Decompositions of diffusion operators and related couplings

Abstract

Results by Cranston, Greven, and Feng-YuWang, on relationships between coupling and shift coupling, and harmonic functions and space time harmonic functions are reviewed. These lead to extensions of a result by Freire on the separate harmonicity of bounded harmonic functions on certain product manifolds. The extensions are to situations where a diffusion operator is decomposed into the sum of two other commuting diffusion operators. This is shown to arise for a class of foliated Riemannian manifolds with totally geodesic leaves. A form of skew product decomposition of Brownianmotions on these foliatedmanifolds is obtained, as are gradient estimates in leaf directions. Relationships between stochastic completeness of the manifold itself and stochastic completeness of its leaves are established. Baudoin and Garafola’s “sub-Riemannian manifolds with transverse symmetries” are shown to be examples.