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 Fundam. Prikl. Mat.: Year: Volume: Issue: Page: Find

 Fundam. Prikl. Mat., 2016, Volume 21, Issue 4, Pages 67–98 (Mi fpm1748)

The Wiener measure on the Heisenberg group and parabolic equations

S. V. Mamon

Lomonosov Moscow State University

Abstract: In this paper, we study questions related to the theory of stochastic processes on Lie nilpotent groups. In particular, we consider the stochastic process on the Heisenberg group $H_3(\mathbb{R})$ whose trajectories satisfy the horizontal conditions in the stochastic sense relative to the standard contact structure on $H_3(\mathbb{R})$. It is shown that this process is a homogeneous Markov process relative to the Heisenberg group operation. There was found a representation in the form of a Wiener integral for a one-parameter linear semigroup of operators for which the Heisenberg sublaplacian generated by basis vector fields of the corresponding Lie algebra $L(H_3)$ is producing. The main method of solving the problem in this paper is using the path integrals technique, which indicates the common direction of further development of the results.

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Document Type: Article
UDC: 512.813.52+517.955.4+517.983.37+517.987.4+519.216.22

Citation: S. V. Mamon, “The Wiener measure on the Heisenberg group and parabolic equations”, Fundam. Prikl. Mat., 21:4 (2016), 67–98

Citation in format AMSBIB
\Bibitem{Mam16} \by S.~V.~Mamon \paper The Wiener measure on the Heisenberg group and parabolic equations \jour Fundam. Prikl. Mat. \yr 2016 \vol 21 \issue 4 \pages 67--98 \mathnet{http://mi.mathnet.ru/fpm1748} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3783797}