The Wiener measure on the Heisenberg group and parabolic equations

S. V. Mamon

Fundamentalʹnaâ i prikladnaâ matematika, Tome 21 (2016) no. 4, pp. 67-98.

Résumé

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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