Estimates of Dirichlet heat kernels
Let \(M\) be a connected Riemannian manifold and \(D\subset M\) a bounded open regular domain. Denote by \(p^D_t(x,y)\) the Dirichlet heat kernel, that is, the fundamental solution of the heat equation \[ \partial_t u(t,x)=Lu(t,x),\quad u|_{\partial D}=0 \] on \(D\) where \(L=\frac {1}{2}(\Delta +Z)\), \(Z\) being a smooth vector field. Explicit lower bounds on \(p^D_t\) and two-side comparison results are obtained by means of logarithmic transformations that allows to derive a Bismut type derivative formula. The Li-Yau type Harnack inequality is also established.
scientific article; zbMATH DE number 1364466
scientific article