Groupe d’études et de recherche en analyse des décisions

# 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.