Abnormal boundary decay for stable operators

2025/10/04 by Soobin Cho, Renming Song, Cho, Soobin +1
Economics, Econometrics and Finance · Engineering · Mathematics · #35K08 #47G20 #60J45 #60J50 #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Probability (math.PR) #Stability and Controllability of Differential Equations #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2510.03961

openalex publication_date 2025/10/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/31

Abstract

Assume α∈ (0, 2) and d≥ 2. Let \mathcal L^α be the generator of a symmetric, but not necessarily isotropic, α-stable process X in \mathbb R^d whose Lévy density is comparable with that of an isotropic α-stable process. In this paper, we show that the C^{1, \rm Dini} regularity assumption on an open set D⊂ \mathbb R^d is optimal for the standard boundary decay property for nonnegative \mathcal L^α-harmonic functions in D, and for the standard boundary decay property of the heat kernel p^D(t,x,y) of the part process X^D of X on D by proving the following: (i) If D is a C^{1, \rm Dini} open set and h is a nonnegative function which is \mathcal L^α-harmonic in D and vanishes near a portion of ∂ D, then the rate at which h(x) decays to 0 near that portion of ∂ D is \rm dist (x, D^c)^α/2. (ii) If D is a C^{1, \rm Dini} open set, then, as x→ ∂ D, the rate at which p^D(t,x,y) tends to 0 is \rm dist (x, D^c)^α/2. (iii) For any non-Dini modulus of continuity ℓ, there exist non-C^{1, \rm Dini} open sets D, with ∂ D locally being the graph of a C^{1, ℓ} function, such that the standard boundary decay properties above do not hold for D.

Citations

• Approximate factorizations for non-symmetric jump processes
• Boundary regularity and Hopf lemma for nondegenerate stable operators
• Optimal boundary regularity and Green function estimates for nonlocal equations in divergence form
• Heat kernel estimates for regional fractional Laplacians with multi-singular critical potentials in C^{1, β} open sets
• Robust near-diagonal Green function estimates
• Two-sided estimates for the transition densities of symmetric Markov processes dominated by stable-like processes in C^1,η open sets
• On notions of harmonicity

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