Heat kernels and analyticity of non-symmetric jump diffusion semigroups

Let d⩾ 1 and α∈ (0, 2). Consider the following non-local and non-symmetric Lévy-type operator on R^d: (Formula presented.), where 0 < κ₀⩽ κ(x, z) ⩽ κ₁, κ(x, z) = κ(x, - z) , and | κ(x, z) - κ(y, z) | ⩽ κ₂|x-y| ^β for some β∈ (0, 1). Using Levi’s method, we construct the fundamental solution (also called heat kernel) pακ(t,x,y) of Lακ, and establish its sharp two-sided estimates as well as its fractional derivative and gradient estimates. We also show that pακ(t,x,y) is jointly Hölder continuous in (t, x). The lower bound heat kernel estimate is obtained by using a probabilistic argument. The fundamental solution of Lακ gives rise a Feller process {X, P_x, x∈ R^d} on R^d. We determine the Lévy system of X and show that P_x solves the martingale problem for (Lακ,Cb2(Rd)). Furthermore, we show that the C₀-semigroup associated with Lακ is analytic in L^p(R^d) for every p∈ [1, ∞). A maximum principle for solutions of the parabolic equation ∂tu=Lακu is also established. As an application of the main result of this paper, sharp two-sided estimates for the transition density of the solution of d X_t= A(X_t-) d Y_t is derived, where Y is a (rotationally) symmetric stable process on R^d and A(x) is a Hölder continuous d× d matrix-valued function on R^d that is uniformly elliptic and bounded.