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Renming Song

Publications and source records attributed to Renming Song.

At least 19 recordsLinked to original sources
arXiv2026

Laws of the iterated logarithm for the range of once-reinforced random walk on $\mathbb{Z}$

In this paper, we study once-reinforced random walk on $\mathbb{Z}$, and prove sharp Chung-type and Hartman-Wintner laws of the iterated logarithm for the range. Let $R_n$ denote the range at time $n$ and let $c>0$ be the reinforcement parameter. Then, almost surely, \begin{equation*} \liminf_{n\to\infty}\frac{R_n}{\sqrt{n/\log\log n}}=\fracπ{\sqrt{2}}, \qquad \limsup_{n\to\infty}\frac{cR_n}{\sqrt{2n\log\log n}}=1. \end{equation*} The proofs are based on the independent increment decomposition of the inverse range and sharp generating function estimates.

math.PR↗
arXiv2026

Local properties for $1$-dimensional critical branching Lévy process

Consider a one dimensional critical branching Lévy process $((Z_t)_{t\geq 0}, \mathbb {P}_x)$. Assume that the offspring distribution either has finite second moment or belongs to the domain of attraction to some $α$-stable distribution with $α\in (1, 2)$, and that the underlying Lévy process $(ξ_t)_{t\geq 0}$ is non-lattice and has finite $2+δ^*$ moment for some $δ^*>0$. We first prove that $$t^{\frac{1}{α-1}}\left(1- \mathbb{E}_{\sqrt{t}y}\left(\exp\left\{-\frac{1}{t^{\frac{1}{α-1}-\frac{1}{2}}}\int h(x) Z_t(\mathrm{d}x) -\frac{1}{t^{\frac{1}{α-1}}} \int g\left(\frac{x}{\sqrt{t}}\right)Z_t(\mathrm{d}x)\right\}\right)\right)$$ converges as $t\to\infty$ for any non-negative bounded Lipschtitz function $g$ and any non-negative directly Riemann integrable function $h$ of compact support. Then for any $y\in \mathbb{R}$ and bounded Borel set of positive Lebesgue measure with its boundary having zero Lebesgue measure, under a higher moment condition on $ξ$, we find the decay rate of the probability $\mathbb {P}_{\sqrt{t}y}(Z_t(A)>0)$. As an application, we prove some convergence results for $Z_t$ under the conditional law $\mathbb {P}_{\sqrt{t}y}(\cdot| Z_t(A)>0).$

math.PR↗
arXiv2026

From killed BBM with drift to BBM: the extremal process

In this paper, we study the asymptotic behavior of the extremal process of a standard one-dimensional branching Brownian motion (BBM) with drift $-ρ>-\sqrt2$ and absorbing barrier at level $-x$. We prove that the two-dimensional point process, with first component being the extremal process of the BBM and the second component being the running minimum of the BBM with drift, converges weakly to a decorated Poisson point process (DPPP) on $\mathbb{R} \times [0, \infty)$. This framework allows us to supply a common limit coupling of the extremal processes for different barrier levels and to explicitly derive the limit, as $t\to\infty$, of the extremal process of the killed BBM killed at level $-x$, demonstrating that the double limit, when $t\to\infty$ first and then $x\to\infty$, of the extremal process of the BBM killed at level $-x$ coincides with the limit of the extremal process of (un-killed) BBM.

math.PR↗
arXiv2026

Favorite sites of one-dimensional transient random walk

In this paper, we study favorite sites of one-dimensional transient random walk on $\mathbb{Z}$ beyond the simple case. Denote by $\mathcal{K}(n)$ the set of favorite sites at time $n$. The goal of this paper is twofold. First, we study $\# \mathcal{K}(n)$, the cardinality of $\mathcal{K}(n)$, and show that, under some mild conditions, for any positive integer $k$, $\mathbf{P}(\#\mathcal{K}(n)=k, {\rm i.o.})=1$, and $\limsup_{n\to\infty}\#\mathcal{K}(n)/\log\log n=-1/\log γ$, where $γ$ is the probability that the random walk never returns to the starting point. Second, we study the escape rate of favorite sites, and give the limsup and liminf behaviors, where the liminf behavior is characterized via an integral test. To the best of our knowledge, this integral test is the first result of this nature for random walks (in non-random environments).

math.PR↗
arXiv2026

Limit behavior of linearly edge-reinforced random walks on the half-line

Motivated by the article [M. Takei, Electron. J. Probab. 26 (2021), article no. 104], we study the limit behavior of linearly edge-reinforced random walks on the half-line $\mathbb{Z}_+$ with reinforcement parameter $δ>0$, and each edge $\{x,x+1\}$ has the initial weight $x^α\ln^βx$ for $x > 1$ and $1$ for $x = 0, 1$. The aim of this paper is to study the almost sure limit behavior of the walk in the recurrent regime, and extend the results of Takei mentioned above.

math.PR↗
arXiv2026

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 $Σ$ of $F$. We assume that $F\setminus Σ$ is a $κ$-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 $J(x,y)=|x-y|^{-d-α}{\mathcal B}(x,y)$, where $α\in (0,2)$ and the function ${\mathcal B}(x,y)$ blows up at a subset $Σ$ 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 $F\setminus Σ$. Examples of processes falling within our general framework include traces of isotropic $α$-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↗
arXiv2026

Abnormal boundary decay for stable operators

Assume $α\in (0, 2)$ and $d\ge 2$. Let $\mathcal L^α$ be the generator of a symmetric, but not necessarily isotropic, $α$-stable process $X$ in $\mathbb R^d$ whose Lévy density is comparable with that of an isotropic $α$-stable process. In this paper, we show that the $C^{1, \rm Dini}$ regularity assumption on an open set $D\subset \mathbb R^d$ is optimal for the standard boundary decay property for nonnegative $\mathcal L^α$-harmonic functions in $D$, and for the standard boundary decay property of the heat kernel $p^D(t,x,y)$ of the part process $X^D$ of $X$ on $D$ by proving the following: (i) If $D$ is a $C^{1, \rm Dini}$ open set and $h$ is a nonnegative function which is $\mathcal L^α$-harmonic in $D$ and vanishes near a portion of $\partial D$, then the rate at which $h(x)$ decays to 0 near that portion of $\partial D$ is ${\rm dist} (x, D^c)^{α/2}$. (ii) If $D$ is a $C^{1, \rm Dini}$ open set, then, as $x\to \partial D$, the rate at which $p^D(t,x,y)$ tends to 0 is ${\rm dist} (x, D^c)^{α/2}$. (iii) For any non-Dini modulus of continuity $\ell$, there exist non-$C^{1, \rm Dini}$ open sets $D$, with $\partial D$ locally being the graph of a $C^{1, \ell}$ function, such that the standard boundary decay properties above do not hold for $D$.

math.AP↗
arXiv2026

The asymptotic behavior of rarely visited edges of the simple random walk

In this paper, we study the asymptotic behavior of the number of rarely visited edges (i.e., edges that visited only once) of a simple symmetric random walk on $\mathbb{Z}$. Let $α(n)$ be the number of rarely visited edges up to time $n$. First, we evaluate $\mathbb{E}(α(n))$, show that $n\to \mathbb{E}(α(n))$ is non-decreasing in $n$ and that $\lim\limits_{n\to+\infty}\mathbb{E}(α(n))=2$. Then we study the asymptotic behavior of $\mathbb{P} (α(n)>a(\log n)^2)$ for any $a>0$ and use it to show that there exists a constant $C\in(1/32,1/2]$ such that $\limsup\limits_{n\to+\infty}\frac{α(n)}{(\log n)^2}=C$ almost surely.

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 $κ$. The jump kernel of $X$ is of the form $J(x,y)={\cal B}(x,y)|x-y|^{-d-α}$, $α\in (0,2)$, 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 $κ$, 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

Law of iterated logarithm for supercritical non-symmetric branching Markov process

Let $\{(X_t)_{t\geq 0}, \mathbb{P}_{δ_x}, x\in E\}$ be a supercritical branching Markov process (which is not necessary symmetric) on a locally compact metric measure space $(E,μ)$ with spatially dependent local branching mechanism. Under some assumptions on the semigroup of the spatial motion, we first prove law of iterated logarithm type results for $\langle f, X_t\rangle$ under the second moment condition on the branching mechanism, where $f$ is a linear combination of eigenfunctions of the mean semigroup $\{T_t, t\geq0\}$ of $X$. Then we prove law of iterated logarithm type results for $\langle f, X_t\rangle$ under the fourth moment condition, where $f$ belongs to a larger class of functions.

math.PR↗
arXiv2025

Favorite sites of one-dimensional asymmetric simple random walk

In this paper, we study favorite sites of one-dimensional asymmetric simple random walks. We show that almost surely, for any fixed integer $r\geq 1$, ``$r$ favorite sites" occurs infinitely often. We also give the asymptotic growth rate of the number of favorite sites.

math.PR↗
arXiv2025

Asymptotic behaviors of subcritical branching killed Lévy processes

In this paper, we investigate the asymptotic behaviors of the survival probability and maximal displacement of a subcritical branching killed Lévy process $X$ in $\mathbb{R}$. Let $ζ$ denote the extinction time, $M_t$ be the maximal position of all the particles alive at time $t$, and $M:=\sup_{t\ge 0}M_t$ be the all-time maximum. Under the assumption that the offspring distribution satisfies the $L\log L$ condition and some conditions on the spatial motion, we find the decay rate of the survival probability $\mathbb{P}_x(ζ>t)$ and the tail behavior of $M_t$ as $t\to\infty$. As a consequence, we establish a Yaglom-type theorem. We also find the asymptotic behavior of $\mathbb{P}_x(M>y)$ as $y\to\infty$.

math.PR↗
arXiv2025

Large deviations and almost sure convergence for the extremes of branching Lévy processes

In this paper, we investigate the asymptotic behavior of supercritical branching Markov processes $\{\mathbb{X}_t, t \ge0\}$ whose spatial motions are Lévy processes with regularly varying tails. Recently, Ren et al. [Appl. Probab. 61 (2024)] studied the weak convergence of the extremes of $\{\mathbb{X}_t, t \ge0\}$. In this paper, we establish the large deviation of $\{\mathbb{X}_t, t \ge0\}$ as well as some almost sure convergence results of the maximum of $\mathbb{X}_t$.

math.PR↗
arXiv2025

Moments of additive martingales of branching Lévy processes and applications

Let $W_t(θ)$ be the Biggins martingale of a supercritical branching Lévy process with non-local branching mechanism, and denote by $W_\infty(θ)$ its limit. In this paper, we first study moment properties of $W_t(θ)$ and $W_\infty(θ)$, and the tail behavior of $W_\infty(θ)$. We then apply these results to establish central limit theorems for $W_t(θ)-W_\infty(θ)$.

math.PR↗
arXiv2025

Approximate factorizations for non-symmetric jump processes

In this paper, we first extend the approximate factorization for purely discontinuous Markov process established in \cite{CKSV20} by getting rid of some of the conditions imposed in \cite{CKSV20}. Then we apply the approximate factorization to obtain sharp two-sided heat kernel estimates for three classes of processes: stable-like processes with critical killings in $C^{1, {\rm Dini}}$ open sets; killed stable-like processes in the setting of \cite{KW24} in $C^{1, \varepsilon}$ open sets; and non-symmetric stable processes in what we call $C^{1,2{\text - \rm Dini}}$ open sets. In particular, we obtain explicit sharp two-sided heat kernel estimates of killed $α$-stable processes in $C^{1, {\rm Dini}}$ open sets for all $α\in (0, 2)$ and of censored $α$-stable processes in $C^{1, {\rm Dini}}$ open sets for all $α\in (1, 2)$.

math.PR↗
arXiv2025

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 $J(x,y)=\mathcal B(x,y)|x-y|^{-α-d}$, $α\in (0,2)$, 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↗
arXiv2025

Tail probability of maximal displacement in critical and subcritical branching stable processes

In this paper, we study critical and subcritical branching $α$-stable processes, $α\in (0, 2)$. We obtain the exact asymptotic behaviors of the tails of the maximal positions of all subcritical branching $α$-stable processes with positive jumps. In the case of subcritical branching spectrally negative $α$-stable processes, we obtain the exact asymptotic behaviors of the tails of the maximal positions under the assumption that the offspring distributions satisfy the $L\log L$ condition. For critical branching $α$-stable processes, we obtain the exact asymptotic behaviors of the tails under the assumption that the offspring distributions belong to the domain of attraction of a $γ$-distribution, $γ\in (1, 2]$.

math.PR↗