a(n)\} first studied by Cochran et al. \cite{cks}. The notion of (nearly) equivalent functions, (nearly) equivalent sequences and dual Legendre functions will be defined in a very natural way. An application to the growth order of holomorphic functions on \ce_c will also be discussed.Comment: To appear in Infinite Dimensional Analysis, Quantum Probability and Related Topics 4 (2001). Universidade da Madeira CCM preprint 37 (1999