Electron. J. Differential Equations, Vol. 2022 (2022), No. 04, pp. 1-15.

Poisson measures on semi-direct products of infinite-dimensional Hilbert spaces

Richard C. Penney, Roman Urban

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 Rn. 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 ν 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.

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Richard C. Penney
Department of Mathematics
Purdue University
150 N. University St
West Lafayette, IN 47907, USA
email: rcp@math.purdue.edu
Roman Urban
Institute of Mathematics
Wroclaw University
Plac Grunwaldzki 2/4
50-384 Wroclaw, Poland
email: roman.urban@math.uni.wroc.pl

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