Asymptotic capacities for finely divided bodies and stopped diffusions
In \({\mathbb{R}}^ d\), \(d\geq 3\), let \((B_ t)\) denote the Brownian motion and let \((D_ n)\) be a sequence of closed sets. Let \((\tau_ n)\) denote the first hitting-time of \(D_ n\). The authors introduced the notion of ''stable convergence'' of \((\tau_ n)\) to a randomized stopping-time T. Consider the following boundary values problem (BVP) for the heat equation: (1)\ \(\partial V(x,t)/\partial t=(1/2)\Delta v(x,t)\) for \(t>0\), \(x\in D_ n^ c\), \(v(x,t)=0\) for \(t>0\), \(x\in \partial D_ n\), \(v(x,0)=f(x)\) for \(x\in D^ c_ n\), where \(f: {\mathbb{R}}^ d\to {\mathbb{R}}\) is a bounded Borel function. It is known that the above BVP admits \(u_ n(x,t)=E^ x(f(B_ t)\chi_{(\tau_ n>t)})\) as solution. The authors show that if \((\tau_ n)\) converges stably to T then, \[ u(x,t)=E^ x(f(B_ t)\chi_{(T>t)}) \] is the solution of the following BVP: \[ \partial v(x,t)/\partial t=(1/2)\Delta v(x,t)- h(x)v(x,t)\quad for\quad t>0,\quad x\in {\mathbb{R}}^ d,\quad v(x,0)=f(x)\quad for\quad x\in {\mathbb{R}}^ d, \] where the killing rate h represents a ''limiting capacity density''. The model could be used to study certain random media problems which consists of many small bodies becoming smaller and more densely distributed as \(n\to \infty\).
scientific article; zbMATH DE number 4003197