Heat kernels on Riemannian manifolds

The course is devoted to the heat equation and heat kernel on arbitrary Riemannian manifolds. The following topics will be considered.

1. Cauchy problem for the heat equation: uniqueness classes.
2. Stochastic completeness: equivalent conditions and volume test.
3. Long time behaviour of the heat kernel and isoperimetric properties.
4. Heat kernel bounds and volume growth.
5. Escape rate of Brownian motion on manifolds with polynomial volume growth.