arXiv 1804.013132018-04-04math.APmath.PR

非对称Lévy型算子的热核

Heat kernels of non-symmetric Lévy-type operators

Tomasz Grzywny, Karol Szczypkowski

摘要：

我们构造方程$\partial_t=\mathcal{L}^κ$的基本解（热核）$p^κ$，其中在某些假设下，算子$\mathcal{L}^κ$采用以下形式之一，\begin{align*} \mathcal{L}^κf(x)&:= \int_{\mathbb{R}^d}( f(x+z)-f(x)- 1_{|z|<1} \left<z,\nabla f(x)\right>)κ(x,z)J(z)\, dz \,, \mathcal{L}^κf(x)&:= \int_{\mathbb{R}^d}( f(x+z)-f(x))κ(x,z)J(z)\, dz\,, \mathcal{L}^κf(x)&:= \frac1{2}\int_{\mathbb{R}^d}( f(x+z)+f(x-z)-2f(x))κ(x,z)J(z)\, dz\,. \end{align*}特别地，$J\colon \mathbb{R}^d \to [0,\infty]$是Lévy密度，即，$\int_{\mathbb{R}^d}(1\land |x|^2)J(x)dx<\infty$。假设函数$κ(x,z)$在满足$0<κ_0\leq κ(x,z)\leq κ_1$的$\mathbb{R}^d\times \mathbb{R}^d$上是Borel可测的，并且对于某些$β\in (0, 1)$，满足$|κ(x,z)-κ(y,z)|\leq κ_2|x-y|^β$.我们证明了$p^κ$的唯一性、估计、正则性和其他定性性质。

英文摘要：

We construct the fundamental solution (the heat kernel) $p^κ$ to the equation $\partial_t=\mathcal{L}^κ$, where under certain assumptions the operator $\mathcal{L}^κ$ takes one of the following forms, \begin{align*} \mathcal{L}^κf(x)&:= \int_{\mathbb{R}^d}( f(x+z)-f(x)- 1_{|z|<1} \left<z,\nabla f(x)\right>)κ(x,z)J(z)\, dz \,, \mathcal{L}^κf(x)&:= \int_{\mathbb{R}^d}( f(x+z)-f(x))κ(x,z)J(z)\, dz\,, \mathcal{L}^κf(x)&:= \frac1{2}\int_{\mathbb{R}^d}( f(x+z)+f(x-z)-2f(x))κ(x,z)J(z)\, dz\,. \end{align*} In particular, $J\colon \mathbb{R}^d \to [0,\infty]$ is a Lévy density, i.e., $\int_{\mathbb{R}^d}(1\land |x|^2)J(x)dx<\infty$. The function $κ(x,z)$ is assumed to be Borel measurable on $\mathbb{R}^d\times \mathbb{R}^d$ satisfying $0<κ_0\leq κ(x,z)\leq κ_1$, and $|κ(x,z)-κ(y,z)|\leq κ_2|x-y|^β$ for some $β\in (0, 1)$. We prove the uniqueness, estimates, regularity and other qualitative properties of $p^κ$.

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摘要：

英文摘要：

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