[Paper Review] Dirichlet heat kernel for unimodal Lévy processes

Krzysztof Bogdan, Tomasz Grzywny+1 more|arXiv (Cornell University)|Feb 19, 2014

advanced mathematical theories31 references4 citations

TL;DR

This paper provides sharp, global two-sided estimates for the Dirichlet heat kernel of isotropic unimodal pure-jump Lévy processes with infinite Lévy measures and weakly scaling Lévy-Khintchine exponents on $C^{1,1}$ domains. By combining estimates of the free heat kernel and boundary superharmonic functions, it establishes explicit approximate factorizations of the Dirichlet heat kernel that are uniform in time and space, extending prior results beyond subordinate Brownian motions to a broader class of unimodal processes.

ABSTRACT

We estimate the heat kernel of the smooth open set for the isotropic unimodal pure-jump Lévy process with infinite Lévy measure and weakly scaling Lévy-Kchintchine exponent.

Motivation & Objective

• To extend sharp heat kernel estimates beyond subordinate Brownian motions to general unimodal Lévy processes with weak scaling conditions.
• To provide global, uniform-in-time and space two-sided estimates for the Dirichlet heat kernel on $C^{1,1}$ open sets.
• To overcome limitations of previous methods based on the boundary Harnack principle, particularly for processes like the truncated stable Lévy process where it fails.
• To establish a synthetic, unified approach using free heat kernel estimates and boundary superharmonic function analysis for bounded, exterior, and halfspace-like $C^{1,1}$ sets.

Proposed method

• The method relies on combining precise estimates of the free transition density $p(t,x,y)$ from prior work [9] for unimodal Lévy processes with weak scaling conditions.
• It uses explicit bounds on superharmonic functions of the process near the boundary of $C^{1,1}$ domains, derived from preparatory work [8].
• The approach constructs approximate factorizations of the Dirichlet heat kernel $p_D(t,x,y)$ in the form $p_D(t,x,y) \approx P^x(\tau_D > t) P^y(\tau_D > t) p(t,x,y)$, valid uniformly across time and space.
• The analysis applies to bounded, exterior, and halfspace-like $C^{1,1}$ sets, leveraging the geometry and scaling properties of the Lévy process.
• It establishes global comparability constants independent of $t$, $x$, and $y$, ensuring uniformity across all time and space scales.
• The method avoids reliance on the boundary Harnack inequality, enabling application to processes where this inequality fails, such as the truncated stable Lévy process.

Experimental results

Research questions

• RQ1 Can sharp, global two-sided estimates for the Dirichlet heat kernel be established for general unimodal Lévy processes beyond subordinate Brownian motions?
• RQ2 How can the Dirichlet heat kernel be factorized in terms of survival probabilities and the free heat kernel for $C^{1,1}$ domains?
• RQ3 What are the conditions under which the boundary Harnack inequality is not required but still sharp estimates can be obtained?
• RQ4 To what extent do global scaling conditions on the Lévy-Khintchine exponent allow for uniform estimates across all time and space scales?
• RQ5 Can the approach be generalized to non-monotone Lévy densities, provided they satisfy weak scaling conditions?

Key findings

• The paper establishes a global approximate factorization of the Dirichlet heat kernel: $p_D(t,x,y) \approx P^x(\tau_D > t) P^y(\tau_D > t) p(t,x,y)$, valid for all $t>0$ and $x,y \in D$ in $C^{1,1}$ domains.
• For exterior $C^{1,1}$ sets, the heat kernel satisfies $p_{\overline{B}_r^c}(t,x,y) \approx \left(1 \wedge \frac{(|x|-r)^{\alpha_1} \wedge (|x|-r)^{\alpha_2}}{t \wedge r^{\alpha_1} \wedge r^{\alpha_2}}\right)^{1/2} \left(1 \wedge \frac{(|y|-r)^{\alpha_1} \wedge (|y|-r)^{\alpha_2}}{t \wedge r^{\alpha_1} \wedge r^{\alpha_2}}\right)^{1/2} p(t,x,y)$, with constants depending only on $d$, $\alpha_1$, and $\alpha_2$.
• The estimates are sharp, with the ratio of upper and lower bounds bounded by a universal constant independent of $t$, $x$, and $y$.
• The method applies to processes with non-continuous Lévy densities, provided they satisfy weak lower and upper scaling conditions, relaxing prior continuity assumptions.
• The approach works for the truncated stable Lévy process, where the boundary Harnack inequality fails, demonstrating broader applicability than previous methods.
• For processes with Lévy density $\nu(x) \approx f(1/|x|)/|x|^d$ where $f$ is nonincreasing and satisfies weak scaling, the method yields uniform estimates, even without continuity of $\nu$.