Consider a catalytic super-Brownian motion $X=X^Γ$ with finite variance branching. Here `catalytic' means that branching of the reactant $X$ is only possible in the presence of some catalyst. Our intrinsic example of a catalyst is a stable random measure $Γ$ on $R$ of index $0< gamma <1$. Consequently, here the catalyst is located in a countable dense subset of $R$. Starting with a finite reactant mass $X_0$ supported by a compact set, $X$ is shown to die in finite time. Our probabilistic argument uses the idea of good and bad historical paths of reactant `particles' during time periods $[T_{n},T_{n+1})$. Good paths have a significant collision local time with the catalyst, and extinction can be shown by individual time change according to the collision local time and a comparison with Feller's branching diffusion. On the other hand, the remaining bad paths are shown to have a small expected mass at time $T_{n+1}$ which can be controlled by the hitting probability of point catalysts and the collision local time spent on them.
We prove a fluctuating limit theorem of a sequence of super-Brownian motions over $\mbb{R}$ with a single point catalyst. The weak convergence of the processes on the space of Schwarz distributions is established. The limiting process is an Ornstein-Uhlenbeck type process solving a Langevin type equation driven by a one-dimensional Brownian motion.
We consider the behaviour of a continuous super-Brownian motion catalysed by a random medium with infinite overall density under the hydrodynamic scaling of mass, time, and space. We show that, in supercritical dimensions, the scaled process converges to a macroscopic heat flow, and the appropriately rescaled random fluctuations around this macroscopic flow are asymptotically bounded, in the sense of log-Laplace transforms, by generalised stable Ornstein-Uhlenbeck processes. The most interesting new effect we observe is the occurrence of an index-jump from a 'Gaussian' situation to stable fluc…
This paper concerns the almost sure time dependent local extinction behavior for super-coalescing Brownian motion $X$ with $(1+β)$-stable branching and Lebesgue initial measure on $\bR$. We first give a representation of $X$ using excursions of a continuous state branching process and Arratia's coalescing Brownian flow. For any nonnegative, nondecreasing and right continuous function $g$, put τ:=\sup \{t\geq 0: X_t([-g(t),g(t)])>0 \}. We prove that $\bP\{τ=\infty\}=0$ or 1 according as the integral $\int_1^\infty g(t)t^{-1-1/β} dt$ is finite or infinite.
We consider continuous state branching processes that are perturbed by a Brownian motion. These processes are constructed as the unique strong solution of a stochastic differential equation. The long-term extinction and explosion behaviours are studied. In the stable case, the extinction and explosion probabilities are given explicitly. We find three regimes for the asymptotic behaviour of the explosion probability and, as in the case of branching processes in random environment, we find five regimes for the asymptotic behaviour of the extinction probability. In the supercritical regime, we st…
In this paper, we study the extinction time of logistic branching processes which are perturbed by an independent random environment driven by a Brownian motion. Our arguments use a Lamperti-type representation which is interesting on its own right and provides a one to one correspondence between the latter family of processes and the family of Feller diffusions which are perturbed by an independent spectrally positive Lévy process. When the independent random perturbation (of the Feller diffusion) is driven by a subordinator then the logistic branching processes in a Brownian environment conv…
We study the maximal displacement and related population for a branching Brownian motion in Euclidean space in terms of the principal eigenvalue of an associated Schrödinger type operator. We first determine their growth rates on the survival event. We then establish the upper deviation for the maximal displacement under the possibility of extinction. Under the non-extinction condition, we further discuss the decay rate of the upper deviation probability and the population growth at the critical phase.
Basic properties of Brownian motion are used to derive two results concerning birth-death chains. First, the probability of extinction is calculated. Second, sufficient conditions on the transition probabilities of a birth-death chain are given to ensure that the expected value of the chain converges to a limit. The theory of Brownian motion local time figures prominently in the proof of the second result.
We consider a branching particle model in which particles move inside a Euclidean domain according to the following rules. The particles move as independent Brownian motions until one of them hits the boundary. This particle is killed but another randomly chosen particle branches into two particles, to keep the population size constant. We prove that the particle population does not approach the boundary simultaneously in a finite time in some Lipschitz domains. This is used to prove a limit theorem for the empirical distribution of the particle family.
We consider critical branching Brownian motion with absorption, in which there is initially a single particle at $x > 0$, particles move according to independent one-dimensional Brownian motions with the critical drift of $-\sqrt{2}$, and particles are absorbed when they reach zero. Here we obtain asymptotic results concerning the behavior of the process before the extinction time, as the position $x$ of the initial particle tends to infinity. We estimate the number of particles in the system at a given time and the position of the right-most particle. We also obtain asymptotic results for the…
We establish finite time extinction with probability one for weak solutions of the Cauchy-Dirichlet problem for the 1D stochastic porous medium equation with Stratonovich transport noise and compactly supported smooth initial datum. Heuristically, this is expected to hold because Brownian motion has average spread rate $O(t^\frac{1}{2})$ whereas the support of solutions to the deterministic PME grows only with rate $O(t^{\frac{1}{m{+}1}})$. The rigorous proof relies on a contraction principle up to time-dependent shift for Wong-Zakai type approximations, the transformation to a deterministic P…
It is well known that, starting with finite mass, the super-Brownian motion dies out in finite time. The goal of this article is to show that with some additional work, one can prove finite time die-out for two types of systems of stochastic differential equations on the lattice Z^d. Our first system involves the heat equation on the lattice Z^d, with a nonlinear noise term u(t,x)^gamma dB_x(t), with 1/2 <= gamma < 1. The B_x are independent Brownian motions. When gamma = 1/2, the measure which puts mass u(t,x) at x is a super-random walk and it is well-known that the process becomes extinct i…
We consider a Fleming-Viot-type particle system consisting of independently moving particles that are killed on the boundary of a domain. At the time of death of a particle, another particle branches. If there are only two particles and the underlying motion is a Bessel process on $(0,\infty)$, both particles converge to 0 at a finite time if and only if the dimension of the Bessel process is less than 0. If the underlying diffusion is Brownian motion with a drift stronger than (but arbitrarily close to, in a suitable sense) the drift of a Bessel process, all particles converge to 0 at a finit…
We construct a measure valued Markov process which we call infinite canonical super-Brownian motion, and which corresponds to the canonical measure of super-Brownian motion conditioned on non-extinction. Infinite canonical super-Brownian motion is a natural candidate for the scaling limit of various random branching objects on $\Z^d$ when these objects are (a) critical; (b) mean-field and (c) infinite. We prove that ICSBM is the scaling limit of the spread-out oriented percolation incipient infinite cluster above 4 dimensions and of incipient infinite branching random walk in any dimension. We…
Extinction and extinguishment properties for a nonlinear predator-prey branching model
Shenzhen MSU-BIT University（深圳莫斯科国立鲍曼技术大学）；Southern University of Science and Technology（南方科技大学）；Peking University（北京大学）；North Minzu University（北方民族大学）
We study a branching Brownian motion $Z$ in $\mathbb{R}^d$, among obstacles scattered according to a Poisson random measure with a radially decaying intensity. Obstacles are balls with constant radius and each one works as a trap for the whole motion when hit by a particle. Considering a general offspring distribution, we derive the decay rate of the annealed probability that none of the particles of $Z$ hits a trap, asymptotically in time $t$. This proves to be a rich problem motivating the proof of a more general result about the speed of branching Brownian motion conditioned on non-extincti…
We provide the exact large-time behavior of the tail distribution of the extinction time of a self-similar fragmentation process with a negative index of self-similarity, improving thus a previous result on the logarithmic asymptotic behavior of this tail. Two factors influence this behavior: the distribution of the largest fragment at the time of a dislocation and the index of self-similarity. As an application we obtain the asymptotic behavior of all moments of the largest fragment and compare it to the behavior of the moments of a tagged fragment, whose decrease is in general significantly …
A general model of catalytic branching process (CBP) with any finite number of catalysis centers in a discrete space is studied. More exactly, it is assumed that particles move in this space according to a specified Markov chain and they may produce offspring only in the presence of catalysts located at fixed points. The asymptotic (in time) behavior of the total number of particles as well as the local particles numbers is investigated. The problems of finding the global extinction probability and local extinction probability are solved. Necessary and sufficient conditions are established for…
This paper studies: (i) the long time behaviour of the empirical distribution of age and normalised position of an age dependent critical branching Markov process conditioned on non-extinction; and (ii) the super-process limit of a sequence of age dependent critical branching Brownian motions.
Motivated by the study of the directed polymer model with mobile Poissonian traps or catalysts and the stochastic parabolic Anderson model with time dependent potential, we investigate the asymptotic behavior of \[\mathbb{E}\otimes\mathbb{E}_0\exp\left\{\pmθ\int^t_0\bar{V}(s,B_s)ds\right\}\qquad (t\to\infty)\] where $þ>0$ is a constant, $\overline{V}$ is the renormalized Poisson potential of the form \[\overline{V}(s,x)=\int_{\mathbb{R}^d}\frac{1}{|y-x|^p}\left(ω_s(dy)-dy\right),\] and $ω_s$ is the measure-valued process consisting of independent Brownian particles whose initial positions form…
We construct a class of superprocesses by taking the high density limit of a sequence of interacting-branching particle systems. The spatial motion of the superprocess is determined by a system of interacting diffusions, the branching density is given by an arbitrary bounded non-negative Borel function, and the superprocess is characterized by a martingale problem as a diffusion process with state space $M(\IR)$, improving and extending considerably the construction of Wang (1997, 1998). It is then proved in a special case that a suitable rescaled process of the superprocess converges to the u…
Branching explosion of Brownian Fleming-Viot systems
School of Mathematical Sciences, South China Normal University（华南师范大学数学科学学院）；Information and Artificial Intelligence, Anhui Agricultural University（安徽农业大学信息与人工智能学院）
We consider a discrete-time branching random walk defined on the real line, which is assumed to be supercritical and in the boundary case. It is known that its leftmost position of the $n$-th generation behaves asymptotically like $\frac{3}{2}\ln n$, provided the non-extinction of the system. The main goal of this paper, is to prove that the path from the root to the leftmost particle, after a suitable normalizatoin, converges weakly to a Brownian excursion in $D([0,1],\r)$.
We investigate the random continuous trees called Lévy trees, which are obtained as scaling limits of discrete Galton-Watson trees. We give a mathematically precise definition of these random trees as random variables taking values in the set of equivalence classes of compact rooted R-trees, which is equipped with the Gromov-Hausdorff distance. To construct Lévy trees, we make use of the coding by the height process which was studied in detail in previous work. We then investigate various probabilistic properties of Lévy trees. In particular we establish a branching property analogous to the w…
Motivated as a null model for comparison with data, we study the following model for a phylogenetic tree on $n$ extant species. The origin of the clade is a random time in the past, whose (improper) distribution is uniform on $(0,\infty)$. After that origin, the process of extinctions and speciations is a continuous-time critical branching process of constant rate, conditioned on having the prescribed number $n$ of species at the present time. We study various mathematical properties of this model as $n \to \infty$ limits: time of origin and of most recent common ancestor; pattern of divergenc…
