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Renming Song

Publications and source records attributed to Renming Song.

At least 37 records · Page 2Linked to original sources
arXiv2025

Heat kernel estimates for Schrödinger operators with supercritical killing potentials

In this paper, we study the Schrödinger operator $Δ-V$, where $V$ is a supercritical non-negative potential belonging to a large class of functions containing functions of the form $b|x|^{-(2+2β)}$, $b, β>0$. We obtain two-sided estimates on the heat kernel $p(t, x, y)$ of $Δ-V$, along with estimates for the corresponding Green function. Unlike the case of the fractional Schrödinger operator $-(-Δ)^{α/2}-V$, $α\in (0, 2)$, 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

Time fractional stochastic differential equations driven by pure jump Lévy noise

In this paper we introduce a variable order time fractional differential equation driven by pure jump Lévy noise, which models the motion of a particle exhibiting memory effect. We prove the well-posedness of this equation without assuming any integrability condition on the initial condition and the large jump coefficient, by using a truncation argument. Under some extra conditions, we also derive some $L^p$ moment estimates on the solutions. As an application of moment estimates, we prove the Hölder regularity of the solutions.

math.PR↗
arXiv2024

Heat kernel estimates for regional fractional Laplacians with multi-singular critical potentials in $C^{1, β}$ open sets

Let $D$ be an open set of $\mathbb{R}^d$, $α\in (0, 2)$ and let $\mathcal{L}_α^D$ be the generator of the censored $α$-stable process in $D$. In this paper, we establish sharp two-sided heat kernel estimates for $\mathcal{L}_α^D-κ$, with $κ$ being a non-negative critical potential and $D$ being a $C^{1, β}$ open set, $β\in ((α-1)_+,1]$. The potential $κ$ can exhibit multi-singularities and our regularity assumption on $D$ is weaker than the regularity assumed in earlier literature on heat kernel estimates of fractional Laplacians.

math.PR↗
arXiv2024

Asymptotic behaviors of subcritical branching killed Brownian motion with drift

In this paper, we study asymptotic behaviors of a subcritical branching killed Brownian motion with drift $-ρ$ and offspring distribution $\{p_k:k\ge 0\}$. Let $\widetildeζ^{-ρ}$ be the extinction time of this subcritical branching killed Brownian motion, $\widetilde{M}_t^{-ρ}$ the maximal position of all the particles alive at time $t$ and $\widetilde{M}^{-ρ}:=\max_{t\ge 0}\widetilde{M}_t^{-ρ}$ the all time maximal position. Let $\mathbb{P}_x$ be the law of this subcritical branching killed Brownian motion when the initial particle is located at $x\in (0,\infty)$. Under the assumption $\sum_{k=1}^\infty k (\log k) p_k <\infty$, we establish the decay rates of $\mathbb{P}_x(\widetildeζ^{-ρ}>t)$ and $\mathbb{P}_x(\widetilde{M}^{-ρ}>y)$ as $t$ and $y$ tend to $\infty$ respectively. We also establish the decay rate of $\mathbb{P}_x(\widetilde{M}_t^{-ρ}>z(t,ρ))$ as $t\to\infty$, where $z(t,ρ)=\sqrt{t}z-ρt$ for $ρ\leq 0$ and $z(t,ρ)=z$ for $ρ>0$. As a consequence, we obtain a Yaglom-type limit theorem.

math.PR↗
arXiv2024

Tails of extinction time and maximal displacement for critical branching killed Lévy process

In this paper, we study asymptotic behaviors of the tails of extinction time and maximal displacement of a critical branching killed Lévy process $(Z_t^{(0,\infty)})_{t\ge 0}$ in $\mathbb{R}$, in which all particles (and their descendants) are killed upon exiting $(0, \infty)$. Let $ζ^{(0,\infty)}$ and $M_t^{(0,\infty)}$ be the extinction time and maximal position of all the particles alive at time $t$ of this branching killed Lévy process and define $M^{(0,\infty)}: = \sup_{t\geq 0} M_t^{(0,\infty)}$. Under the assumption that the offspring distribution belongs to the domain of attraction of an $α$-stable distribution, $α\in (1, 2]$, and some moment conditions on the spatial motion, we give the decay rates of the survival probabilities $$ \mathbb{P}_{y}(ζ^{(0,\infty)}>t), \quad \mathbb{P}_{\sqrt{t}y}(ζ^{(0,\infty)}>t) $$ and the tail probabilities $$ \mathbb{P}_{y}(M^{(0,\infty)}\geq x), \quad \mathbb{P}_{xy}(M^{(0,\infty)}\geq x). $$ We also study the scaling limits of $M_t^{(0,\infty)}$ and the point process $Z_t^{(0,\infty)}$ under $\mathbb{P}_{\sqrt{t}y}(\cdot |ζ^{(0,\infty)}>t)$ and $\mathbb{P}_y(\cdot |ζ^{(0,\infty)}>t)$. The scaling limits under $\mathbb{P}_{\sqrt{t}y}(\cdot |ζ^{(0,\infty)}>t)$ are represented in terms of super killed Brownian motion.

math.PR↗
arXiv2024

On the anti-concentration functions of some familiar families of distributions

Let $\{X_α\}$ be a family of random variables following a certain type of distributions with finite expectation $\mathbf{E}[X_α]$ and finite variance ${\rm Var}(X_α)$, where $α$ is a parameter. Motivated by the recent paper of Hollom and Portier (arXiv: 2306.07811v1), we study the anti-concentration function $(0, \infty)\ni y\to \inf_α\mathbf{P}\left(|X_α-\mathbf{E}[X_α]|\geq y \sqrt{{\rm Var}(X_α)}\right)$ and find its explicit expression. We show that, for certain familiar families of distributions, including uniform distributions, exponential distributions, non-degenerate Gaussian distributions and student's $t$-distribution, the anti-concentration function is not identically zero, while for some other familiar families of distributions, including binomial, Poisson, negative binomial, hypergeometric, Gamma, Pareto, Weibull, log-normal and Beta distributions, the anti-concentration function is identically zero.

math.PR↗
arXiv2024

Fractional Laplacian with supercritical killings

In this paper, we study Feynman-Kac semigroups of symmetric $α$-stable processes with supercritical killing potentials belonging to a large class of functions containing functions of the form $b|x|^{-β}$, where $b>0$ and $β>α$. We obtain two-sided estimates on the densities $p(t, x, y)$ of these semigroups for all $t>0$, along with estimates for the corresponding Green functions.

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 $$ L^{\mathcal{B}}_αf(x)=\lim_{ε\to 0} \int_{D,\, |y-x|>ε}\big(f(y)-f(x)\big) \mathcal{B}(x,y)|x-y|^{-d-α}\,dy, $$ and $$ L f(x)=L^{\mathcal{B}}_αf(x) - κ(x) f(x), $$ 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 $κ$ may be subcritical or critical. From a probabilistic point of view we study the reflected process on the closure $\overline{D}$ with infinitesimal generator $L^{\mathcal{B}}_α$, and its part process on $D$ obtained by either killing at the boundary $\partial D$, or by killing via the killing potential $κ(x)$. 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 Hölder 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↗
arXiv2023

$1$-stable fluctuation of the derivative martingale of branching random walk

In this paper, we study the functional convergence in law of the fluctuations of the derivative martingale of branching random walk on the real line. Our main result strengthens the results of Buraczewski et. al. [Ann. Probab., 2021] and is the branching random walk counterpart of the main result of Maillard and Pain [Ann. Probab., 2019] for branching Brownian motion.

math.PR↗
arXiv2023

Asymptotic expansion for additive measure of branching Brownian motion

Let $N(t)$ be the collection of particles alive at time $t$ in a branching Brownian motion in $\mathbb{R}^d$, and for $u\in N(t)$, let $\mathbf{X}_u(t)$ be the position of particle $u$ at time $t$. For $θ\in \mathbb{R}^d$, we define the additive measures of the branching Brownian motion by$$μ_t^θ(\mathrm{d}\mathbf{x}):= e^{-(1+\frac{\Vertθ\Vert^2}{2})t}\sum_{u\in N(t)} e^{-θ\cdot \mathbf{X}_u(t)} δ_{\left(\mathbf{X}_u(t)+θt\right)}(\mathrm{d}\mathbf{x}).$$ In this paper, under some conditions on the offspring distribution, we give asymptotic expansions of arbitrary order for $μ_t^θ((\mathbf{a}, \mathbf{b}])$ and $μ_t^θ((-\infty, \mathbf{a}])$ for $θ\in \mathbb{R}^d$ with $\Vert θ\Vert <\sqrt{2}$. These expansions sharpen the asymptotic results of Asmussen and Kaplan (1976) and Kang (1999), and are analogs of the expansions in Gao and Liu (2021) and Révész, Rosen and Shi (2005) for branching Wiener processes (a particular class of branching random walks) corresponding to $θ=\mathbf{0}$.

math.PR↗
arXiv2023

Tail probability of maximal displacement in critical branching Lévy process with stable branching

Consider a critical branching Lévy process $\{X_t, t\ge 0\}$ with branching rate $β>0, $ offspring distribution $\{p_k:k\geq 0\}$ and spatial motion $\{ξ_t, Π_x\}$. For any $t\ge 0$, let $N_t$ be the collection of particles alive at time $t$, and, for any $u\in N_t$, let $X_u(t)$ be the position of $u$ at time $t$. We study the tail probability of the maximal displacement $M:=\sup_{t>0}\sup_{u\in N_t} X_u(t)$ under the assumption $\lim_{n\to\infty} n^α\sum_{k=n}^\infty p_k =κ\in(0,\infty)$ for some $α\in (1,2)$, $Π_0(ξ_1)=0$ and $Π_0 (|ξ_1|^r)\in (0,\infty)$ for some $r> 2α/(α-1)$. Our main result is a generalization of the main result of Sawyer and Fleischman (1979) for branching Brownian motions and that of Lalley and Shao (2015) for branching random walks, both of which are proved under the assumption $\sum_{k=0}^\infty k^3 p_k<\infty$.

math.PR↗
arXiv2023

Some New Results on Gaussian Product Inequalities

The long-standing Gaussian product inequality (GPI) conjecture states that, for any centered $\mathbb{R}^n$-valued Gaussian random vector $(X_1, \dots, X_n)$ and any positive reals $α_1, \dots, α_n$, ${\bf E}[\prod_{j=1}^{n}|X_j|^{α_j}]\ge \prod_{j=1}^{n}{\bf E}[|X_j|^{α_j}]$. In this paper, we present some related inequalities for centered $\mathbb{R}^n$-valued Gaussian random vector $(X_1, \dots, X_n)$ when $\{α_1, \dots, α_n\}$ contains both positive and negative numbers.

math.PR↗
arXiv2023

Asymptotic expansion for branching killed Brownian motion with drift

Let $Z_t^{(0,\infty)}$ be the point process formed by the positions of all particles alive at time $t$ in a branching Brownian motion with drift and killed upon reaching 0. We study the asymptotic expansions of $Z_t^{(0,\infty)}(A)$ for $A= (a,b)$ and $A=(a,\infty)$ under the assumption that $\sum_{k=1}^\infty k(\log k)^{1+λ} p_k <\infty$ for large $λ$ in the regime of $θ\in [0,\sqrt{2})$. These results extend and sharpen the results of Louidor and Saglietti [J. Stat. Phys, 2020] and Kesten [Stochastic Process. Appl., 1978].

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↗
arXiv2023

Positive self-similar Markov processes obtained by resurrection

In this paper we study positive self-similar Markov processes obtained by (partially) resurrecting a strictly $α$-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↗
arXiv2023

Extremal process for irreducible multitype branching Brownian motion

We first study the convergence of solutions of a system of F-KPP equations related to irreducible multitype branching Brownian motions with Heaviside-type initial conditions to traveling wave solutions. Then we apply this convergence result to prove that the extremal processes of irreducible multitype branching Brownian motions converges weakly to a cluster point process.

math.PR↗
arXiv2023

Homogenization of Non-symmetric Jump Processes

We study the homogenization for a class of non-symmetric pure jump Feller processes. The jump intensity involves periodic and aperiodic constituents, as well as oscillating and non-oscillating constituents. This means that the noise can come both from the underlying periodic medium and from external environments, and is allowed to have different scales. It turns out that the Feller process converges in distribution, as the scaling parameter goes to zero, to a Lévy process. As special cases of our result, some homogenization problems studied in previous works can be recovered. We also generalize the approach to the homogenization of symmetric stable-like processes with variable order. Moreover, we present some numerical experiments to demonstrate the usage of our homogenization results in the numerical approximation of first exit times.

math.PR↗
arXiv2023

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↗