Heat kernels for non-symmetric diffusion operators with jumps

Abstract

For d⩾2, we prove the existence and uniqueness of heat kernels to the following time-dependent second order diffusion operator with jumps:Lt:=12∑i,j=1daij(t,x)∂ij2+∑i=1dbi(t,x)∂i+Ltκ, where a=(aij) is a uniformly bounded, elliptic, and Hölder continuous matrix-valued function, b belongs to some suitable Kato's class, and Ltκ is a non-local α-stable-type operator with bounded kernel κ. Moreover, we establish sharp two-sided estimates, gradient estimate and fractional derivative estimate for the heat kernel under some mild conditions.