Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 04, pp. 1-15.

Title: Poisson measures on semi-direct products of infinite-dimensional Hilbert spaces
       
Authors: Richard C. Penney (Purdue Univ., West Lafayette, IN, USA)
         Roman Urban (Wroclaw Univ.,  Wroclaw, Poland)

Abstract:
Let $G=X\rtimes A$ where X and A are Hilbert spaces considered as additive groups
and the A-action on G is diagonal in some orthonormal basis.
We consider a particular second order left-invariant differential operator L on
G which is analogous to the Laplacian on R<sup>n</sup>.
We  prove the existence of  "heat kernel" for L and give a probabilistic formula for it.
We then prove that X is a "Poisson boundary" in a sense of Furstenberg for L with
a (not necessarily) probabilistic measure &nu; on X called the "Poisson measure"
for the operator L.

Submitted March 23, 2021. Published January 10, 2022
Math Subject Classifications: 35C05, 60J25, 60J60, 60J45.
Key Words: Poisson measure; Gaussian measure; Hilbert space; Brownian motion;
           evolution kernel; diffusion processes.