Sharp estimates on the heat kernels and Green functions of subordinate Brownian motions in smooth domains

A subordinate Brownian motion is a Lévy process which can obtained by replacing the time of Brownian motion by an independent increasing Lévy process. The infinitesimal generator of a subordinate Brownian motion is \(-\phi(-\Delta)\), where \(\phi\) is the Laplace exponent of the subordinator. When \(\phi(\lambda)=\lambda^{\alpha/2}\) for some \(\alpha\in (0, 2)\), we get the fractional Laplacian \(-(-\Delta)^{\alpha/2}\) as a special case. In this talk, I will give a survey of some recent results on sharp two-sided estimates on the Dirichlet heat kernels and Green functions of \(-\phi(-\Delta)\) in smooth domains.