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Kyung-Youn Kim

Publications and source records attributed to Kyung-Youn Kim.

9 recordsLinked to original sources
arXiv2024

Dirichlet Heat kernel estimates for a large class of anisotropic Markov processes

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.

math.PR↗
arXiv2021

Variational formulas for the exit time of Hunt processes generated by semi-Dirichlet forms

Variational formulas for the Laplace transform of the exit time from an open set of a Hunt process generated by a regular lower bounded semi-Dirichlet form are established. While for symmetric Markov processes, variational formulas are derived for the exponential moments of the exit time. As applications, we provide some comparison theorems and quantitative relations of the exponential moments and Poincaré inequalities.

math.PR↗
arXiv2021

Heat kernel bounds for nonlocal operators with singular kernels

We prove sharp two-sided bounds of the fundamental solution for an integro-differential operator of order $α\in (0,2)$ that generates a $d$-dimensional Markov process. The corresponding Dirichlet form is comparable to that of $d$ independent copies of one-dimensional jump processes, i.e., the jumping measure is singular with respect to the $d$-dimensional Lebesgue measure.

math.AP↗
arXiv2020

Heat kernel bounds for a large class of Markov process with singular jump

Let $Z=(Z^{1}, \ldots, Z^{d})$ be the $d$-dimensional Lévy processes where $Z^{i}$'s are independent $1$-dimensional Lévy processes with jump kernel $J^{ϕ, 1}(u,w) =|u-w|^{-1}ϕ(|u-w|)^{-1}$ for $u, w\in \mathbb R$. Here $ϕ$ is an increasing function with weak scaling condition of order $\underline α, \overline α\in (0, 2)$. Let $J(x,y) \asymp J^ϕ(x,y)$ be the symmetric measurable function where \begin{align*} J^ϕ(x,y):=\begin{cases} J^{ϕ, 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 jump kernel $J$, we show the existence of non-isotropic Markov processes $X:=(X^{1}, \ldots, X^{d})$ and obtain sharp two-sided heat kernel estimates for the transition density functions.

math.PR↗
arXiv2020

Capacity and Exit Time for Non-reversible Diffusions

Capacity is an important quantity in potential theory and in the study of Markov processes. We give equivalent conditions between the capacity, the mean exit time, and the Green function for non-reversible diffusions.

math.PR↗
arXiv2019

Estimates of Dirichlet heat kernel for symmetric Markov processes

We consider a large class of symmetric pure jump Markov processes dominated by isotropic unimodal Lévy processes with weak scaling conditions. First, we establish sharp two-sided heat kernel estimates for these processes in $C^{1,1}$ open sets. As corollaries of our main results, we obtain sharp two-sided Green function estimates and a scale invariant boundary Harnack inequality with explicit decay rates in $C^{1,1}$ open sets.

math.PR↗
arXiv2015

Global heat kernel estimates for symmetric Markov processes dominated by stable-like processes in exterior $C^{1,η}$ open sets

In this paper, we establish sharp two-sided heat kernel estimates for a large class of symmetric Markov processes in exterior $C^{1,η}$ open sets for all $t> 0$. The processes are symmetric pure jump Markov processes with jumping kernel intensity $$κ(x, y)ψ(|x-y|)^{-1}|x-y|^{-d-α}$$ where $α\in(0,2)$, $ψ$ is an increasing function on $[ 0, \infty)$ with $ψ(r)=1$ on $0 1$ for $β\in[0, \infty]$. A symmetric function $κ(x, y)$ is bounded by two positive constants and $|κ(x, y)-κ(x,x)|\le c_5 |x-y|^ρ$ for $|x-y|<1$ and $ρ>α/2$. As a corollary of our main result, we estimates sharp two-sided Green function for this process in $C^{1,η}$ exterior open sets.

math.PR↗
arXiv2014

Two-sided estimates for the transition densities of symmetric Markov processes dominated by stable-like processes in $C^{1,η}$ open sets

In this paper, we study sharp Dirichlet heat kernel estimates for a large class of symmetric Markov processes in $C^{1,η}$ open sets. The processes are symmetric pure jump Markov processes with jumping intensity $κ(x,y) ψ_1 (|x-y|)^{-1} |x-y|^{-d-α}$, where $α\in (0,2)$. Here, $ψ_1$ is an increasing function on $[ 0, \infty )$, with $ψ_1(r)=1$ on $0 1$ for $β\in [0,\infty]$, and $ κ( x, y)$ is a symmetric function confined between two positive constants, with $|κ(x,y)-κ(x,x)|\leq c_5|x-y|^ρ$ for $|x-y|<1$ and $ρ>α/2$. We establish two-sided estimates for the transition densities of such processes in $C^{1,η}$ open sets when $η\in (α/2, 1]$. In particular, our result includes (relativistic) symmetric stable processes and finite-range stable processes in $C^{1,η}$ open sets when $η\in (α/2, 1]$.

math.PR↗