2014/02/26 by Jie-Ming Wang, Wang, Jie-Ming
Mathematics · #47G20 #60J75 #FOS: Mathematics #Primary 60J35 #Probability (math.PR) #Secondary 47D07 #math.PR #msc:47D07 #msc:47G20 #msc:60J35 #msc:60J75
paper · pdf · doi:10.48550/arxiv.1402.6477
arXiv admin note: substantial text overlap with arXiv:1312.7594
arxiv created 2014/02/26 · arxiv updated 2016/09/30
Suppose that d≥ 1 and 0<β<2. We establish the existence and uniqueness of the fundamental solution qb(t, x, y) to the operator Lb=Δ+Sb, where Sbf(x) := ∫ℝd ( f(x+z) - f(x) - ∇ f(x) ⋅ z\mathbb1_\|z| ≤ 1\ ) \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 x, z∈ ℝd. We show that if for each x∈ℝd, b(x, z) ≥ 0 for a.e. z∈ℝd, 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. Furthermore, sharp two-sided estimates on qb(t, x, y) are derived.