Department of Mathematics
Purdue University
150 N. University St
West Lafayette, IN 47907, USA
email: rcp@math.purdue.edu
Institute of Mathematics
Wroclaw University
Plac Grunwaldzki 2/4
50-384 Wroclaw, Poland
email: roman.urban@math.uni.wroc.pl
Richard C. Penney, Roman Urban
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 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.
Show me the PDF file (350 KB), TEX file for this article.
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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 |
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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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