Optimal Trapping of Brownian Motion: A Nonlinear Analogue of the Torsion Function

Torsional rigidity for cylinders with a Brownian fracture

We obtain bounds for the expected loss of torsional rigidity of a cylinder $Ω_L=(-L/2,L/2) \times Ω\subset \R^3$ of length $L$ due to a Brownian fracture that starts at a random point in $Ω_L,$ and runs until the first time it exits $Ω_L$. These bounds are expressed in terms of the geometry of the cross-section $Ω\subset \R^2$. It is shown that if $Ω$ is a disc with radius $R$, then in the limit as $L \rightarrow \infty$ the expected loss of torsional rigidity equals $cR^5$ for some $c\in (0,\infty)$. We derive bounds for $c$ in terms of the expected Newtonian capacity of the trace of a Browni…

布朗运动的陷阱行为

Trap behaviors for Brownian motions

研究领域几何性质与布朗运动扩散动力学的关系，重点探讨随机过程中的'陷阱'现象。

Torsional rigidity for regions with a Brownian boundary

Let $T^m$ be the $m$-dimensional unit torus, $m \in N$. The torsional rigidity of an open set $Ω\subset T^m$ is the integral with respect to Lebesgue measure over all starting points $x \in Ω$ of the expected lifetime in $Ω$ of a Brownian motion starting at $x$. In this paper we consider $Ω= T^m \backslash β[0,t]$, the complement of the path $β[0,t]$ of an independent Brownian motion up to time $t$. We compute the leading order asymptotic behaviour of the expectation of the torsional rigidity in the limit as $t \to \infty$. For $m=2$ the main contribution comes from the components in $T^2 \bac…

Towards Optimal Gradient Bounds for the Torsion Function in the Plane

Let $Ω\subset \mathbb{R}^2$ be a bounded, convex domain and let $u$ be the solution of $-Δu = 1$ vanishing on the boundary $\partial Ω$. The estimate $$ \| \nabla u\|_{L^{\infty}(Ω)} \leq c |Ω|^{1/2}$$ is classical. We use the P-functional, the stability theory of the torsion function and Brownian motion to establish the estimate for a universal $c < (2π)^{-1/2}$. We also give a numerical construction showing that the optimal constant satisfies $c \geq 0.358$. The problem is important in different settings: (1) as the maximum shear stress in Saint Venant Elasticity Theory, (2) as an optimal co…

非局部边值问题、随机重置与图上的布朗运动

Non-local Boundary Value Problems, stochastic resetting and Brownian motions on graphs

本文研究了涉及非局部算子的动态边界条件，详细描述了此类算子及其与随机时间和泛函的关系，基于非局部算子与边值问题的相互作用刻画了布朗运动的边界行为，并重点分析了带跳跃（重置/重启）的 Feller-Wentzell 扩散，将结果从实数轴推广至带陷阱点或排斥顶点的星图。

星图上的黏性布朗运动

Sticky Brownian motions on star graphs

本文在星图上构造随机过程，用奇异测度刻画陷阱并定义非局部算子，引入非局部动态条件，通过半马尔可夫过程获得全图运动的概率表示，并可推广至一般图结构。