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Longmin Wang

Publications and source records attributed to Longmin Wang.

13 recordsLinked to original sources
arXiv2023

The growth of the Green function for random walks and Poincar{\'e} series

Given a probability measure $\mu$ on a finitely generated group $\Gamma$, the Green function $G(x,y|r)$ encodes many properties of the random walk associated with $\mu$. Finding asymptotics of $G(x,y|r)$ as $y$ goes to infinity is a common thread in probability theory and is usually referred as renewal theory in literature. Endowing $\Gamma$ with a word distance, we denote by $H_r(n)$ the sum of the Green function $G(e,x|r)$ along the sphere of radius $n$. This quantity appears naturally when studying asymptotic properties of branching random walks driven by $\mu$ on $\Gamma$ and the behavior of $H_r(n)$ as $n$ goes to infinity is intimately related to renewal theory. Our motivation in this paper is to construct various examples of particular behaviors for $H_r(n)$. First, our main result exhibits a class of relatively hyperbolic groups with convergent Poincar{\'e} series generated by $H_r(n)$, which answers some questions raised in a previous paper of the authors. Along the way, we investigate the behavior of $H_r(n)$ for several classes of finitely generated groups, including abelian groups, certain nilpotent groups, lamplighter groups, and Cartesian products of free groups.

math.GR↗
arXiv2022

Branching Random Walks on relatively hyperbolic groups

Let $\Gamma$ be a non-elementary relatively hyperbolic group with a finite generating set. Consider a finitely supported admissible and symmetric probability measure $\mu$ on $\Gamma$ and a probability measure $\nu$ on $\mathbb{N}$ with mean $r$. Let $\mathrm{BRW}(\Gamma,\nu,\mu)$ be the branching random walk on $\Gamma$ with offspring distribution $\nu$ and base motion given by the random walk with step distribution $\mu$. It is known that for $1 < r \leq R$ with $R$ the radius of convergence for the Green function of the random walk, the population of $\mathrm{BRW}(\Gamma,\nu,\mu)$ survives forever, but eventually vacates every finite subset of $\Gamma$. We prove that in this regime, the growth rate of the trace of the branching random walk is equal to the growth rate $\omega_\Gamma(r)$ of the Green function of the underlying random walk. We also prove that the Hausdorff dimension of the limit set $\Lambda(r)$, which is the random subset of the Bowditch boundary consisting of all accumulation points of the trace of $\mathrm{BRW}(\Gamma,\nu,\mu)$, is equal to a constant times $\omega_\Gamma(r)$.

math.PR↗
arXiv2020

Limit set of branching random walks on hyperbolic groups

Let $Γ$ be a nonelementary hyperbolic group with a word metric $d$ and $\partialΓ$ its hyperbolic boundary equipped with a visual metric $d_a$ for some parameter $a>1$. Fix a superexponential symmetric probability $μ$ on $Γ$ whose support generates $Γ$ as a semigroup, and denote by $ρ$ the spectral radius of the random walk $Y$ on $Γ$ with step distribution $μ$. Let $ν$ be a probability on $\{1,\, 2, \, 3, \, \ldots\}$ with mean $λ=\sum\limits_{k=1}^\infty kν(k)<\infty$. Let $\mathrm{BRW}(Γ, \, ν, \, μ)$ be the branching random walk on $Γ$ with offspring distribution $ν$ and base motion $Y$ and $H(λ)$ the volume growth rate for the trace of $\mathrm{BRW}(Γ, \, ν, \, μ)$. We prove for $λ\in [1, \, ρ^{-1})$ that the Hausdorff dimension of the limit set $Λ$, which is the random subset of $(\partial Γ, \, d_a)$ consisting of all accumulation points of the trace of $\mathrm{BRW}(Γ, \, ν, \, μ)$, is given by $\log_a H(λ)$. Furthermore, we prove that $H(λ)$ is almost surely a deterministic, strictly increasing and continuous function of $λ\in [1, \, ρ^{-1}]$, is bounded by the square root of the volume growth rate of $Γ$, and has critical exponent $1/2$ at $ρ^{-1}$ in the sense that \[ H(ρ^{-1}) - H(λ) \sim C \sqrt{ρ^{-1} - λ} \quad \text{as } λ\uparrow ρ^{-1} \] for some positive constant $C$. We conjecture that the Hausdorff dimension of $Λ$ in the critical case $λ=ρ^{-1}$ is $\log_aH(ρ^{-1})$ almost surely. This has been confirmed on free groups or the free product (by amalgamation) of finitely many finite groups equipped with the word metric $d$ defined by the standard generating set.

math.PR↗
arXiv2019

Extremes of Vector-Valued Gaussian Processes

The seminal papers of Pickands [1,2] paved the way for a systematic study of high exceedance probabilities of both stationary and non-stationary Gaussian processes. Yet, in the vector-valued setting, due to the lack of key tools including Slepian's Lemma, Borell-TIS and Piterbarg inequalities there has not been any methodological development in the literature for the study of extremes of vector-valued Gaussian processes. In this contribution we develop the uniform double-sum method for the vector-valued setting obtaining the exact asymptotics of the exceedance probabilities for both stationary and non-stationary Gaussian processes. We apply our findings to the operator fractional Brownian motion and the operator fractional Ornstein-Uhlenbeck process.

math.PR↗
arXiv2018

Uniform spanning forests associated with biased random walks on Euclidean lattices

The uniform spanning forest measure ($\mathsf{USF}$) on a locally finite, infinite connected graph $G$ with conductance $c$ is defined as a weak limit of uniform spanning tree measure on finite subgraphs. Depending on the underlying graph and conductances, the corresponding $\mathsf{USF}$ is not necessarily concentrated on the set of spanning trees. Pemantle~\cite{PR1991} showed that on $\mathbb{Z}^d$, equipped with the unit conductance $ c=1$, $\mathsf{USF}$ is concentrated on spanning trees if and only if $d \leq 4$. In this work we study the $\mathsf{USF}$ associated with conductances induced by $λ$--biased random walk on $\mathbb{Z}^d$, $d \geq 2$, $0 < λ< 1$, i.e. conductances are set to be $c(e) = λ^{-|e|}$, where $|e|$ is the graph distance of the edge $e$ from the origin. Our main result states that in this case $\mathsf{USF}$ consists of finitely many trees if and only if $d = 2$ or $3$. More precisely, we prove that the uniform spanning forest has $2^d$ trees if $d = 2$ or $3$, and infinitely many trees if $d \geq 4$. Our method relies on the analysis of the spectral radius and the speed of the $λ$--biased random walk on $\mathbb{Z}^d$.

math.PR↗
arXiv2016

Yaglom limit for stable processes in cones

We give the asymptotics of the tail of the distribution of the first exit time of the isotropic $α$-stable Lévy process from the Lipschitz cone in $\mathbb{R}^d$. We obtain the Yaglom limit for the killed stable process for the cone. We construct and estimate entrance laws for the process from the vertex into the cone. For the symmetric Cauchy process and the positive half-line we give a spectral representation of the Yaglom limit. Our approach relies on the scalings of the stable process and the cone, which allow to express the temporal asymptotics of the distribution of the process at infinity by means of the spatial asymptotics of harmonic functions of the process at the vertex; on the representation of the probability of survival of the process in the cone as a Green potential; and on the approximate factorization of the heat kernel of the cone, which secures compactness and yields a limiting (Yaglom) measure by means of Prokhorov's theorem.

math.PR↗
arXiv2016

On the exact asymptotics of exit time from a cone of an isotropic $α$-self-similar Markov process with a skew-product structure

In this paper we identify the asymptotic tail of the distribution of the exit time $τ_C$ from a cone $C$ of an isotropic $α$-self-similar Markov process $X_t$ with a skew-product structure, that is $X_t$ is a product of its radial process and independent time changed angular component $Θ_t$. Under some additional regularity assumptions, the angular process $Θ_t$ killed on exiting from the cone $C$ has the transition density that could be expressed in terms of a complete set of orthogonal eigenfunctions with corresponding eigenvalues of an appropriate generator. Using this fact and some asymptotic properties of the exponential functional of a killed Lévy process related with Lamperti representation of the radial process, we prove that $$\mathbb{P}_x(τ_C>t)\sim h(x)t^{-κ_1}$$ as $t\rightarrow\infty$ for $h$ and $κ_1$ identified explicitly. The result extends the work of DeBlassie (1988) and Bañuelos and Smits (1997) concerning the Brownian motion.

math.PR↗
arXiv2013

Uniqueness of Stable Processes with Drift

Suppose that $d\geq1$ and $α\in (1, 2)$. Let $Y$ be a rotationally symmetric $α$-stable process on $\R^d$ and $b$ a $\R^d$-valued measurable function on $\R^d$ belonging to a certain Kato class of $Y$. We show that $\rd X^b_t=\rd Y_t+b(X^b_t)\rd t$ with $X^b_0=x$ has a unique weak solution for every $x\in \R^d$. Let $\sL^b=-(-Δ)^{α/2} + b \cdot \nabla$, which is the infinitesimal generator of $X^b$. Denote by $C^\infty_c(\R^d)$ the space of smooth functions on $\R^d$ with compact support. We further show that the martingale problem for $(\sL^b, C^\infty_c(\R^d))$ has a unique solution for each initial value $x\in \R^d$.

math.PR↗
arXiv2011

Isotropic self-similar Markov processes

We show that an isotropic self-similar Markov process in ${\Bbb R}^d$ has a skew product structure if and only if its radial and angular parts do not jump at the same time.

math.PR↗