Blog

Heat kernel is the fundamental solution of the heat equation, and it is also the transition density of the diffusion process corresponding to the Laplacian. These relation is also ture for non-local operators, and corresponding process is a discontinuous Markov process. In this talk, we consider anisotropic Markov processes. Let $L_i$, $i=1,\ldots,d$, be identical and independent $1$-dimensional L\’evy processes. Define a Markov process $M:=(M_1, \ldots, M_d)$ whose jumping kernel is comparable to that of $L:=(L_1, \ldots, L_d)$. Then $M$ is a pure jump process that jumps parallel to the coordinate axes. We discuss the sharp two-sided heat kernel bounds for $M$ on $C^{1,1}$ open set $D\subset {\rm I\!R}^d$ as well as the potential theory of anisotropic Markov process. This is the joint work with Lidan Wang.

[1] Dirichlet Heat kernel estimates for a large class of anisotropic Markov process (with Lidan Wang).