Asymptotic of the Green's Function of a Riemannian Manifold and Ito's Stochastic Integrals

Abstract

Quantitative estimates are obtained by comparison with ordinary differential equations associated to a subharmonic exhaustion function q. We associate with q a ratio a, which can be considered as the heat flow in an intrinsic time, and the sup and the inf of a, namely a ⁺ and a ^-, on the level hypersurfaces of q. Then a ⁺ and a ^- define heat flows on the real line. Comparison between the heat flow on the manifold and heat flows on the line are obtained by stochastic integrals.