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