Subexponential behaviour of the Dirichlet heat kernel
Let \(D\) be an open and connected set in \(\mathbb{R}^m\), and let \(p_D(x,y; t)\), \(x\in D\), \(y\in D\), \(t> 0\) denote the Dirichlet heat kernel associated to the parabolic operator \(-\Delta_D+ {\partial\over\partial t}\), where \(-\Delta_D\) is the Dirichlet Laplacian for \(D\). Let (\(B(s)\), \(s\geq 0\); \(P_x\), \(x\in\mathbb{R}^m\)) be a Brownian motion associated to \(-\Delta+{\partial\over\partial t}\), and let \(T_D\) be the first exit time from \(D\). In this paper a relation between the decay properties of \(p_D(x,y; t)\) and \(P_x[T_D> t]\) for large time is established, under assumption that \(P_x[T_D> t]\) decays subexponentially. Variational lower bounds for \(p_D(x,x; t)\) and \(P_x[T_D> 1]\) are given, too.
scientific article; zbMATH DE number 1901762