Global heat kernel estimates for symmetric jump processes

Abstract

Abstract. In this paper, we study sharp heat kernel estimates for a large class of symmetric jump-type processes in R d for all t > 0. A prototype of the processes under consideration are symmetric jump processes on R d with jumping intensitywhere ν is a probability measure on [α 1 , α 2 ] ⊂ (0, 2), Φ is an increasing function on [0, ∞) with c 1 e c 2 r β ≤ Φ(r) ≤ c 3 e c 4 r β with β ∈ (0, ∞), and c(α, x, y) is a jointly measurable function that is bounded between two positive constants and is symmetric in (x, y). They include, in particular, mixed relativistic symmetric stable processes on R d with different masses. We also establish the parabolic Harnack principle.

Fall-Off of Eigenfunctions for Non-Local Schrödinger Operators with Decaying Potentials

“…Note that the condition γ 1 = γ 2 in part (3) of the corollary is a technical assumption required only for an application of the large-time estimates of transition densities in [13]. The above result covers a large family of Lévy processes with jump intensities exponentially localized at infinity.…”

Global Heat Kernel Estimates for Relativistic Stable Processes in Half-space-like Open Sets

“…Until very recently, even in R d , large time sharp two-sided heat kernel estimates were not available for jump processes with Lévy densities decaying exponentially at infinity. In a recent paper [6], the first two named authors, together with Kumagai, succeeded in establishing global sharp two-sided estimates for the heat kernel of a large class of jump processes, including the relativistic α-stable process X 1 . This result provides us a guideline in getting the correct interior estimates of the Dirchlet heat kernel for X 1 in a half-space.…”

Global Heat Kernel Estimates for Symmetric Markov Processes Dominated by Stable-Like Processes in Exterior C 1,η Open Sets

“…Then there exists a positive constant c = c(a, β, T , λ 1 , λ 2) (c = c(a, λ 1 , λ 2 ) when β = 0, respectively) such that for all t ∈ [T , ∞) (t > 0 when β = 0, respectively) and x, y ∈ D with δ D (x)∧δ D (y) ≥ aϕ −1 (t) ≥ 2|x − y|, we have p D (t, x, y) ≥ c /ϕ −d (t).Proof By the same proof as that of [10, Proposition 3.4], we deduce the proposition using the parabolic Harnack inequality(see[15, Theorem 4.12] for β = 0 and[6, Theorem 4.11] for β ∈ (0, ∞]) and Lemma 3.2.…”

“…Then there exists a positive constant c = c(a, β, T ) (c = c(a) when β = 0, respectively) such that for all t ∈ [T , ∞) (t > 0 when β = 0, respectively), we have inf y∈R d P y τ B(y,aϕ −1 (t)) > t ≥ c .Proof When β = 0, using[15, Theorem 4.12 and Proposition 4.9], the proof is almost identical to that of[9, Lemma 3.1]. When β ∈ (0, ∞], using[6, Theorem 4.8], the proof is the same as that of[10, Lemma 3.2]. So we omit the proof detail.…”