2015/03/18 by Zhen-Qing Chen, Ting Yang, Chen, Zhen-Qing +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Applied mathematics #Dirichlet distribution #Economics #FOS: Mathematics #Fractional Laplacian #Heat kernel #Mathematical analysis #Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations #Perturbation (astronomy) #Physics #Probability (math.PR) #math.PR
paper · pdf · doi:10.48550/arxiv.1503.05302
arxiv created 2015/03/18 · openalex publication_date 2015/03/18 · arxiv updated 2015/03/19 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
For d≥ 2 and 0ε\ (f(x+z)-f(x))\fracb(x,z)|z|d+β dz, and b(x,z) is a bounded measurable function on ℝd×ℝd with b(x,z)=b(x,-z) for every x,z∈ℝd. Here \cal A(d, -β) is a normalizing constant so that Sb=-(-Δ)β/2 when b(x, z)≡ 1. It was recently shown in Chen and Wang [arXiv:1312.7594 [math.PR]] that when b(x, z) ≥ -\fracA(d, -α) A(d, -β) |z|β-α, then Lb admits a unique fundamental solution pb(t, x, y) which is strictly positive and continuous. The kernel pb(t, x, y) uniquely determines a conservative Feller process Xb, which has strong Feller property. The Feller process Xb is also the unique solution to the martingale problem of (Lb, S(ℝd)), where S(ℝd) denotes the space of tempered functions on ℝd. In this paper, we are concerned with the subprocess Xb,D of Xb killed upon leaving a bounded C1,1 open set D⊂ ℝd. We establish explicit sharp two-sided estimates for the transition density function of Xb, D.