Brownian motion can feel the shape of a drum

Feeling boundary by Brownian motion in a ball

We provide short-time asymptotics with rates of convergence for the Laplace Dirichlet heat kernel in a ball. The boundary behaviour is precisely described. Presented results may be considered as a complement or a generalization of the famous "principle of not feeling the boundary" in case of a ball.

关于布朗运动的首达时间问题——重探镜像逆方法

On first passage time problems of Brownian motion -- The inverse method of images revisited

本文研究布朗运动的首达时间问题，重探镜像法，提出对偶方法给出表示测度存在性的充分条件，并引入高效近似算法及理论基础。

The scaling limit of the weakly self-avoiding walk on a high-dimensional torus

We prove that the scaling limit of the weakly self-avoiding walk on a $d$-dimensional discrete torus is Brownian motion on the continuum torus if the length of the rescaled walk is $o(V^{1/2})$ where $V$ is the volume (number of points) of the torus and if $d>4$. We also prove that the diffusion constant of the resulting torus Brownian motion is the same as the diffusion constant of the scaling limit of the usual weakly self-avoiding walk on $\mathbb{Z}^d$. This provides further manifestation of the fact that the weakly self-avoiding walk model on the torus does not feel that it is on the toru…