Electron. J. Differential Equations, Vol. 2022 (2022), No. 04, pp. 115.
Poisson measures on semidirect products of infinitedimensional Hilbert spaces
Richard C. Penney, Roman Urban
Abstract:
Let "$G=X\rtimes A$
where X and A are Hilbert spaces considered as additive groups
and the Aaction on G is diagonal in some orthonormal basis.
We consider a particular second order leftinvariant differential operator L on
G which is analogous to the Laplacian on R^{n}.
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
50384 Wroclaw, Poland
email: roman.urban@math.uni.wroc.pl

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