Dirichlet Heat kernel estimates for a large class of anisotropic Markov processes
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866917728117325824 |
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| author | Kim, Kyung-Youn Wang, Lidan |
| author_facet | Kim, Kyung-Youn Wang, Lidan |
| contents | Let $Z=(Z^{1}, \ldots, Z^{d})$ be the d-dimensional Lévy {process} where {$Z^i$'s} are independent 1-dimensional Lévy {processes} with identical jumping kernel $ ν^1(r) =r^{-1}ϕ(r)^{-1}$. Here $ϕ$ is {an} increasing function with weakly scaling condition of order $\underline α, \overline α\in (0, 2)$. We consider a symmetric function $J(x,y)$ comparable to \begin{align*}
\begin{cases} ν^1(|x^i - y^i|)\qquad&\text{ if $x^i \ne y^i$ for some $i$ and $x^j = y^j$ for all $j \ne i$}\\ 0\qquad&\text{ if $x^i \ne y^i$ for more than one index $i$}. \end{cases} \end{align*} Corresponding to the jumping kernel $J$, there exists an anisotropic Markov process $X$, see \cite{KW22}. In this article, we establish sharp two-sided Dirichlet heat kernel estimates for $X$ in $C^{1,1}$ open set, under certain regularity conditions. As an application of the main results, we derive the Green function estimates. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_11225 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Dirichlet Heat kernel estimates for a large class of anisotropic Markov processes Kim, Kyung-Youn Wang, Lidan Probability 31B25, 60J50 Let $Z=(Z^{1}, \ldots, Z^{d})$ be the d-dimensional Lévy {process} where {$Z^i$'s} are independent 1-dimensional Lévy {processes} with identical jumping kernel $ ν^1(r) =r^{-1}ϕ(r)^{-1}$. Here $ϕ$ is {an} increasing function with weakly scaling condition of order $\underline α, \overline α\in (0, 2)$. We consider a symmetric function $J(x,y)$ comparable to \begin{align*} \begin{cases} ν^1(|x^i - y^i|)\qquad&\text{ if $x^i \ne y^i$ for some $i$ and $x^j = y^j$ for all $j \ne i$}\\ 0\qquad&\text{ if $x^i \ne y^i$ for more than one index $i$}. \end{cases} \end{align*} Corresponding to the jumping kernel $J$, there exists an anisotropic Markov process $X$, see \cite{KW22}. In this article, we establish sharp two-sided Dirichlet heat kernel estimates for $X$ in $C^{1,1}$ open set, under certain regularity conditions. As an application of the main results, we derive the Green function estimates. |
| title | Dirichlet Heat kernel estimates for a large class of anisotropic Markov processes |
| topic | Probability 31B25, 60J50 |
| url | https://arxiv.org/abs/2210.11225 |