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

Abstract

We study a spatial asymptotic behavior at infinity of kernels p t ( x ) p_t(x) for convolution semigroups of nonlocal pseudo-differential operators. We give general and sharp sufficient conditions under which the limits lim r → ∞ p t ( r θ − y ) t ν ( r θ ) , t ∈ T , θ ∈ E , y ∈ R d \begin{equation*} \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 \end{equation*} exist and can be effectively computed. Here ν \nu is the corresponding Lévy density, T ⊂ ( 0 , ∞ ) T \subset (0,\infty ) is a bounded time-set, and E E is a subset of the unit sphere in R d \mathbb {R}^d , d ≥ 1 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) Lévy 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 Lévy 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 − Δ + m 2 − m \sqrt {-\Delta +m^2} - m , m > 0 m>0 . As a byproduct, we also obtain sharp two-sided estimates of the kernels p t p_t in generalized cones, away from the origin.