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Shifted generalized Mehler semigroups on white noise functionals

Cig Un, Lee Mi Ra

Probability and Mathematical Statistics

2023

Vol. 43, Fasc. 2

185--205

We study shifted generalized Mehler semigroups on white noise functionals. We prove characterizations of invariant (white noise) distribution and the covariance property for shifted generalized Mehler semigroups. Also, we prove a Liouville type property of a shifted generalized Mehler semigroup or its infinitesimal generator.

Wick Calculus for Vector-valued Gaussian White Noise Functionals

Ji Un Cig, Ma Peng Cheng

Probability and Mathematical Statistics

2021

Vol. 41, Fasc. 2

283--302

Based on a Gel’fand triple (N) ⊗Ɛ ⊂Γ(H) ⊗h ⊂((N) ⊗Ɛ)∗, we introduce a new notion of Wick type product of generalized Gaussian white noise functionals which is associated with a continuous bilinear mapping B : Ɛ∗ ×Ɛ∗ → Ɛ∗. Then we study Wick type differential equations for vector-valued generalized Gaussian white noise functionals and, as a simple application, we study Wick type differential equations for matrix-valued generalized Gaussian white noise functionals. For our purposes, we make a systematic study of equicontinuity of the left and right Wick type multiplication operators.

Convolutions of generalized white noise functionals

Ji U. C., Kim Y. Y., Park Y. J.

Probability and Mathematical Statistics

2013

Vol. 33, Fasc. 2

435--450

We study a general definition of convolution products of test white noise functionals, of which the consistency property is examined. As an application of the consistency property of the convolution product we study an extension of the convolution to generalized white noise functionals. We also study relations between the convolution and generalized Fourier-Gauss and generalized Fourier-Mehler transforms.