Point-wise estimates for nonlocal heat kernel of convolution type operators

The paper deals with point-wise estimates for the heat kernel of a nonlocal convolution type operator with a kernel that decays at least exponentially at infinity. It is shown that the large time behaviour of the heat kernel depends essentially on whether $\mathrm{\mid }x\mathrm{\mid }\ll t$, or $\mathrm{\mid }x\mathrm{\mid }\sim t$, or $\mathrm{\mid }x\mathrm{\mid }\gg t$. We obtain sharp point-wise upper bounds for the heat kernel in all these regions. In the first region the nonlocal heat kernel behaves like the classical one, while in the other regions we observe an essential difference.