Heat kernels for non-symmetric diffusion operators with jumps

For d >= 2, we prove the existence and uniqueness of heat kernels to the following time-dependent second order diffusion operator with jumps: L-t := 1/2 Sigma(d)(i, j=1) a(ij)(t,x)partial derivative(2)(ij) + Sigma(d)(i=1) b(i)(t, x)partial derivative(i) + L-t(K), where a = (a(ij)) is a uniformly bounded, elliptic, and Holder continuous matrix-valued function, b belongs to some suitable Kato's class, and L-t(K) is a non-local alpha-stable-type operator with bounded kernel kappa. Moreover, we establish sharp two-sided estimates, gradient estimate and fractional derivative estimate for the heat kernel under some mild conditions. (C) 2017 Elsevier Inc. All rights reserved.