Spectral heat content for Lévy processes

Abstract

In this paper we study the spectral heat content for various Lévy processes. We establish the small time asymptotic behavior of the spectral heat content for Lévy processes of bounded variation in Rd, d≥1. We also study the spectral heat content for arbitrary open sets of finite Lebesgue measure in double-struckR with respect to symmetric Lévy processes of unbounded variation under certain conditions on their characteristic exponents. Finally, we establish that the small time asymptotic behavior of the spectral heat content is stable under integrable perturbations to the Lévy measure.

Small time asymptotics of spectral heat contents for subordinate killed Brownian motions related to isotropic α‐stable processes

“…The heat content with respect to Lévy processes, especially Brownian motions, has been studied extensively, see, for instance, . The spectral heat content with respect to Brownian motion has also been studied a lot (see ). In , a two‐term small time expansion for was established for bounded domains and in a three‐term small time expansion for was obtained for bounded domains with boundary.…”

“…Upper and lower bounds for , , were established in , while explicit expressions for the second term in the asymptotic behavior of , , in dimension 1 for bounded open intervals were obtained in . In the recent paper , the results of were generalized in several directions.…”

On the heat content for the Poisson kernel over the unit ball in the euclidean space

“…Based on the result presented in Theorem 1.1, we conjecture that the spectral heat content defined in (1.5) will satisfy an expansion of the form: for some positive constants and depending on the geometry of the unit ball (they are not expected to have a nice and closed form) where our limit stated in Theorem 1.1 might provide either an upper or lower bound for the constant as has been done for in [3, 13].…”

“…It is noteworthy that the study of the spectral heat content and heat content started by considering first the Gaussian kernel and its connection with the Laplace operator Δ, whose expansions contain geometric information about the underlying sets where these functions are defined. We refer the interested reader to [4][5][6]14] for further details on these topics and [8,13] for results concerning the heat content and spectral heat content associated to other Lévy processes besides that of Brownian motion.…”

Higher Order Terms of the Spectral Heat Content for Killed Subordinate and Subordinate Killed Brownian Motions Related to Symmetric α-Stable Processes in ${\mathbb {R}}$

“…When the Brownian motions are replaced by other Lévy processes, the corresponding quantity is called a spectral heat content for the Lévy processes. It was recently studied intensively in [1,2,9].…”