论文标题
DIRICHLET热内核估计一大类各向异性马尔可夫工艺
Dirichlet Heat kernel estimates for a large class of anisotropic Markov processes
论文作者
论文摘要
令$ z =（z^{1}，\ ldots，z^{d}）$为d-dimensionallévy{process}，其中{$ z^i $'s}是独立的一维lévyvy{processes}，具有相同的跳跃kernel $ nbernel $ nbernel $ nbernel $ nbernel $ n kernel $ n kernel $ c = r^r^r^r^r^r^{ - 1} $ vance。这里$ ϕ $是{an}的增加功能，其较弱的缩放条件$ \ usewessinlineα，\operlineα\ in（0，2）$。我们考虑一个对称函数$ j（x，y）$可与\ begin {align*}相当 \ begin {cases}ν^1（| x^i -y^i |）\ qquad＆\ text {如果$ x^i \ ne y^i^i $ for Some $ i $和$ x^j = y^j $ for ALL $ j \ ne I $} \\ 0 $ j \ ne i $} \ \ \ \ \ \ 0 \ qquad＆qquad＆qquad＆\ qquad＆\ qquad＆\ text {if $ x^i $ y y y y y $ y $ i $ i $ i $ i $ i $ \ end {cases} \ end {align*}与跳跃内核$ j $相对应，存在各向异性马可分子进程$ x $，请参见\ cite {kw22}。在本文中，我们在某些规律性条件下建立了$ x $ in $ c^{1,1} $开放式$ x $ in $ x $ in $ x $的尖锐的双面迪利奇热内核估计。作为主要结果的应用，我们得出了绿色函数估计。
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.
