arXiv subjects

Haojie Hou

Publications and source records attributed to Haojie Hou.

At least 19 recordsLinked to original sources

arXiv2026

Small value probabilities of additive and derivative martingales in supercritical branching Brownian motions and super Brownian motions

In this paper, we establish asymptotics for the small value probabilities of additive and derivative martingales in both supercritical branching Brownian motions and super Brownian motions, thereby extending the corresponding results for Galton--Watson processes and continuous-state branching processes. For the derivative martingale in branching Brownian motion, our result also agrees with the findings in the arXiv version of Arguin et al. [arXiv:1008.4386 v1] and with those of Hu [Ann. Inst. H. Poincaré Probab. Stat., 2016].

math.PR↗

arXiv2026

Law of the iterated logarithm for supercritical non-local spatial branching processes

Suppose that $X=(X_{t})_{t\ge 0}$ is either a general supercritical non-local branching Markov process, or a general supercritical non-local superprocess, on a Luzin space. Here, by ``supercritical" we mean that the mean semigroup of $X$ exhibits a Perron-Frobenius type behaviour with a positive principal eigenvalue. In this paper, we study the almost sure behaviour of a family of martingales naturally associated with the real or complex-valued eigenpairs of the mean semigroup. Under a fourth-moment condition, we establish limit theorems of the iterated logarithm type for these martingales. In particular, we discover three regimes, each resulting in different scaling factors and limits. Furthermore, we obtain a law of the iterated logarithm for the linear functional $\langle \mathrm{Re}(f),X_{t}\rangle$ where $f$ is a sum of finite terms of eigenfunctions and $\mathrm{Re}(f)$ denotes its real part. In the context of branching Markov processes, our results improve on existing literature by complementing the known results for multitype branching processes in Asmussen [Trans. Amer. Math. Soc. 231 (1) (1977) 233--248] and generalizing the recent work of Hou, Ren and Song [arXiv: 2505.12691v1, arXiv: 2505.12691v2] to allow for non-local branching mechanisms. For superprocesses, as far as we know, our results are new.

math.PR↗

arXiv2026

Asymptotic behaviours of critical branching random walk in $\mathbb{R}^d$

In this paper, we study the asymptotic behaviours of a critical branching random walk in $\mathbb{R}^d$ under the assumption that the offspring distribution belongs to the domain of attraction of an $α$-stable law with $α\in(1,2]$, and that the jump distribution has a finite $\frac{2α}{α-1}$-th moment. First, we establish the precise decay rate for the tail probability of the all-time maximal displacement $M^d$. Next, we investigate the maximal displacement $M_n^d$ at generation $n$ and prove a conditional limit theorem for the distribution of $M_n^d$ given that the process survives up to generation $n$. These results extend the corresponding 1-dimensional results of Lalley and Shao (2015) to the case $d\ge2$. Finally, we study the asymptotic behaviour of the total progeny $ζ$. In particular, we show that, conditioned on the event $\{M^d\ge x\}$, $ζ$ converges in distribution under an appropriate normalization. This result reveals a quantitative relationship between the maximal displacement and the total progeny size.

math.PR↗

arXiv2026

On the maximal displacement of subcritical branching random walks with stretched exponential tail

We study the maximal displacement of a one-dimensional subcritical branching random walk with offspring distribution $\{p_k\}$ and step size $X$ such that $m := \sum_{k=1}^\infty k p_k \in (0,1)$. Let $M_n$ denote the maximal position of all particles alive at time $n$ and let $M := \sup_{n \in \mathbb{N}} M_n$. First, we show that \[ \lim_{x \to +\infty} \frac{e^{λx^b}}{\ell(x) x^a } \, \mathbb{P}(M > x) = \frac{1 - p_0}{1 - m} \] whenever $\mathbb{P}(X > x) = \ell(x) x^a e^{-λx^b}$ for some slowly varying function $\ell$, $b \in [0,1)$, and under further assumptions on $a$. Next, we prove that \[ \lim_{x \to +\infty} \frac{e^{λx^b+γx}}{\ell(x) x^a } \, \mathbb{P}(M > x) \quad \text{exists and belongs to } (0, \infty) \] provided that $\sum_{k=1}^\infty k (\log k) p_k < \infty$ and for some $x_*>0$, $\mathbb{P}(X > x) = \int_x^\infty \ell(y) y^a e^{-λy^b - γy} \, \mathrm{d}y$ for all $x > x_*$. Here, $\ell$ is a slowly varying function, $m \mathbb{E}(e^{γX}) < 1$, $b \in [0,1)$, and $a$ satisfies certain conditions.

math.PR↗

arXiv2026

On the subcritical self-catalytic branching Brownian motions

The self-catalytic branching Brownian motions (SBBM) are extensions of the classical one-dimensional branching Brownian motions by incorporating pairwise branchings catalyzed by the intersection local times of the particle pairs. These processes naturally arise as the moment duals of certain reaction-diffusion equations perturbed by multiplicative space-time white noise. For the subcritical case of the catalytic branching mechanism, we construct the SBBM allowing an infinite number of initial particles. Additionally, we establish the coming down from infinity (CDI) property for these systems and characterize their CDI rates.

math.PR↗

arXiv2026

Susceptible-Infected Epidemics on Evolving Graphs at Critical Infection Rate

Consider an SI process on a graph $G$ where each S--I connection becomes I--I at rate $λ$. Here S and I stand for ``susceptible'' and ``infected'' respectively. The evoSI model is a modification of the SI model in which S--I edges are broken at rate $ρ$ and the ``S'' connects to a randomly chosen vertex. It is proven in Durrett and Yao [2022, Electron. J. Probab.] that, for the supercritical evoSI process on the configuration model, there exists a quantity $Δ$ depending on the first three moments of the degree distribution such that the sign of $Δ$ governs the continuity of the phase transition of the final epidemic size near the critical infection rate $λ_c$. In this paper, we consider the critical evoSI model on the configuration model, i.e., $λ=λ_c$. We show that, if $Δ>0$, then the probability of a major outbreak starting from a single infected individual is $Cn^{-1/3}(1+o(1))$ for some explicit constant $C>0$, where $n$ is the size of the graph. On the contrary, if $Δ<0$, then this probability is $o(n^{-1/3})$. The case $Δ<0$ is reminiscent of the critical {\ER} graphs, where the probability for the size of the largest component to be of order $n$ decays exponentially in $n$.

math.PR↗

arXiv2026

Minimum and extremal process for a branching random walk outside the boundary case

This work extends the studies on the minimum and extremal process of a supercritical branching random walk outside the boundary case which cannot be reduced to the boundary case. We study here the situation where the log-generating function explodes at $1$ and the random walk associated to the spine possesses a stretched exponential tail with exponent $b\in(0,\frac12)$. Under suitable conditions, we confirm the conjecture of Barral, Hu and Madaule [Bernoulli 24(2) 2018 801-841], and obtain the weak convergence for the minimum and the extremal process. We also establish an a.s. infimum result over all infinity rays of this system.

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

On the maximal displacement of subcritical branching random walks with or without killing

Consider a subcritical branching random walk $\{Z_k\}_{k\geq 0}$ with offspring distribution $\{p_k\}_{k\geq 0}$ and step size $X$. Let $M_n$ denote the rightmost position reached by $\{Z_k\}_{k\geq 0}$ up to generation $n$, and define $M := \sup_{n\geq 0} M_n$. In this paper we give asymptotics of tail probability of $M$ under optimal assumptions $\sum^{\infty}_{k=1}(k\log k) p_k<\infty$ and $\mathbb{E}[Xe^{γX}]<\infty$, where $γ>0$ is a constant such that $\mathbb{E}[e^{γX}]=\frac{1}{m}$ and $m=\sum_{k=0}^\infty kp_k\in (0,1)$. Moreover, we confirm the conjecture of Neuman and Zheng [Probab. Theory Related Fields. 167 (2017) 1137--1164] by establishing the existence of a critical value $m\mathbb{E}[X e^{γX}]$ such that \begin{align*} \lim_{n\to\infty}e^{γcn}\mathbb{P}(M_n\geq cn)= \left\{ \begin{aligned} &κ\in(0,1], &c\in\big(0,m\mathbb{E}[Xe^{γX}]\big); &0, &c\in\big(m\mathbb{E}[Xe^{γX}],\infty\big), \end{aligned} \right. \end{align*} where $κ$ represents the non-zero limit. Finally, we extend these results to the maximal displacement of branching random walks with killing. Interestingly, this limit can be characterized through both the global minimum of a random walk with positive drift and the maximal displacement of the branching random walk without killing.

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↗

arXiv2024

Heat kernel estimates for nonlocal kinetic operators

In this paper, we employ probabilistic techniques to derive sharp, explicit two-sided estimates for the heat kernel of the nonlocal kinetic operator $$ Δ^{α/2}_v + v \cdot \nabla_x, \quad α\in (0, 2),\ (x,v)\in {\mathbb R}^{d}\times{\mathbb R}^d,$$ where $ Δ^{α/2}_v $ represents the fractional Laplacian acting on the velocity variable $v$. Additionally, we establish logarithmic gradient estimates with respect to both the spatial variable $x$ and the velocity variable $v$. In fact, the estimates are developed for more general non-symmetric stable-like operators, demonstrating explicit dependence on the lower and upper bounds of the kernel functions. These results, in particular, provide a solution to a fundamental problem in the study of \emph{nonlocal} kinetic operators.

math.PR↗

arXiv2024

Strain-modulated Valley Polarization and Piezomagnetic Effects in Altermagnetic Cr$_2$S$_2$

Altermagnetism exhibits advantages over both ferromagnetic and antiferromagnetic counterparts by enabling spin splitting within antiferromagnetic materials. Currently, it is established that valley polarization in altermagnetism remains largely insensitive to spin-orbit coupling and spin. Here, using Cr$_2$S$_2$ as a case study, we investigate the mechanism through which an external field modulates valley polarization in altermagnetism. This effect arises from the external field's disruption of diagonal mirror symmetry $M_{xy}$, consequently inducing valley polarization within the material. Strain not only induces valley polarization but also generates an almost uniform magnetic field, which can reach as high as 118.39 T under 5% uniaxial strain. In addition, this symmetry breaking in Cr$_2$S$_2$ monolayers results in significant piezomagnetic properties, merging piezomagnetic and altermagnetic characteristics in two-dimensional materials.

cond-mat.mtrl-sci↗

arXiv2024

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 $y\in \R$ 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↗

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↗

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↗