Branching Brownian motion with "mild" Poissonian obstacles

Quenched convergence of a sequence of superprocesses in R^d among Poissonian obstacles

We prove a convergence theorem for a sequence of super-Brownian motions moving among hard Poissonian obstacles, when the intensity of the obstacles grows to infinity but their diameters shrink to zero in an appropriate manner. The superprocesses are shown to converge in probability for the law $\mathbf{P}$ of the obstacles, and $\mathbf{P}$-almost surely for a subsequence, towards a superprocess with underlying spatial motion given by Brownian motion and (inhomogeneous) branching mechanism $ψ(u,x)$ of the form $ψ(u,x)= u^2+ κ(x)u$, where $κ(x)$ depends on the density of the obstacles. This wor…

关于泊松障碍物空置集中球内最大可见性的研究

On the maximum visibility in a ball through the vacant set of Poissonian obstacles

本文研究三种障碍物模型中球内最大可见距离，证明其除以特定对数因子后依概率收敛于显式常数，并给出维度3时布朗交错的修正因子。

通过泊松障碍物的空集实现多跳可见性

Multi-hop visibility through the vacant set of Poissonian obstacles

该研究针对d维欧几里得空间中三种不同障碍物模型的空集，推导了多跳可见性概率的精确界，观察到其在两个特征尺度上的普适行为，推广了早期针对单跳的相关结果。

Asymptotics for the Wiener sausage among Poissonian obstacles

We consider the Wiener sausage among Poissonian obstacles. The obstacle is called hard if Brownian motion entering the obstacle is immediately killed, and is called soft if it is killed at certain rate. It is known that Brownian motion conditioned to survive among obstacles is confined in a ball near its starting point. We show the weak law of large numbers, large deviation principle in special cases and the moment asymptotics for the volume of the corresponding Wiener sausage. One of the consequence of our results is that the trajectory of Brownian motion almost fills the confinement ball.

Replica overlap and covering time for the Wiener sausages among Poissonian obstacles

We study two objects concerning the Wiener sausage among Poissonian obstacles. The first is the asymptotics for the \textit{replica overlap}, which is the intersection of two independent Wiener sausages. We show that it is asymptotically equal to their union. This result confirms that the localizing effect of the media is so strong as to completely determine the motional range of particles. The second is an estimate on the \textit{covering time}. It is known that the Wiener sausage avoiding Poissonian obstacles up to time $t$ is confined in some `clearing' ball near the origin and almost fills…

Conditional survival distributions of Brownian trajectories in a one dimensional Poissonian environment in the critical case

In this work we consider a one-dimensional Brownian motion with constant drift moving among a Poissonian cloud of obstacles. Our main result proves convergence of the law of processes conditional on survival up to time $t$ as $t$ converges to infinity in the critical case where the drift coincides with the intensity of the Poisson process. The complements a previous result of T. Povel, who considered the same question in the case where the drift is strictly smaller than the intensity.