arXiv subjects

Hyunchul Park

Publications and source records attributed to Hyunchul Park.

16 recordsLinked to original sources

arXiv2026

Small-time heat decay for stable processes on fractal drums

In this paper, we study the spectral heat content for isotropic stable processes on fractal drums (namely, open sets with fractal boundaries). The spectral heat content for subordinate killed Brownian motions by stable subordinators was investigated in \cite{PX23}, and the present work serves as a natural extension of \cite{PX23} for the spectral heat content for stable processes. Under suitable geometric conditions on the underlying domains, we show that the decay rate of the spectral heat content for stable processes differs substantially from that for subordinate killed Brownian motions when $\alpha=d-\b$, where $\b$ is the interior Minkowski dimension of the boundary of the underlying open set.

math.PR↗

arXiv2025

Dichotomy in the small-time asymptotics of spectral heat content for L\'evy processes

We establish a dichotomy in the small-time asymptotic behavior of the spectral heat content (SHC) for symmetric, but not necessarily isotropic, L\'evy processes whose L\'evy density satisfies a weak lower scaling condition near zero. This dichotomy is governed by whether the process has unbounded or bounded variation. In the unbounded variation case, the leading asymptotic behavior of the SHC is determined by the expected supremum of the process projected in the normal direction near the boundary. In contrast, for processes with bounded variation, the SHC decays linearly in time. Our main result, Theorem \ref{thm:main}, extends and unifies key results from \cite{GPS19}, \cite{KP24}, and \cite{PS22}, covering a broader class of non-isotropic L\'evy processes and offering a streamlined proof.

math.PR↗

arXiv2024

On the principal eigenvalue for compound Poisson processes

We investigate the explicit expression for the principal eigenvalue $λ_{1}^{X}(D)$ for a large class of compound Poisson processes $X$ on a bounded open set $D$ by examining its spectral heat content. When the jump density of the compound Poisson process is radially symmetric and strictly decreasing, we demonstrate that balls are the unique minimizers for $λ_{1}^{X}(D)$ among all sets with equal Lebesgue measure. Furthermore, we show that this uniqueness fails if the jump density is not strictly decreasing.

math.PR↗

arXiv2024

Heat content for Gaussian processes: small-time asymptotic analysis

This paper establishes the small-time asymptotic behaviors of the regular heat content and spectral heat content for general Gaussian processes in both one-dimensional and multi-dimensional settings, where the boundary of the underlying domain satisfies some smoothness condition. For the amount of heat loss associated with the spectral heat content, the exact asymptotic behavior with rate function being the expected supremum process is obtained, whereas for the regular heat content, the exact asymptotic behavior is described in terms of the standard deviation function.

math.PR↗

arXiv2024

A unified approach to the small-time behavior of the spectral heat content for isotropic Lévy processes

This paper establishes the precise small-time asymptotic behavior of the spectral heat content for isotropic Lévy processes on bounded $C^{1,1}$ open sets of $\mathbb{R}^{d}$ with $d\ge 2$, where the underlying characteristic exponents are regularly varying at infinity with index $α\in (1,2]$, including the case $α=2$. Moreover, this asymptotic behavior is shown to be stable under an integrable perturbation of its Lévy measure. These results cover a wide class of isotropic Lévy processes, including Brownian motions, stable processes, and jump diffusions, and the proofs provide a unified approach to the asymptotic behavior of the spectral heat content for all of these processes.

math.PR↗

arXiv2023

Spectral heat content for α-stable processes in C1,1 open sets

In this paper we study the asymptotic behavior, as $t\downarrow 0$, of the spectral heat content $Q^{(α)}_{D}(t)$ for isotropic $α$-stable processes, $α\in [1,2)$, in bounded $C^{1,1}$ open sets $D\subset \R^{d}$, $d\geq 2$. Together with the results from \cite{Val2017} for $d=1$ and \cite{GPS19} for $α\in (0,1)$, the main theorem of this paper establishes the asymptotic behavior of the spectral heat content up to the second term for all $α\in (0,2)$ and $d\geq1$, and resolves the conjecture raised in \cite{Val2017}.

math.PR↗

arXiv2022

Large-time and small-time behaviors of the spectral heat content for time-changed stable processes

We study the large-time and small-time asymptotic behaviors of the spectral heat content for time-changed stable processes, where the time change belongs to a large class of inverse subordinators. For the large-time behavior, the spectral heat content decays polynomially with the decay rate determined by the Laplace exponent of the underlying subordinator, which is in sharp contrast to the exponential decay observed in the case when the time change is a subordinator. On the other hand, the small-time behavior exhibits three different decay regimes, where the decay rate is determined by both the Laplace exponent and the index of the stable process.

math.PR↗

arXiv2021

Spectral heat content for time-changed killed Brownian motions

The spectral heat content is investigated for time-changed killed Brownian motions on C1,1 open sets, where the time change is given by either a subordinator or an inverse subordinator, with the underlying Laplace exponent being regularly varying at \infty with index β\in (0, 1). In the case of inverse subordinators, the asymptotic limit of the spectral heat content is shown to involve a probabilistic term depending only on β\in (0, 1). In contrast, in the case of subordinators, this universality holds only when β\in ( 1/2 , 1).

math.PR↗

arXiv2021

Spectral heat content on a class of fractal sets for subordinate killed Brownian motions

We study the spectral heat content for a class of open sets with fractal boundaries determined by similitudes in $\mathbb{R}^{d}$, $d\geq 1$, with respect to subordinate killed Brownian motions via $α/2$-stable subordinators and establish the asymptotic behavior of the spectral heat content as $t\to 0$ for the full range of $α\in (0,2)$. Our main theorems show that these asymptotic behaviors depend on whether the sequence of logarithms of the coefficients of the similitudes is arithmetic when $α\in [d-\b,2)$, where $\b$ is the interior Minkowski dimension of the boundary of the open set. The main tools for proving the theorems are the previous results on the spectral heat content for Brownian motions and the renewal theorem.

math.PR↗

arXiv2021

Higher order terms of the spectral heat content for killed subordinate and subordinate killed Brownian motions related to symmetric α-stable processes in R

We investigate the 3rd term of spectral heat content for killed subordinate and subordinate killed Brownian motions on a bounded open interval D = (a, b) in a real line when the underlying subordinators are stable subordinators with index αis in (1, 2) or α= 1. We prove that in the 3rd term of spectral heat content, one can observe the length b-a of the interval D.

math.PR↗

arXiv2019

Uniform dimension results for the inverse images of symmetric Lévy processes

We prove uniform Hausdorff and packing dimension results for the inverse images of a large class of real-valued symmetric Lévy processes. Our main result for the Hausdorff dimension extends that of Kaufman (1985) for Brownian motion and that of Song, Xiao, and Yang (2018) for $α$-stable Lévy processes with $1<α<2$. Along the way, we also prove an upper bound for the uniform modulus of continuity of the local times of these processes.

math.PR↗

arXiv2018

Small time asymptotics of spectral heat contents for subordinate killed Brownian motions related to isotropic α-stable processes

In this paper we study the small time asymptotic behavior of the spectral heat content $\widetilde{Q}_D^{(α)}(t)$ of an arbitrary bounded $C^{1,1}$ domain $D$ with respect to the \textit{subordinate killed Brownian motion} in $D$ via an $(α/2)$-stable subordinator. For all $α\in (0,2)$, we establish a two-term small time expansion for $\widetilde{Q}_D^{(α)}(t)$ in all dimensions. When $α\in (1,2)$ and $d\geq 2$, we establish a three-term small time expansion for $\widetilde{Q}_D^{(α)}(t)$.

math.PR↗

arXiv2017

Spectral Heat Content for Lévy Processes

In this paper we study the spectral heat content for various Lévy processes. We establish the asymptotic behavior of the spectral heat content for Lévy processes of bounded variation in $\mathbb{R}^{d}$, $d\geq 1$. We also study the spectral heat content for arbitrary open sets of finite Lebesgue measure in $\mathbb{R}$ with respect to Lévy processes of unbounded variation under certain conditions on their characteristic exponents. Finally we establish that the asymptotic behavior of the spectral heat content is stable under integrable perturbations to the Lévy measure.

math.PR↗

arXiv2017

Fatou's theorem for subordinate Brownian motions with Gaussian components on $C^{1,1}$ open sets

We prove Fatou's theorem for nonnegative harmonic functions with respect to subordinate Brownian motions with Gaussian components on bounded $C^{1,1}$ open sets $D$. We prove that nonnegative harmonic functions with respect to such processes on $D$ converge nontangentially almost everywhere with respect to the surface measure as well as the harmonic measure restricted to the boundary of the domain. In order to prove this, we first prove that the harmonic measure restricted to $\partial D$ is mutually absolutely continuous with respect to the surface measure. We also show that tangential convergence fails on the unit ball.

math.PR↗

arXiv2015

Fatou and relative Fatou theorem for subordinate Brownian motions with Gaussian components on smooth domains

We prove relative Fatou's theorem for nonnegative harmonic functions with respect to a large class of killed subordinate Brownian motions with Gaussian components in bounded $C^{1,1}$ open sets in $\mathbb{R}^{d}$, $d\geq 2$, which asserts the existence of nontangential limit of the ratio of two harmonic functions with respect to the killed processes. When $D=B(x_{0},r)$ is a ball we prove Fatou theorem. That is, we establish the existence of nontangential limit of a single nonnegative harmonic function. We also prove this is the best result possible by showing that there is a nonnegative harmonic function which does not have a tangential limit a.e. when $d=2$ and $D=B(0,1)$.

math.PR↗

arXiv2012

Trace estimates for relativistic stable processes

In this paper, we study the asymptotic behavior, as the time $t$ goes to zero, of the trace of the semigroup of a killed relativistic $α$-stable process in bounded $C^{1,1}$ open sets and bounded Lipschitz open sets. More precisely, we establish the asymptotic expansion in terms of $t$ of the trace with an error bound of order $t^{2/α}t^{-d/α}$ for $C^{1,1}$ open sets and of order $t^{1/α}t^{-d/α}$ for Lipschitz open sets. Compared with the corresponding expansions for stable processes, there are more terms between the orders $t^{-d/α}$ and $t^{(2-d)/α}$ for $C^{1,1}$ open sets, and, when $α\in (0, 1]$, between the orders $t^{-d/α}$ and $t^{(1-d)/α}$ for Lipschitz open sets.

math.PR↗