2013/12/29 by Zhen-Qing Chen, Chen, Zhen-Qing, Jie-Ming Wang +1 · 3 citations
Mathematics · #47G20 #60J75 #Analysis of PDEs (math.AP) #FOS: Mathematics #Primary 60J35 #Probability (math.PR) #Secondary 47D07 #math.AP #math.PR #msc:47D07 #msc:47G20 #msc:60J35 #msc:60J75
paper · pdf · doi:10.48550/arxiv.1312.7594
arxiv created 2016/09/29 · arxiv updated 2016/09/30
Suppose that d≥ 1 and 0<β<α<2. We establish the existence and uniqueness of the fundamental solution qb(t, x, y) to a class of (possibly nonsymmetric) non-local operators Lb=Δα/2+Sb, where Sbf(x):=A(d, -β) ∫Rd ( f(x+z)-f(x)- ∇ f(x) ⋅ z 1_\|z|≤ 1\ ) \fracb(x, z)|z|d+βdz and b(x, z) is a bounded measurable function on Rd× Rd with b(x, z)=b(x, -z) for x, z∈ Rd. Here A(d, -β) is a normalizing constant so that Sb=Δβ/2 when b(x, z)≡ 1. We show that if b(x, z) ≥ -\frac\cal A(d, -α)A(d, -β) |z|β-α, then qb(t, x, y) is a strictly positive continuous function and it uniquely determines a conservative Feller process Xb, which has strong Feller property. The Feller process Xb is the unique solution to the martingale problem of (Lb, \cal S (Rd)), where \cal S(Rd) denotes the space of tempered functions on Rd. Furthermore, sharp two-sided estimates on qb(t, x, y) are derived. In stark contrast with the gradient perturbations, these estimates exhibit different behaviors for different types of b(x, z). The model considered in this paper contains the following as a special case. Let Y and Z be (rotationally) symmetric α-stable process and symmetric β-stable processes on Rd, respectively, that are independent to each other. Solution to stochastic differential equations dXt=dYt + c(Xt-)dZt has infinitesimal generator Lb with b(x, z)=| c(x)|β.