Estimates of Dirichlet heat kernels

By using logarithmic transformations and stochastic analysis, an explicit lower bound of Dirichlet heat kernels is obtained, which can be sharp for both short time and long time. Next, a two-side comparison theorem is presented for Dirichlet heat kernels and some closed ones, from which we derive the Bismut's type derivative formula for Dirichlet heat kernels. Moreover, the Li-Yau's type Harnack inequality is established for Dirichlet heat semigroups. Finally, the integration estimate of Dirichlet heat kernels is also studied.