José Hermenegildo Ramírez-González

arXiv2026

Extinction and Survival in an Interval-Activation Frog Model on \mathbb{Z} with Random Survival Parameters and Symmetric Random Walks

We study an interval-activation frog model on $\mathbb Z$ with i.i.d.\ initial numbers of frogs $(η_x)_{x\in\mathbb Z}$, satisfying $0<\mathbb{E}[η_0]<\infty$. Frogs at the origin are initially active and all others are sleeping. Each frog performs a symmetric integer-valued random walk and has a random lifetime $L$ determined by an i.i.d.\ survival parameter $π\in(0,1)$, with $\mathbb{P}(L\ge k\mid π=p)=p^k$. Every jump activates all sleeping frogs at the integer sites between its endpoints. Let $D^\to$ denote the maximal rightward displacement of a single frog before death. We derive survival and extinction criteria from the tail behavior of $D^\to$. If $\mathbb{P}(|ξ_1|\ge n)\sim n^{-α}L_ξ(n)$, with $L_ξ$ slowly varying, then survival holds with positive probability for $0<α<1$, while for $α=1$ both survival and almost sure extinction may occur. For $1<α<2$, assume $\mathbb{P}(|ξ_1|>n)\sim c_ξn^{-α}$; in the finite-variance case assume $\mathbb{E}[ξ_1]=0$ and $\operatorname{Var}(ξ_1)=σ^2\in(0,\infty)$. Setting $r=α$ in the stable case and $r=2$ in the finite-variance case, if the law of $π$ has density $f_π(u)\sim(1-u)^{β-1}\ell((1-u)^{-1})$ as $u\uparrow1$, then, for $0<β<1$, $n\mathbb{P}(D^\to\ge n)\sim C_βn^{1-rβ}\ell(n^r)$, with explicit $C_β$. Hence the sharp off-critical threshold is $β_c=1/r$: survival holds for $β<1/r$, extinction holds almost surely for $β>1/r$, and explicit sufficient conditions on the critical line leave a factor-four gap.

math.PR↗

arXiv2024

Occupation time fluctuations of an age-dependent critical binary branching particle system

We study the limit fluctuations of the rescaled occupation time process of a branching particle system in $\mathbb{R}^d$, where the particles are subject to symmetric $α$-stable migration ($0<α\leq2$), critical binary branching, and general non-lattice lifetime distribution. We focus on two different regimes: lifetime distributions having finite expectation, and Pareto-type lifetime distributions, i.e. distributions belonging to the normal domain of attraction of a $γ$-stable law with $γ\in(0,1)$. In the latter case we show that, for dimensions $αγ<d<α(1+γ)$, the rescaled occupation time fluctuations converge weakly to a centered Gaussian process whose covariance function is explicitly calculated, and we call it {\em weighted sub-fractional Brownian motion.} Moreover, in the case of lifetimes with finite mean, we show that for $α<d<2α$ the fluctuation limit turns out to be the same as in the case of exponentially distributed lifetimes studied by Bojdecki et al. [7,8,9]. We also investigate the maximal parameter range allowing existence of the weighted sub-fractional Brownian motion and provide some of its fundamental properties, such as path continuity, long-range dependence, self-similarity and the lack of Markov property.

math.PR↗