A construction of diffusion processes associated with sub-Laplacian on CR manifolds and its applications

TL;DR

This paper constructs a diffusion process linked to the sub-Laplacian on CR manifolds using geometric and probabilistic methods, enabling analysis of heat kernels, Dirichlet problems, and stochastic integrals.

Contribution

It introduces a novel probabilistic construction of diffusion processes associated with the sub-Laplacian on CR manifolds, utilizing the Tanaka-Webster connection and Malliavin calculus.

Findings

Heat kernel estimates derived via stochastic analysis

Probabilistic solutions to Dirichlet problems for sub-Laplacian

Distributional properties of stochastic line integrals

Abstract

A diffusion process associated with the real sub-Laplacian $\Delta_b$, the real part of the complex Kohn-Spencer laplacian $\square_b$, on a strictly pseudoconvex CR manifold is constructed via the Eells-Elworthy-Malliavin method by taking advantage of the metric connection due to Tanaka-Webster. Using the diffusion process and the Malliavin calculus, the heat kernel and the Dirichlet problem for $\Delta_b$ are studied in a probabilistic manner. Moreover, distributions of stochastic line integrals along the diffusion process will be investigated.