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 Lévytype operator on Rd: L κ α f (x) := p.v. ∫ Rd ( f (x + z) − f (x)) z) |z|d+α dz, where 0 < κ0 6 κ(x, z) 6 κ1, κ(x, z) = κ(x,−z), and |κ(x, z)−κ(y, z)| 6 κ2|x−y| for some β ∈ (0, 1). Using Levi’s method, we construct the fundamental solution (also called heat kernel) pα(t, x, y) of L κ α , and establish its sharp two-sided estimates as well as its fractional derivative and gradient estimates of the heat kernel. We also show that pα(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 L κ α gives rise a Feller process {X,Px, x ∈ Rd} on Rd. We determine the Lévy system of X and show that Px solves the martingale problem for (L κ α ,C 2 b(R d)). Furthermore, we obtain the analyticity of the non-symmetric semigroup associated with L κ α in L p-spaces for every p ∈ [1,∞). A maximum principle for solutions of the parabolic equation ∂tu = L κ αu is also established.