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