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- W1582731318 abstract "Let $$dgeqslant 1$$ and $$alpha in (0, 2)$$ . Consider the following non-local and non-symmetric Lévy-type operator on $${mathbb R}^d$$ : $$begin{aligned} {fancyscript{L}}^kappa _{alpha }f(x):=hbox {p.v.}int _{{mathbb R}^d}(f(x+z)-f(x)) frac{kappa (x,z)}{ |z|^{d+alpha }} {mathord {mathrm{d}}}z, end{aligned}$$ where $$0<kappa _0leqslant kappa (x,z)leqslant kappa _1, kappa (x,z)=kappa (x,-z)$$ , and $$|kappa (x,z)-kappa (y,z)|leqslant kappa _2|x-y|^beta $$ for some $$beta in (0,1)$$ . Using Levi’s method, we construct the fundamental solution (also called heat kernel) $$p^kappa _alpha (t, x, y)$$ of $${fancyscript{L}}^kappa _alpha $$ , and establish its sharp two-sided estimates as well as its fractional derivative and gradient estimates. We also show that $$p^kappa _alpha (t, x, y)$$ is jointly Hölder continuous in $$(t, x)$$ . The lower bound heat kernel estimate is obtained by using a probabilistic argument. The fundamental solution of $${fancyscript{L}}^kappa _{alpha }$$ gives rise a Feller process $${X, {mathbb P}_x, xin {mathbb R}^d}$$ on $${mathbb R}^d$$ . We determine the Lévy system of $$X$$ and show that $${mathbb P}_x$$ solves the martingale problem for $$({fancyscript{L}}^kappa _{alpha }, C^2_b({mathbb R}^d))$$ . Furthermore, we show that the $$C_0$$ -semigroup associated with $${fancyscript{L}}^kappa _alpha $$ is analytic in $$L^p ({mathbb R}^d)$$ for every $$pin [1,infty )$$ . A maximum principle for solutions of the parabolic equation $$partial _t u ={fancyscript{L}}^kappa _alpha u$$ 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 $${mathord {mathrm{d}}}X_t = A(X_{t-}) {mathord {mathrm{d}}}Y_t$$ is derived, where $$Y$$ is a (rotationally) symmetric stable process on $${mathbb R}^d$$ and $$A(x)$$ is a Hölder continuous $$dtimes d$$ matrix-valued function on $${mathbb R}^d$$ that is uniformly elliptic and bounded." @default.
- W1582731318 created "2016-06-24" @default.
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- W1582731318 date "2015-05-10" @default.
- W1582731318 modified "2023-10-01" @default.
- W1582731318 title "Heat kernels and analyticity of non-symmetric jump diffusion semigroups" @default.
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- W1582731318 doi "https://doi.org/10.1007/s00440-015-0631-y" @default.
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