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Heat kernels of non-symmetric jump processes: beyond the stable case Song, Renming


Let $J$ be the Lévy density of a symmetric Lévy process in $\bf{R}^d$ with its Lévy exponent satisfying a weak lower scaling condition at infinity. Consider the non-symmetric and non-local operator \[ {\cal L}^{\kappa}f(x):= \lim_{\epsilon \downarrow 0} \int_{\{z \in \bf{R}^d: |z|>\epsilon\}}(f(x+z)-f(z))\kappa(x,z)J(z)\, dz\, , \] where $\kappa(x,z)$ is a Borel measurable function on $\bf{R}^d\times \bf{R}^d$ satisfying $0<\kappa_0\le \kappa(x,z)\le \kappa_1$, $\kappa(x,z)=\kappa(x,-z)$ and $|\kappa(x,z)-\kappa(y,z)|\le \kappa_2|x-y|^{\beta}$ for some $\beta\in (0, 1)$. We construct the heat kernel $p^\kappa(t, x, y)$ of ${\cal L}^\kappa$, establish its upper bound as well as its fractional derivative and gradient estimates. Under an additional weak upper scaling condition at infinity, we also establish a lower bound for the heat kernel $p^\kappa$. This talk is based on a joint paper with Panki Kim and Zoran Vondracek.

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