Capacitary Trapping and Brownian Escape: A Sharp Hardy Principle for Absorbing Media

Abstract

We study Brownian motion killed upon exiting arbitrary open sets $\Omega\subset\mathbb R^n$, $n\in[3,\infty)\cap\mathbb N$. Let $\alpha\in(0,1]$. The infimal radius $d_\alpha(x)$ at which the absorber near $x$ carries an $\alpha$-fraction of the Newtonian capacity of the ball of the same radius provides a local measure of electrostatic trapping. We prove that this capacitary loading forces an absorption probability of at least $c_n\alpha$ within a dimension-dependent multiple of the diffusive time scale $d_\alpha(x)^2$. Independently, if $p\in (0,1)$ and $\tau_p(x)$ is the time by which Brownian motion started at $x$ has been absorbed with probability at least $p$, then \begin{align*} \int_\Omega\frac{|u\left(x\right)|^2}{\tau_p\left(x\right)}\,\mathrm d x \le\frac{2}{p^2}\int_\Omega|\nabla u\left(x\right)|^2\,\mathrm d x \end{align*} for every $u\in C_{\mathrm{c}}^\infty(\Omega)$. Together these statements give a capacitary Hardy inequality with constant $C_n\alpha^{-2}$ on every open set, answering a problem of Maz'ya, where $C_n$ is a positive constant and, moreover, the exponent of $\alpha^{-2}$ is sharp. As applications, we derive corresponding estimates for survival probabilities, a corresponding spectral lower bound (including a recovery of the Maz'ya--Shubin lower bound), heat dissipation, Schr\"odinger forms, and Fisher information.