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arXiv 1606.02005math.PR
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非对称跳过程的热核：超越稳定情形

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

Panki Kim, Renming Song, Zoran Vondraček

备注：
Gradient estimate improved, several errors corrected; 57 pages
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摘要：

设$J$是$\mathbb{R}^d$中对称Lévy过程的Lévy密度，且其Lévy指数在无穷远处满足弱下标度条件.考虑非对称非局部算子$$ {\mathcal L}^κf(x):= \lim_{ε\downarrow 0} \int_{\{z \in \mathbb{R}^d: |z|>ε\}}(f(x+z)-f(x))κ(x,z)J(z)\, dz\, , $$，其中$κ(x,z)$是$\mathbb{R}^d\times \mathbb{R}^d$上的Borel可测函数，对于某些$β\in (0, 1)$，满足$0<κ_0\le κ(x,z)\le κ_1$，$κ(x,z)=κ(x,-z)$和$|κ(x,z)-κ(y,z)|\le κ_2|x-y|^β$.我们构造了${\mathcal L}^κ$的热核$p^κ(t, x, y)$，建立了它的上界以及它的分数阶导数和梯度估计.在一个额外的弱上标度条件下，在无穷远，我们还建立了一个下界的热核$p^κ$。

英文摘要：

Let $J$ be the Lévy density of a symmetric Lévy process in $\mathbb{R}^d$ with its Lévy exponent satisfying a weak lower scaling condition at infinity. Consider the non-symmetric and non-local operator $$ {\mathcal L}^κf(x):= \lim_{ε\downarrow 0} \int_{\{z \in \mathbb{R}^d: |z|>ε\}}(f(x+z)-f(x))κ(x,z)J(z)\, dz\, , $$ where $κ(x,z)$ is a Borel measurable function on $\mathbb{R}^d\times \mathbb{R}^d$ satisfying $0<κ_0\le κ(x,z)\le κ_1$, $κ(x,z)=κ(x,-z)$ and $|κ(x,z)-κ(y,z)|\le κ_2|x-y|^β$ for some $β\in (0, 1)$. We construct the heat kernel $p^κ(t, x, y)$ of ${\mathcal L}^κ$, 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 $p^κ$.

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