Asymptotic capacities for finely divided bodies and stopped diffusions (Q1822417)

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 \(n\to \infty\).