arXiv subjects

Panki Kim

Publications and source records attributed to Panki Kim.

At least 19 recordsLinked to original sources

arXiv2025

Heat kernel estimates for Markov processes with blowing-up jump kernels

In this paper, we establish sharp two-sided heat kernel estimates for a large class of purely discontinuous symmetric Markov processes on closed subsets $F$ of $\mathbb{R}^d$, whose jump kernels blow up on a Borel subset $\Sigma$ of $F$. We assume that $F\setminus \Sigma$ is a $\kappa$-fat set and is dense in $F$. To the best of our knowledge, this is the first work establishing sharp heat kernel estimates for jump processes whose jump kernels blow up on part of the state space. The jump kernels under consideration take the form $J(x,y)=|x-y|^{-d-\alpha}{\mathcal B}(x,y)$, where $\alpha\in (0,2)$ and the function ${\mathcal B}(x,y)$ blows up at a subset $\Sigma$ of $F$. A fundamental obstacle is that the tails of the jump measures are not uniformly bounded, and hence standard techniques in heat kernel analysis do not provide a priori off-diagonal estimates. To overcome this difficulty, we develop a new approach based on weighted integral estimates for the heat kernel that are sensitive to both the blow-up behavior of the jump kernel and the geometry of $F\setminus \Sigma$. Examples of processes falling within our general framework include traces of isotropic $\alpha$-stable processes in $C^{1,\rm Dini}$ sets, processes in Lipschitz sets arising in connection with the nonlocal Neumann problem, and a large class of resurrected self-similar processes in the closed upper half-space.

math.PR↗

arXiv2025

Heat kernel estimates for Markov processes in bounded sets with jump kernels decaying at the boundary

In this paper, we study two types of purely discontinuous symmetric Markov processes $X$ in bounded smooth subsets of $\mathbb R^d$: conservative processes and processes killed either upon approaching the boundary of the set or by a killing potential $\kappa$. The jump kernel of $X$ is of the form $J(x,y)={\cal B}(x,y)|x-y|^{-d-\alpha}$, $\alpha\in (0,2)$, where the function ${\cal B}(x,y)$ decays to 0 at the boundary and is described in terms of two $O$-regularly varying functions and one slowly varying function. Under the conditions, introduced in \cite{CKSV24}, on ${\cal B}(x,y)$ and on the killing potential $\kappa$, we establish sharp two-sided estimates on the heat kernel of $X$: in Lipschitz sets when $X$ is conservative, and in $C^{1,1}$ open sets for the killed process.

math.PR↗

arXiv2025

Stability of H\"older regularity and weighted functional inequalities

We study symmetric Dirichlet forms on metric measure spaces, which may possess both strongly local and pure-jump parts. We introduce a new formulation of a tail condition for jump measures and weighted functional inequalities. Our framework accommodates Dirichlet forms with singular jump measures and those associated with trace processes of mixed-type stable processes. Using these new weighted functional inequalities, we establish stable, equivalent characterizations of H\"older regularity for caloric and harmonic functions. As an application of our main result, we prove the H\"older continuity of caloric functions for a large class of symmetric Markov processes exhibiting boundary blow-up behavior, among other results.

math.PR↗

arXiv2025

Heat kernel estimates for Schr\"odinger operators with supercritical killing potentials

In this paper, we study the Schr\"odinger operator $\Delta-V$, where $V$ is a supercritical non-negative potential belonging to a large class of functions containing functions of the form $b|x|^{-(2+2\beta)}$, $b, \beta>0$. We obtain two-sided estimates on the heat kernel $p(t, x, y)$ of $\Delta-V$, along with estimates for the corresponding Green function. Unlike the case of the fractional Schr\"odinger operator $-(-\Delta)^{\alpha/2}-V$, $\alpha\in (0, 2)$, with supercritical killing potential dealt with in [11], in the present case, the heat kernel $p(t, x, y)$ decays to 0 exponentially as $x$ or $y$ tends to the origin.

math.PR↗

arXiv2024

Markov processes with jump kernels decaying at the boundary

The goal of this work is to develop a general theory for non-local singular operators of the type $$ L^{\mathcal{B}}_{\alpha}f(x)=\lim_{\epsilon\to 0} \int_{D,\, |y-x|>\epsilon}\big(f(y)-f(x)\big) \mathcal{B}(x,y)|x-y|^{-d-\alpha}\,dy, $$ and $$ L f(x)=L^{\mathcal{B}}_{\alpha}f(x) - \kappa(x) f(x), $$ in case $D$ is a $C^{1,1}$ open set in $\mathbb{R}^d$, $d\ge 2$. The function $\mathcal{B}(x,y)$ above may vanish at the boundary of $D$, and the killing potential $\kappa$ may be subcritical or critical. From a probabilistic point of view we study the reflected process on the closure $\overline{D}$ with infinitesimal generator $L^{\mathcal{B}}_{\alpha}$, and its part process on $D$ obtained by either killing at the boundary $\partial D$, or by killing via the killing potential $\kappa(x)$. The general theory developed in this work (i) contains subordinate killed stable processes in $C^{1,1}$ open sets as a special case, (ii) covers the case when $\mathcal{B}(x,y)$ is bounded between two positive constants and is well approximated by certain H\"older continuous functions, and (iii) extends the main results known for the half-space in $\mathbb{R}^d$. The main results of the work are the boundary Harnack principle and its possible failure, and sharp two-sided Green function estimates. Our results on the boundary Harnack principle completely cover the corresponding earlier results in the case of half-space. Our Green function estimates extend the corresponding earlier estimates in the case of half-space to bounded $C^{1, 1}$ open sets.

math.PR↗

arXiv2022

Heat kernel estimates for Dirichlet forms degenerate at the boundary

The goal of this paper is to establish sharp two-sided estimates on the heat kernels of two types of purely discontinuous symmetric Markov processes in the upper half-space of $\mathbb R^d$ with jump kernels degenerate at the boundary. The jump kernels are of the form $J(x,y)=\mathcal B(x,y)|x-y|^{-\alpha-d}$, $\alpha\in (0,2)$, where the function $\mathcal B$ depends on four parameters and may vanish at the boundary. Our results are the first sharp two-sided estimates for the heat kernels of non-local operators with jump kernels degenerate at the boundary. The first type of processes are conservative Markov processes on $\overline{\mathbb R}^d_+$ with jump kernel $J(x,y)$. Depending on the regions where the parameters belong, the heat kernels estimates have three different forms, two of them are qualitatively different from all previously known heat kernel estimates. The second type of processes are the processes above killed either by a critical potential or upon hitting the boundary of the half-space. We establish that their heat kernel estimates have the approximate factorization property with survival probabilities decaying as a power of the distance to the boundary, where the power depends on the constant in the critical potential.

math.PR↗

arXiv2022

Laws of the iterated logarithm for occupation times of Markov processes

In this paper, we discuss the laws of the iterated logarithm (LIL) for occupation times of Markov processes $Y$ in general metric measure space both near zero and near infinity under some minimal assumptions. We first establish LILs of (truncated) occupation times on balls $B(x,r)$ of radii $r$ up to an function $\Phi (r)$, which is an iterated logarithm of mean exit time of $Y$, by showing that the function $\Phi$ is optimal. Our first result on LILs of occupation times covers both near zero and near infinity regardless of transience and recurrence of the process. Our assumptions are truly local in particular at zero and the function $\Phi$ in our truncated occupation times $r \mapsto\int_0^{ \Phi (x,r)} {\bf 1}_{B(x,r)}(Y_s)ds$ depends on space variable $x$ too. We also prove that a similar LIL for total occupation times $r \mapsto\int_0^\infty {\bf 1}_{B(x,r)}(Y_s)ds$ holds when the process is transient. Then we establish LIL concerning large time behaviors of occupation times $t \mapsto \int_0^t {\bf 1}_{A}(Y_s)ds$ under an additional condition that guarantees the recurrence of the process. Our results cover a large class of Feller (Levy-like) processes, random conductance models with long range jumps, jump processes with mixed polynomial local growths and jump processes with singular jumping kernels.

math.PR↗

arXiv2022

Potential theory of Dirichlet forms with jump kernels blowing up at the boundary

In this paper we study the potential theory of Dirichlet forms on the half-space $\mathbb{R}^d_+$ defined by the jump kernel $J(x,y)=|x-y|^{-d-\alpha}\mathcal{B}(x,y)$ and the killing potential $\kappa x_d^{-\alpha}$, where $\alpha\in (0, 2)$ and $\mathcal{B}(x,y)$ can blow up to infinity at the boundary. The jump kernel and the killing potential depend on several parameters. For all admissible values of the parameters involved and all $d \ge 1$, we prove that the boundary Harnack principle holds, and establish sharp two-sided estimates on the Green functions of these processes.

math.PR↗

arXiv2022

Harnack inequality and interior regularity for Markov processes with degenerate jump kernels

In this paper we study interior potential-theoretic properties of purely discontinuous Markov processes in proper open subsets $D\subset \mathbb{R}^d$. The jump kernels of the processes may be degenerate at the boundary in the sense that they may vanish or blow up at the boundary. Under certain natural conditions on the jump kernel, we show that the scale invariant Harnack inequality holds for any proper open subset $D\subset \mathbb{R}^d$ and prove some interior regularity of harmonic functions. We also prove a Dynkin-type formula and several other interior results.

math.PR↗

arXiv2022

General Law of iterated logarithm for Markov processes: Liminf laws

Continuing from arXiv:2102.01917v2, in this paper, we discuss general criteria and forms of liminf laws of iterated logarithm (LIL) for continuous-time Markov processes. Under some minimal assumptions, which are weaker than those in arXiv:2102.01917v2, we establish liminf LIL at zero (at infinity, respectively) in general metric measure spaces. In particular, our assumptions for liminf law of LIL at zero and the form of liminf LIL are truly local so that we can cover highly space-inhomogenous cases. Our results cover all examples in arXiv:2102.01917v2 including random conductance models with long range jumps. Moreover, we show that the general form of liminf law of LIL at zero holds for a large class of jump processes whose jumping measures have logarithmic tails and Feller processes with symbols of varying order which are not covered before.

math.PR↗

arXiv2022

Positive self-similar Markov processes obtained by resurrection

In this paper we study positive self-similar Markov processes obtained by (partially) resurrecting a strictly $\alpha$-stable process at its first exit time from $(0,\infty)$. We construct those processes by using the Lamperti transform. We explain their long term behavior and give conditions for absorption at 0 in finite time. In case the process is absorbed at 0 in finite time, we give a necessary and sufficient condition for the existence of a recurrent extension. The motivation to study resurrected processes comes from the fact that their jump kernels may explode at zero. We establish sharp two-sided jump kernel estimates for a large class of resurrected stable processes.

math.PR↗

arXiv2022

Stability of estimates for fundamental solutions under Feynman-Kac perturbations for symmetric Markov processes

In this paper, when a given symmetric Markov process X satisfies the stability of global heat kernel two-sided (upper) estimates by Markov perturbations, we give a necessary and sufficient condition on the stability of global two-sided (upper) estimates for fundamental solution of Feynman-Kac semigroup of X. As a corollary, under the same assumptions, a weak type global two-sided (upper) estimates holds for the fundamental solution of Feynman-Kac semigroup with (extended) Kato class conditions for measures. This generalizes all known results on the stability of global integral kernel estimates by symmetric Feynman-Kac perturbations with Kato class conditions in the framework of symmetric Markov processes.

math.PR↗

arXiv2022

Self-similar solution for Hardy operator

We describe the large-time asymptotics of solutions to the heat equation for the fractional Laplacian with added subcritical or even critical Hardy-type potential. The asymptotics is governed by a self-similar solution of the equation, obtained as a normalized limit at the origin of the kernel of the corresponding Feynman-Kac semigroup.

math.AP↗

arXiv2022

Heat kernel estimates for subordinate Markov processes and their applications

In this paper, we establish sharp two-sided estimates for transition densities of a large class of subordinate Markov processes. As applications, we show that the parabolic Harnack inequality and Hölder regularity hold for parabolic functions of such processes, and derive sharp two-sided Green function estimates.

math.PR↗

arXiv2021

Potential theory of Dirichlet forms degenerate at the boundary: the case of no killing potential

In this paper we consider the Dirichlet form on the half-space $\mathbb{R}^d_+$ defined by the jump kernel $J(x,y)=|x-y|^{-d-\alpha}\mathcal{B}(x,y)$, where $\mathcal{B}(x,y)$ can be degenerate at the boundary. Unlike our previous works [6,7] where we imposed critical killing, here we assume that the killing potential is identically zero. In case $\alpha\in (1,2)$ we first show that the corresponding Hunt process has finite lifetime and dies at the boundary. Then, as our main contribution, we prove the boundary Harnack principle and establish sharp two-sided Green function estimates. Our results cover the case of the censored $\alpha$-stable process, $\alpha\in (1,2)$, in the half-space studied in [2].

math.PR↗

arXiv2021

Two-sided Dirichlet heat estimates of symmetric stable processes on horn-shaped regions

In this paper, we consider symmetric $α$-stable processes on (unbounded) horn-shaped regions which are non-uniformly $C^{1,1}$ near infinity. By using probabilistic approaches extensively, we establish two-sided Dirichlet heat estimates of such processes for all time. The estimates are very sensitive with respect to the reference function corresponding to each horn-shaped region. Our results also cover the case that the associated Dirichlet semigroup is not intrinsically ultracontractive. A striking observation from our estimates is that, even when the associated Dirichlet semigroup is intrinsically ultracontractive, the so-called Varopoulos-type estimates do not hold for symmetric stable processes on horn-shaped regions.

math.PR↗

arXiv2021

Heat kernels for reflected diffusions with jumps on inner uniform domains

In this paper, we study sharp two-sided heat kernel estimates for a large class of symmetric reflected diffusions with jumps on the closure of an inner uniform domain $D$ in a length metric space. The length metric is the intrinsic metric of a strongly local Dirichlet form. When $D$ is an inner uniform domain in the Euclidean space, a prototype for a special case of the processes under consideration are symmetric reflected diffusions with jumps on $D$, whose infinitesimal generators are non-local (pseudo-differential) operators $L$ on $D$ of the form $$ L u(x) =\frac12 \sum_{i, j=1}^d \frac{\partial}{\partial x_i} \left(a_{ij}(x) \frac{\partial u(x)}{\partial x_j}\right) + \lim_{\eps \downarrow 0} \int_{\{y\in D: \, ρ_D(y, x)>\eps\}} (u(y)-u(x)) J(x, y)\, dy $$ satisfying "Neumann boundary condition". Here, $ρ_D(x,y)$ is the length metric on $D$, $A(x)=(a_{ij}(x))_{1\leq i,j\leq d}$ is a measurable $d\times d$ matrix-valued function on $D$ that is uniformly elliptic and bounded, and $$ J(x,y):= \frac{1}{Φ(ρ_D(x,y))} \int_{[α_1, α_2]} \frac{c(α, x,y)} {ρ_D(x,y)^{d+α}} \,ν(dα) , $$ where $ν$ is a finite measure on $[α_1, α_2] \subset (0, 2)$, $Φ$ is an increasing function on $[ 0, \infty )$ with $c_1e^{c_2r^β} \le Φ(r) \le c_3 e^{c_4r^β}$ for some $β\in [0,\infty]$, and $c(α, x, y)$ is a jointly measurable function that is bounded between two positive constants and is symmetric in $(x, y)$.

math.PR↗

arXiv2021

General Law of iterated logarithm for Markov processes: Limsup law

In this paper, we discuss general criteria of limsup law of iterated logarithm (LIL) for continuous-time Markov processes. We consider minimal assumptions for LILs to hold at zero(at infinity, respectively) in general metric measure spaces. We establish LILs under local assumptions near zero (near infinity, respectively) on uniform bounds of the expectations of first exit times from balls in terms of a function $\phi$ and uniform bounds on the tails of the jumping kernel in terms of a function $\psi$. The main result is that a simple ratio test in terms of the functions $\phi$ and $\psi$ completely determines whether there exists a positive non-decreasing function $\Psi$ such that $\limsup |X_t|/\Psi(t)$ is positive and finite a.s., or not. Our results cover a large class of subordinate diffusions, jump processes with mixed polynomial local growths, jump processes with singular jumping kernels and random conductance models with long range jumps.

math.PR↗