Heat kernels and analyticity of non-symmetric jump diffusion semigroups

Let $d\geq 1$ and $α\in (0, 2)$. Consider the following non-local and non-symmetric Lévy-type operator on $\mR^d$: $$ \sL^κ_αf(x):=\mbox{p.v.}\int_{\mR^d}(f(x+z)-f(x))\frac{κ(x,z)}{|z|^{d+α}} \dif z, $$ where $0<κ_0\leq κ(x,z)\leq κ_1$, $κ(x,z)=κ(x,-z)$, and $|κ(x,z)-κ(y,z)|\leqκ_2|x-y|^β$ for some $β\in(0,1)$. Using Levi's method, we construct the fundamental solution (also called heat kernel) $p^κ_α(t, x, y)$ of $\sL^κ_α$, and establish its sharp two-sided estimates as well as its fractional derivative and gradient estimates of the heat kernel. We also show that $p^κ_α(t, x, y)$ is jointly Hölder continuous in $(t, x)$. The lower bound heat kernel estimate is obtained by using a probabilistic argument. The fundamental solution of $\sL^κ_α$ gives rise a Feller process $\{X, \mP_x, x\in \mR^d\}$ on $\mR^d$. We determine the Lévy system of $X$ and show that $\mP_x$ solves the martingale problem for $(\sL^κ_α, C^2_b(\mR^d))$. Furthermore, we obtain the analyticity of the non-symmetric semigroup associated with $\sL^κ_α$ in $L^p$-spaces for every $p\in[1,\infty)$. A maximum principle for solutions of the parabolic equation $\partial_t u =\sL^κ_αu$ is also established.

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