Searcharxiv⌕ Search

arXiv subjects

Jieliang Hong

Publications and source records attributed to Jieliang Hong.

16 recordsLinked to original sources
arXiv2026

Boundary behavior and optimal pointwise H\"older exponent of the total local time of $(1+\beta)$-stable super-Brownian motion

Let $L^x$ be the total local time of one-dimensional super-Brownian motion with $(1+\beta)$-stable branching mechanism, $0<\beta<1$. We prove that $\{x\in\mathbb R:L^x>0\}$ is almost surely a bounded open interval $(\mathsf L,\mathsf R)$ and that \[ h_L(\mathsf L)=h_L(\mathsf R)=1+\frac{2}{\beta} \] almost surely, where $h_L$ denotes the pointwise H\"older exponent. Thus the pointwise $\gamma$-H\"older condition holds at both endpoints for every $\gamma<1+2/\beta$ and fails for every $\gamma>1+2/\beta$. The same conclusions hold under the canonical excursion measure.

math.PR↗
arXiv2025

Occupation times for superprocesses in random environments

Let $X=(X_t, t\geq 0)$ be a superprocess in a random environment governed by a Gaussian noise $W=\{W(t, x),t\geq 0,x\in\mathbb{R}^d\}$ white in time and colored in space with correlation kernel $g$. We consider the occupation time process of the model starting from a finite measure. It is shown that the occupation time process of $X$ is absolutely continuous with respect to Lebesgue measure in $d\leq 3$, whereas it is singular with respect to Lebesgue measure in $d\geq 4$. Regarding the absolutely continuous case in $d\leq 3$, we further prove that the associated density function is jointly H\"older continuous based on the Tanaka formula and moment formulas, and derive the H\"older exponents with respect to the spatial variable $x$ and the time variable $t$.

math.PR↗
arXiv2025

On the differentiability of the local time of the ($1+\beta$)-stable super-Brownian motion

We consider the local times of $(1+\beta)$-stable $d$-dimensional super-Brownian motion with $0<\beta<1$. Mytnik and Perkins (2003) proved that the local time, denoted by $L(t, x)$, is jointly continuous for $d=1$, whereas it is locally unbounded in $x$ for $d\geq 2$ where it exists. This paper strengthens the results of Mytnik and Perkins for $d=1$ by showing that when $X_0$ is atomless, $L(t,\cdot)$ is differentiable at every fixed deterministic point almost surely and is differentiable Lebesgue-a.e. However, with probability one, the local time is almost surely not differentiable at every spatial point simultaneously.

math.PR↗
arXiv2024

Some quenched and annealed limit theorems of superprocesses in random environments

Let $X=(X_t, t\geq 0)$ be a superprocess in a random environment described by a Gaussian noise $W=\{W(t,x), t\geq 0, x\in \mathbb{R}^d\}$ white in time and colored in space with correlation kernel $g(x,y)$. When $d\geq 3$, under the condition that the correlation function $g(x,y)$ is bounded above by some appropriate function $\bar{g}(x-y)$, we present the quenched and annealed Strong Law of Large Numbers and the Central Limit Theorems regarding the weighted occupation measure $\int_0^t X_s ds$ as $t\to \infty$.

math.PR↗
arXiv2024

Stochastic partial differential equations for superprocesses in random environments

Let $X=(X_t, t\geq 0)$ be a superprocess in a random environment described by a Gaussian noise $W^g=\{W^g(t,x), t\geq 0, x\in \mathbb{R}^d\}$ white in time and colored in space with correlation kernel $g(x,y)$. We show that when $d=1$, $X_t$ admits a jointly continuous density function $X_t(x)$ that is a unique in law solution to a stochastic partial differential equation \begin{align*} \frac{\partial }{\partial t}X_t(x)=\fracΔ{2} X_t(x)+\sqrt{X_t(x)} \dot{V}(t,x)+X_t(x)\dot{W}^g(t, x) , \quad X_t(x)\geq 0, \end{align*} where $V=\{V(t,x), t\geq 0, x\in \mathbb{R}\}$ is a space-time white noise and is orthogonal with $W^g$. When $d\geq 2$, we prove that $X_t$ is singular and hence density does not exist.

math.PR↗
arXiv2023

Exceptional times for the instantaneous propagation of superprocess

For a Dawson-Watanabe superprocess $X$ on $\mathbb{R}^d$, it is shown in Perkins (1990) that if the underlying spatial motion belongs to a certain class of Lévy processes that admit jumps, then with probability one the closed support of $X_t$ is the whole space for almost all $t>0$ before extinction, the so-called ``instantaneous propagation'' property. In this paper for superprocesses on $\mathbb{R}^1$ whose spatial motion is the symmetric stable process of index $α\in (0,2/3)$, we prove that there exist exceptional times at which the support is compact and nonempty. Moreover, we show that the set of exceptional times is dense with full Hausdorff dimension. Besides, we prove that near extinction, the support of the superprocess is concentrated arbitrarily close to the distinction point, thus upgrading the corresponding results in Tribe (1992) from $α\in (0,1/2)$ to $α\in (0,2/3)$, and we further show that the set of such exceptional times also admits a full Hausdorff dimension.

math.PR↗
arXiv2023

Exceptional times for the instantaneous propagation of superprocess

For a Dawson-Watanabe superprocess $X$ on $\mathbb{R}^d$, it is shown in Perkins (1990) that if the underlying spatial motion belongs to a certain class of Lévy processes that admit jumps, then with probability one the closed support of $X_t$ is the whole space for almost all $t>0$ before extinction, the so-called ``instantaneous propagation'' property. In this paper for superprocesses on $\mathbb{R}^1$ whose spatial motion is the symmetric stable process of index $α\in (0,2/3)$, we prove that there exist exceptional times at which the support is compact and nonempty. Moreover, we show that the set of exceptional times is dense with full Hausdorff dimension. Besides, we prove that near extinction, the support of the superprocess is concentrated arbitrarily close to the distinction point, thus upgrading the corresponding results in Tribe (1992) from $α\in (0,1/2)$ to $α\in (0,2/3)$, and we further show that the set of such exceptional times also admits a full Hausdorff dimension.

math.PR↗
arXiv2023

Rescaled SIR epidemic processes converge to super-Brownian motion in four or more dimensions

In dimensions $d\geq 4$, by choosing a suitable scaling parameter, we show that the rescaled spatial SIR epidemic process converges to a super-Brownian motion with drift, thus complementing the previous results by Lalley (Probab. Theory Related Fields,144(2009),429--469) and Lalley-Zheng (Prob. Th. Rel. Fields,148(2010),527--566) on the convergence of SIR epidemics in $d\leq 3$. The scaling parameters we choose also agree with the corresponding asymptotics for the critical probability $p_c$ of the range-$R$ bond percolation on $\mathbb{Z}^d$ as $R\to \infty$.

math.PR↗
arXiv2023

A lower bound for p_c in range-R bond percolation in four, five and six dimensions

For the range-R bond percolation in d=4,5,6, we obtain a lower bound for the critical probability p_c for R large, agreeing with the conjectured asymptotics and thus complementing the corresponding results of Van der Hofstad-Sakai (2005) for d>6, and Frei-Perkins (2016), Hong (2021) for d<4. The proof follows by showing the extinction of the associated SIR epidemic model and introducing a self-avoiding branching random walk where births onto visited sites are suppressed and the total range of which dominates that of the SIR epidemic process.

math.PR↗
arXiv2021

An upper bound for $p_c$ in range-$R$ bond percolation in two and three dimensions

An upper bound for the critical probability of long range bond percolation in $d=2$ and $d=3$ is obtained by connecting the bond percolation with the SIR epidemic model, thus complementing the lower bound result in Frei and Perkins arXiv:arch-ive/1603.04130. A key ingredient is that we establish a uniform bound for the local times of branching random walk by calculating their exponential moments and by using the discrete versions of Tanaka's formula and Garsia's Lemma.

math.PR↗
arXiv2020

On the boundary local time measure of super-Brownian motion

If $L^x$ is the total occupation local time of $d$-dimensional super-Brownian motion, $X$, for $d=2$ and $d=3$, we construct a random measure $\mathcal{L}$, called the boundary local time measure, as a rescaling of $L^x e^{-λL^x} dx$ as $λ\to \infty$, thus confirming a conjecture of \cite{MP17} and further show that the support of $\mathcal{L}$ equals the topological boundary of the range of $X$, $\partial\mathcal{R}$. This latter result uses a second construction of a boundary local time $\widetilde{\mathcal{L}}$ given in terms of exit measures and we prove that $\widetilde{\mathcal{L}}=c\mathcal{L}$ a.s. for some constant $c>0$. We derive reasonably explicit first and second moment measures for $\mathcal{L}$ in terms of negative dimensional Bessel processes and use it with the energy method to give a more direct proof of the lower bound of the Hausdorff dimension of $\partial\mathcal{R}$ in \cite{HMP18}. The construction requires a refinement of the $L^2$ upper bounds in \cite{MP17} and \cite{HMP18} to exact $L^2$ asymptotics. The methods also refine the left tail bounds for $L^x$ in \cite{MP17} to exact asymptotics. We conjecture that the Minkowski content of $\partial\mathcal{R}$ is equal to the total mass of the boundary local time $\mathcal{L}$ up to some constant.

math.PR↗
arXiv2020

Exit measure, local time and a boundary local time of super-Brownian motion

We use a renormalization of the total mass of the exit measure from the complement of a small ball centered at $x\in \mathbb{R}^d$ for $d\leq 3$ to give a new construction of the total local time $L^x$ of super-Brownian motion at $x$. In \cite{Hong20} a more singular renormalization of the total mass of the exit measure concentrating on $x$, where the exit measure is positive but unusually small, is used to build a boundary local time supported on the topological boundary of the range of super-Brownian motion. Our exit measure construction of $L^x$ motivates this renormalization. We give an important step of this construction here by establishing the convergence of the associated mean measure to an explicit limit; this will be used in the construction of the boundary local time in \cite{Hong20}. Both our results rely on the behaviour of solutions to the associated semilnear elliptic equation with singular initial data and on Le Gall's special Markov property for exit measures.

math.PR↗
arXiv2018

Renormalization of local times of super-Brownian motion

For the local time $L_t^x$ of super-Brownian motion $X$ starting from $δ_0$, we study its asymptotic behavior as $x\to 0$. In $d=3$, we find a normalization $ψ(x)=(1/(2π^2) \log (1/|x|))^{1/2}$ such that $(L_t^x-1/(2π|x|))/ψ(x)$ converges in distribution to standard normal as $x\to 0$. In $d=2$, we show that $L_t^x-(1/π)\log (1/|x|)$ converges a.s. as $x\to 0$. We also consider general initial conditions and get similar renormalization results. The behavior of the local time allows us to derive a second order term in the asymptotic behavior of a related semilinear elliptic equation.

math.PR↗
arXiv2018

On the topological boundary of the range of super-Brownian motion-extended version

We show that if $\partial\mathcal{R}$ is the boundary of the range of super-Brownian motion and dim denotes Hausdorff dimension, then with probability one, for any open set $U$, $\partial\mathcal{R}\cap U\neq\emptyset$ implies $$\text{dim}(U\cap\partial\mathcal{R})=\begin{cases} 4-2\sqrt2\approx1.17&\text{ if }d=2\\ \frac{9-\sqrt{17}}{2}\approx 2.44&\text{ if }d=3. \end{cases}$$ This improves recent results of the last two authors (arxiv:1711.03486) by working with the actual topological boundary, rather than the boundary of the zero set of the local time, and establishing a local result for the dimension.

math.PR↗
arXiv2017

Local behavior of local times of super Brownian motion

For $x\in R^d- \{0\}$, in dimension $d=3$, we study the asymptotic behavior of the local time $L_t^x$ of super-Brownian motion $X$ starting from $δ_0$ as $x \to 0$. Let $ψ(x)=((1/2π^2) \log (1/|x|))^{1/2}$ be a normalization, Theorem 1 implies that $(L_t^x-(1/2π|x|))/ψ(x)$ converges in distribution to a standard normal distributed random variable as $x\to 0$. For dimension $d=2$, Theorem 2 implies that $L^x_t-(1/π)\log(1/|x|)$ is $L^1$ bounded as $x\to 0$. To do this, we prove a Tanaka formula for the local time which refines a result in Barlow, Evans and Perkins.

math.PR↗