Spatial asymptotics at infinity for heat kernels of integro-differential operators

We study a spatial asymptotic behaviour at infinity of kernels $p_t(x)$ for convolution semigroups of nonlocal pseudo-differential operators. We give general and sharp sufficient conditions under which the limits $$\lim_{r \to \infty} \frac{p_t(r\theta-y)}{t \, \nu(r\theta)}, \quad t \in T, \ \ \theta \in E, \ \ y \in \mathbb R^d,$$ exist and can be effectively computed. Here $\nu$ is the corresponding L\'evy density, $T \subset (0,\infty)$ is a bounded time-set and $E$ is a subset of the unit sphere in $\mathbb R^d$, $d \geq 1$. Our results are local on the unit sphere. They apply to a wide class of convolution semigroups, including those corresponding to highly asymmetric (finite and infinite) L\'evy measures. Key examples include fairly general families of stable, tempered stable, jump-diffusion and compound Poisson semigroups. A main emphasis is put on the semigroups with L\'evy measures that are exponentially localized at infinity, for which our assumptions and results are strongly related to the existence of the multidimensional exponential moments. Here a key example is the evolution semigroup corresponding to the so-called quasi-relativistic Hamiltonian $\sqrt{-\Delta+m^2} - m$, $m>0$. As a byproduct, we also obtain sharp two-sided estimates of the kernels $p_t$ in generalized cones, away from the origin.