Perturbation by Non-Local Operators

Suppose that d ≥ 1 and 0 < β < α < 2. We establish the existence and uniqueness of the fundamental solution q(t, x, y) to a class of (possibly nonsymmetric) non-local operators L = ∆ + S, where Sf(x) := A(d,−β) ∫ R ( f(x+ z)− f(x)−∇f(x) · z1{|z|≤1} ) b(x, z) |z|d+β dz and b(x, z) is a bounded measurable function on R×R with b(x, z) = b(x,−z) for x, z ∈ R. Here A(d,−β) is a normalizing constant so that S = ∆ when b(x, z) ≡ 1. We show that if b(x, z) ≥ − A(d,−β) |z| , then q(t, x, y) is a strictly positive continuous function and it uniquely determines a conservative Feller process X, which has strong Feller property. The Feller process X is the unique solution to the martingale problem of (L,S(R)), where S(R) denotes the space of tempered functions on R. Furthermore, sharp two-sided estimates on q(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 R, respectively, that are independent to each other. Solution to stochastic differential equations dXt = dYt + c(Xt−)dZt has infinitesimal generator L with b(x, z) = |c(x)| . AMS 2010 Mathematics Subject Classification: Primary 60J35, 47G20, 60J75; Secondary 47D07