Heat Kernels for reflected jump diffusions on Ahlfors regular domains 

In this talk, we study two-sided heat kernel estimates for symmetric reflected jump diffusions on Ahlfors regular domains in metric measure spaces. For symmetric jump processes on metric measure spaces, it has been established that mixed stable-like heat kernel estimates hold if and only if a two-sided jump kernel condition and a cut-off Sobolev inequality are satisfied. So the key is to show that the cut-off Sobolev inequality holds for reflected jump diffusions on Ahlfors regular domains. For this, we show that there is  a suitable extension operator from the Ahlfors regular domain to the whole metric space.  

This talk is based on joint work with Zhen-Qing Chen.