An annihilating--branching particle model for the heat equation with average temperature zero

Coexistence in locally regulated competing populations and survival of branching annihilating random walk

We propose two models of the evolution of a pair of competing populations. Both are lattice based. The first is a compromise between fully spatial models, which do not appear amenable to analytic results, and interacting particle system models, which do not, at present, incorporate all of the competitive strategies that a population might adopt. The second is a simplification of the first, in which competition is only supposed to act within lattice sites and the total population size within each lattice point is a constant. In a special case, this second model is dual to a branching annihilati…

分支湮灭随机游走的存活与完全收敛

Survival and complete convergence for a branching annihilating random walk

研究了$d$维格点上的离散时间分支湮灭随机游走（BARW），证明了当移动距离$R$足够大时过程存活并具有唯一非平凡遍历平衡，且$μ$值过小或过大时过程消亡，同时证明了该情形下的完全收敛。

Extinction window of mean field branching annihilating random walk

We study a model of growing population that competes for resources. At each time step, all existing particles reproduce and the offspring randomly move to neighboring sites. Then at any site with more than one offspring, the particles are annihilated. This is a nonmonotone model, which makes the analysis more difficult. We consider the extinction window of this model in the finite mean-field case, where there are $n$ sites but movement is allowed to any site (the complete graph). We show that although the system survives for exponential time, the extinction window is logarithmic.

Dependent Double Branching Annihilating Random Walk

Double (or parity conserving) branching annihilating random walk, introduced by Sudbury in '90, is a one-dimensional non-attractive particle system in which positive and negative particles perform nearest neighbor hopping, produce two offsprings to neighboring lattice points and annihilate when they meet. Given an odd number of initial particles, positive recurrence as seen from the leftmost particle position was first proved by Belitsky, Ferrari, Menshikov and Popov in '01 and, subsequently in a much more general setup, in the article by Sturm and Swart (Tightness of voter model interfaces) i…

Voter models with heterozygosity selection

This paper studies variations of the usual voter model that favor types that are locally less common. Such models are dual to certain systems of branching annihilating random walks that are parity preserving. For both the voter models and their dual branching annihilating systems we determine all homogeneous invariant laws, and we study convergence to these laws started from other initial laws.

Product structure of heat phase space and branching Brownian motion

A generical formalism for the discussion of Brownian processes with non-constant particle number is developed, based on the observation that the phase space of heat possesses a product structure that can be encoded in a commutative unit ring. A single Brownian particle is discussed in a Hilbert module theory, with the underlying ring structure seen to be intimately linked to the non-differentiability of Brownian paths. Multi-particle systems with interactions are explicitly constructed using a Fock space approach. The resulting ring-valued quantum field theory is applied to binary branching Br…

关于半线性热方程分支扩散求解器的稳定性分析

Stability analysis of a branching diffusion solver for semilinear heat equations

本文研究了半线性热方程分支扩散求解器的稳定性，通过分析随机分支过程的可积性条件，证明了解的非爆炸性和弱解的唯一性。

Connectivity properties of Branching Interlacements

We consider connectivity properties of the Branching Interlacements model in $\mathbb{Z}^d,~d\ge5$, recently introduced by Angel, Ráth and Zhu in 2016. Using stochastic dimension techniques we show that every two vertices visited by the branching interlacements are connected via at most $\lceil d/4\rceil$ conditioned critical branching random walks from the underlying Poisson process, and that this upper bound is sharp. In particular every such two branching random walks intersect if and only if $5\le d\le 8$. The stochastic dimension of branching random walk result is of independent interest.…

Branching random walks in random environment: a survey

We consider branching particle processes on discrete structures like the hypercube in a random fitness landscape (i.e., random branching/killing rates). The main question is about the location where the main part of the population sits at a late time, if the state space is large. For answering this, we take the expectation with respect to the migration (mutation) and the branching/killing (selection) mechanisms, for fixed rates. This is intimately connected with the parabolic Anderson model, the heat equation with random potential, a model that is of interest in mathematical physics because of…

具有不规则漂移项的 Wright-Fisher 随机热方程

Wright-Fisher stochastic heat equations with irregular drifts

本文研究具有不规则漂移项的 Wright-Fisher 随机热方程，利用分支-聚生布朗运动对偶技术证明了在噪声系数非 Lipschitz 且退化时乘性白噪声的正则化效应及解的弱存在唯一性。