Integral inequalities for the fundamental solutions of diffusions on manifolds with divergence-free drift

Abstract

The main purpose of this article is to present uniform integral inequalities for the fundamental solutions of diffusions on compact manifolds with divergence free drift vector fields. The method relies on the fact that the heat flow depends on the isoperimetric function. The isoperimetric function is used to construct a suitable comparison manifold. The heat kernel of this comparison manifold gives uniform bounds for the fundamental solutions of the original diffusion problem. The results presented here can be used to solve some open problem of Bhattacharya and Gotze about diffusions with periodic, divergence free drift vector fields (see [Bhagot1] and [Bhagot2]).