The proposed research is related to several areas of probability theory: the theory of symmetric diffusion processes and Dirichlet forms, stochastic differential geometry, and the analysis of heat kernels. The principal investigator plans to extend the results of Carmona and Zheng about the small time asymptotics of the probability that Brownian paths visiting the sets of a given finite sequence. These extensions will be used to solve several questions: the monotonicity of heat kernels associated to Newmann boundary conditions (Chavel's conjecture) for all times, the study of Riemannian manifolds with piecewise constant metrics via the precise path behavior of Brownian motions on these manifolds, and the study of general non-smooth Riemannian manifolds and in particular the Harnack's inequality by using the above concrete results and weak convergence. There have been a lot of studies about the propagation of heat on some geometric body with smooth surfaces made by some unitary material. The situation becomes more complicated when the surface is not smooth or the body is made with materials of different thermal conductivities. This research is aimed at a microscopic analysis of the latter case using probability theory.
