A free boundary problem arising from branching Brownian motion with selection

The stochastic Fisher-KPP Equation with seed bank and on/off-branching-coalescing Brownian motion

We introduce a new class of stochastic partial differential equations (SPDEs) with seed bank modeling the spread of a beneficial allele in a spatial population where individuals may switch between an active and a dormant state. Incorporating dormancy and the resulting seed bank leads to a two-type coupled system of equations with migration between both states. We first discuss existence and uniqueness of seed bank SPDEs and provide an equivalent delay representation that allows a clear interpretation of the age structure in the seed bank component. The delay representation will also be crucial…

具有代际依赖扩散率的分支布朗运动与非局部偏微分方程

Branching Brownian motion with generation-dependent diffusivity and nonlocal partial differential equations

本文研究具有代际扩散率的分支布朗运动投票模型，揭示长期分布随γ发生相变，并为非局部PDE提供概率解释。

Non local branching Brownians with annihilation and free boundary problems

We study a system of branching Brownian motions on $\mathbb R$ with annihilation: at each branching time a new particle is created and the leftmost one is deleted. In [7] it has been studied the case of strictly local creations (the new particle is put exactly at the same position of the branching particle), in [10] instead the position $y$ of the new particle has a distribution $p(x,y)dy$, $x$ the position of the branching particle, however particles in between branching times do not move. In this paper we consider Brownian motions as in [7] and non local branching as in [10] and prove conver…

On a Class of Non-linear Differential Equations Arising from Branching Diffusions

A non-linear differential equation arising from a stochastic process known as branching Brownian motion is considered. We find an explicit solution and show the uniqueness of the solution under some boundedness conditions using probabilistic ideas. We discuss non-negative solutions. We also generalize this idea to a class of non-linear parabolic differential equations which we call probabilistic parabolic equations.

角度依赖的二维分支布朗运动中的多项式减速

Polynomial slowdown in an angle-dependent 2d branching Brownian motion

研究角度依赖的二维分支布朗运动中最大距离随时间的变化，推导出多项式衰减的渐进行为及关键参数影响。

布朗-弗莱明-维奥特系统的分支爆炸

Branching explosion of Brownian Fleming-Viot systems

该研究解决有界域布朗-弗莱明-维奥特系统的非灭绝问题，构造特定R³有界域证明N≥3时分支爆炸以正概率发生，通过标记谱系转移定理等技术推导精确种群阈值与相关电流性质。

基于秩相互作用的分支 Brown 粒子系统的收敛性与波传播

Convergence and Wave Propagation for a System of Branching Rank-Based Interacting Brownian Particles

本文研究按秩相互作用的分支 Brown 粒子系统，证明 K→∞ 时的加权种群收敛，并数值揭示传播速度阈值及推动/拉动型波前分类。

Brownian bees in the infinite swarm limit

The Brownian bees model is a branching particle system with spatial selection. It is a system of $N$ particles which move as independent Brownian motions in $\mathbb{R}^d$ and independently branch at rate 1, and, crucially, at each branching event, the particle which is the furthest away from the origin is removed to keep the population size constant. In the present work we prove that as $N \to \infty$ the behaviour of the particle system is well approximated by the solution of a free boundary problem (which is the subject of a companion paper), the hydrodynamic limit of the system. We then sh…

Measure-valued branching processes associated with Neumann nonlinear semiflows

We construct a measure-valued branching Markov process associated with a nonlinear boundary value problem, where the boundary condition has a nonlinear pseudo monotone branching mechanism term $-β$, which includes as a limit case $β(u) = - u^{m}$, with $0 < m < 1$. The process is then used in the probabilistic representation of the solution of the parabolic problem associated with a nonlinear Neumann boundary value problem. In this way the classical association of the superprocesses to the Dirichlet boundary value problems also holds for the nonlinear Neumann boundary value problems. It turns …

Speed and fluctuations of N-particle branching Brownian motion with spatial selection

We consider branching Brownian motion on the real line with the following selection mechanism: Every time the number of particles exceeds a (large) given number $N$, only the $N$ right-most particles are kept and the others killed. After rescaling time by $\log^3N$, we show that the properly recentred position of the $\lceil αN\rceil$-th particle from the right, $α\in(0,1)$, converges in law to an explicitly given spectrally positive Lévy process. This behaviour has been predicted to hold for a large class of models falling into the universality class of the FKPP equation with weak multiplicat…

1-stable fluctuations in branching Brownian motion at critical temperature I: the derivative martingale

Let $(Z_t)_{t\geq 0}$ denote the derivative martingale of branching Brownian motion, i.e.\@ the derivative with respect to the inverse temperature of the normalized partition function at critical temperature. A well-known result by Lalley and Sellke [\textit{Ann. Probab.}, 15(3):1052--1061, 1987] says that this martingale converges almost surely to a limit $Z_\infty$, positive on the event of survival. In this paper, our concern is the fluctuations of the derivative martingale around its limit. A corollary of our results is the following convergence, confirming and strengthening a conjecture b…