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^b(t, x, y) to a class of (typically non-symmetric) non-local operators L^b = ^α/2 + S^b, where S^bf (x) := A(d, −β) _Rd f (x + z) − f (x) − ∇f (x) · z1_{|z|≤1_{} |}^b(x _z|d+^z)_β dz and b(x, z) is a bounded measurable function on R^d × R^d with b(x, z) = b(x, −z) for x, z ∈ R^d. Here A(d, −β) is a normalizing constant so that S^b = ^β/2 when b(x, z) ≡ 1. We show that if b(x, z) ≥ − _A^A(d (d ₋⁻_β)^α)|z|^β−^α, then q^b(t, x, y) is a strictly positive continuous function and it uniquely determines a conservative Feller process X^b, which has strong Feller property. The Feller process X^b is the unique solution to the martingale problem of (L^b, S(R^d)), where S(R^d) denotes the space of tempered functions on R^d. Furthermore, sharp two-sided estimates on q^b(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^d, respectively, that are independent to each other. Solution to stochastic differential equations dX_t = dY_t + c(X_t−)dZ_t has infinitesimal generator L^b with b(x, z) = |c(x)|^β.