Positively and negatively excited random walks on integers, with branching processes

Joint invariance principles for random walks with positively and negatively reinforced steps

Given a random walk $(S_n)$ with typical step distributed according to some fixed law and a fixed parameter $p \in (0,1)$, the associated positively step-reinforced random walk is a discrete-time process which performs at each step, with probability $1-p$, the same step as $(S_n)$ while with probability $p$, it repeats one of the steps it performed previously chosen uniformly at random. The negatively step-reinforced random walk follows the same dynamics but when a step is repeated its sign is also changed. In this work, we shall prove functional limit theorems for the triplet of a random walk…

完全正相关共生分支模型的唯一性与长期行为

Uniqueness and Longtime Behavior of the Completely Positively Correlated Symbiotic Branching Model

本文利用共生分支模型与抛物Anderson模型的新对偶性，解决了$ρ=1$时完全正相关共生分支模型的弱唯一性开放问题，并研究了其长期行为，证明了在合适初始条件下其中一个种群最终会灭绝。

Longtime behavior of completely positively correlated Symbiotic Branching Model

We study the longtime behavior of a continuous state Symbiotic Branching Model (SBM). SBM can be seen as a unified model generalizing the Stepping Stone Model, Mutually Catalytic Branching Processes, and the Parabolic Anderson Model. It was introduced by Etheridge and Fleischmann in 2004. The key parameter in these models is the local correlation $ρ$ between the driving Brownian Motions. The longtime behavior of all SBM exhibits a dichotomy between coexistence and non-coexistence of the two populations depending on the recurrence and transience of the migration and also in many cases on the br…

Branching processes in correlated random environment

We consider the critical branching processes in correlated random environment which is positively associated and study the probability of survival up to the n-th generation. Moreover, when the environment is given by fractional Brownian motion, we estimate also the tail of progeny as well as the tail of width.

Asymptotic behavior of critical primitive multi-type branching processes with immigration

Under natural assumptions a Feller type diffusion approximation is derived for critical multi-type branching processes with immigration when the offspring mean matrix is primitive (in other words, positively regular). Namely, it is proved that a sequence of appropriately scaled random step functions formed from a sequence of critical primitive multi-type branching processes with immigration converges weakly towards a squared Bessel process supported by a ray determined by the Perron vector of the offspring mean matrix.

步长增强随机游走的强极限定理

Strong limit theorems for step-reinforced random walks

本文研究具有长程记忆的步长增强随机游走，通过鞅差序列不变原理和截断论证，为正负步长增强随机游走建立了强大数定律与强不变原理，并得出重对数律和泛函中心极限定理。