Sharp Two-Sided Estimates on the Heat Kernels and Green Functions of Subordinate Brownian Motions in Smooth Domains

A subordinate Brownian motion is a L'evy process which can obtained by replacing the time of Brownian motion by an independent increasing L'evy process. The infinitesimal generator of a subordinate Brownian motion is φ(Δ), where φ is the Laplace exponent of the subordinator. When φ(λ)=λ^^α/2} for some α in (0, 2), we get the fractional Laplacian Δ^^{α/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 φ(Δ) in smooth domains.