Hui-Hsiung Kuo

Lévy Laplacian of generalized functions on a nuclear space

Abstract The Levy Laplacian ΔF(ξ) = limN→∞N−1∑n = 1N 〈F″(ξ),en⊗ en〉 is shown to be equal to (i) ∝TF″s″(ξ;t)dt, where Fs″ is the singular part of F″, and (ii) 2limϱ→0ϱ−2(MF(ξ,ϱ)−F(ξ)), where MF is the spherical mean of F. It is proved that regular polynomials are Δ-harmonic and possess the mean value property. A relation between the Levy Laplacian Δ and the Gross Laplacian ΔGF(ξ) = ∑n = 1∞=〈F″(ξ),en⊗ en〉 is obtained. An application to white noise calculus is discussed.

A New Class of White Noise Generalized Functions

The S-transform is studied as a mapping from a space of tensors to a space of functions over a complex space. The range of this transform is characterized in terms of analyticity and growth. These results are applied to a broad class of generalized functions in white noise analysis. These correspond to completions of the Gaussian L2-space which preserve orthogonality of Hermite polynomials. The S-transform is defined for the new generalized functions, and the range of this S-transform is identified in terms of analyticity and growth. Examples of the new spaces of generalized functions are given; these include distributions considered by Kondratiev and Streit, as well as new classes of distributions whose S-transforms have growth bounded by iterated exponentials.

Brownian functionals and applications

Hidas theory of generalized Brownian functionals is surveyed with the applications to: (1) stochastic partial differential equations, (2) Feynman integral, (3) an extension of Itôs lemma, and (4) infinite dimensional Fourier transform.

White noise approach to stochastic integration

We extend to white noise integrals the scalar type integrator inequalities introduced by Accardi, Fagnola and Quaegebeur [1] as a generalization of the Hudson–Parthasarathy basic estimates on stochastic integrals. We use these estimates as ”regularity results”, showing that some Hida distributions are in fact elements of the Fock space. We also use them to prove an analogue regularity result for solutions of white noise equations with bounded coefficients. The white noise approach to stochastic calculus emerged, between 1993 and 1995, from the stochastic limit of quantum theory as a new approach [7] to both classical and quantum stochastic calculus [9], [16]. The main achievements of the new approach have been: (i) The identification of both classical and quantum stochastic equations with white noise Hamiltonian equations. (ii) The explanation of the emergence of the unitarity conditions of Hudson and Parthsarathy as expression of the symmetricity of the associated Hamiltonian equation. (iii) The explicit expression of the coefficients of the stochastic equation as (nonlinear) functions of the coefficients of the associated Hamiltonian equation. (iv) The emergence of a natural nonlinear extension of stochastic calculus. The deep and surprising results obtained in this direction in the quadratic case suggest that the completion of this programme for the higher powers of white noise is one of the most challenging and fascinating problems of contemporary stochastic analysis. None of these results could have been even formulated in the framework of the usual (classical or quantum) stochastic analysis. However the exciting new developments emerged from the white noise approach to stochastic calculus delayed a systematic exposition of its basic analytical tools such as white noise integrals, white noise equations, the theory of distributions on the standard simplex, the causal normal order, ... A large literature existed in the first two of the above mentioned directions, in the framework of Hida’s white noise analysis. However to create a bridge between these results and those of classical and quantum stochastic analysis, one needs some “regularity results”, i.e. conditions which assure that objects, which a priori are Hida distributions, are in fact vectors in a Hilbert space. Such regularity conditions will be formulated here in terms of estimates on white noise integrals. Estimates of this type were developed in [3] and they were strong enough to prove an existence theorem for a multidimensional white noise integral equation. However, as we shall see in a forthcoming paper [8], these estimates were not strong enough to prove the fundamental result of the theory, i.e. the unitarity condition. The main results of the present paper are: (i): The extension of the white noise estimates of [3] to a larger domain (the “maximal algebraic” domain introduced in Section 1) on the lines of [5]. (ii): The introduction of the notion of white noise adaptedness and the proof, under this assumption (which requires a white noise with 1–dimensional parameter), of the white noise analogue of the “scalar type integrator” inequalities of Accardi, Fagnola and Quaegebeur [1] which, in their turn, generalize the basic Hudson–Parthasarathy estimate on stochastic integrals (cf. Proposition 25.6 in [15]).

Diffusion and Brownian motion on infinite-dimensional manifolds

The purpose of this paper is to construct certain diffusion processes, in particular a Brownian motion, on a suitable kind of infinite-dimensional manifold. This manifold is a Banach manifold modelled on an abstract Wiener space. Roughly speaking, each tangent space Tx is equipped with a norm and a densely defined inner product g(x). Local diffusions are constructed first by solving stochastic differential equations. Then these local diffusions are pieced together in a certain way to get a global diffusion. The Brownian motion is completely determined by g and its transition probabilities are proved to be invariant under d -isometries. Here d is the almost-metric (in the sense that two points may have g g . . infinite distance) associated with g. The generalized Beltrami-Laplace operator is defined by means of the Brownian motion and will shed light on the study of potential theory over such a marîifold.

CHARACTERIZATION OF PROBABILITY MEASURES THROUGH THE CANONICALLY ASSOCIATED INTERACTING FOCK SPACES

We continue our program of coding the whole information of a probability measure into a set of commutation relations canonically associated to it by presenting some characterization theorems for the symmetry and factorizability of a probability measure on ℝd in terms of the canonically associated interacting creation, annihilation and number operators.

Bell Numbers, Log-Concavity, and Log-Convexity

AbstractLet {bk(n)}n=0∞ be the Bell numbers of order k. It is proved that the sequence {bk(n)/n!}n=0∞ is log-concave and the sequence {bk(n)}n=0∞ is log-convex, or equivalently, the following inequalities hold for all n⩾0,

DIAGONALIZATION OF THE LÉVY LAPLACIAN AND RELATED STABLE PROCESSES

1 \leqslant \frac{{b_k (n + 2)b_k (n)}}{{b_k (n + 1)^2 }} \leqslant \frac{{n + 2}}{{n + 1}}