Spectral heat content for α-stable processes in C1,1 open sets

Abstract

In this paper we study the asymptotic behavior, as t ↓ 0, of the spectral heat contentTogether with the results from [2] for d = 1 and [7] for α ∈ (0, 1), the main theorem of this paper establishes the asymptotic behavior of the spectral heat content up to the second term for all α ∈ (0, 2) and d ≥ 1, and resolves the conjecture raised in [2].

Spectral heat content on a class of fractal sets for subordinate killed Brownian motions

“…G is the first exit time of the Brownian motions out of G, is barely non-integrable with respect to the law of stable subordinator S (α/2) t . Note that this occurrence of extra logarithm term ln(1/t) is observed for smooth open sets in [1,13,14], but it happens at a different index α. More specifically, for the spectral heat content on smooth sets this phenomenon happens when α = 1, whereas for open sets with fractal boundaries as in our case, this happens when α = d − b, which is strictly less than 1.…”

“…Even though the spectral heat content for killed subordinate Brownian motions is a natural object to study as it covers a large class of the spectral heat content for killed Lévy processes, the spectral heat content for subordinate killed Brownian motions is also important as it oftentimes gives useful information on the spectral heat content for the killed subordinate Brownian motions. For example, in [14] the asymptotic behavior of the spectral heat content for subordinate killed Brownian motions with respect to stable subordinators provides crucial information on the spectral heat content for killed stable processes.…”

Large-time and small-time behaviors of the spectral heat content for time-changed stable processes

“…Recently, there have been growing interests in studying the spectral heat content for jump processes, which is defined by replacing the Brownian motions with other jump processes and putting the zero exterior condition outside Ω in order to take into account the fact that jump processes could exit the domain by jumping into R d \ Ω. In [1,2,4,6,8,9,10], the spectral heat content for stable processes and other Lévy processes are studied for both subordinate killed processes and killed subordinate processes. In particular, in [10], the two-term asymptotic expansion for the spectral heat content for isotropic stable processes on bounded C 1,1 open sets was investigated.…”

“…In [1,2,4,6,8,9,10], the spectral heat content for stable processes and other Lévy processes are studied for both subordinate killed processes and killed subordinate processes. In particular, in [10], the two-term asymptotic expansion for the spectral heat content for isotropic stable processes on bounded C 1,1 open sets was investigated.…”

Large-time and small-time behaviors of the spectral heat content for time-changed stable processes

“…In [1,2,4,6,8,9,10], the spectral heat content for stable processes and other Lévy processes are studied for both subordinate killed processes and killed subordinate processes. In particular, in [10], the two-term asymptotic expansion for the spectral heat content for isotropic stable processes on bounded C 1,1 open sets was investigated.…”