Capacities in Wiener Space, Quasi-Sure Lower Functions, and Kolmogorov's Epsilon-Entropy

Abstract

We propose a set-indexed family of capacities $\{{\cap }_G{\}}_{G\subseteq {\mathbb{R}}_{+}}$ on the classical Wiener space $C({\mathbb{R}}_{+})$. This family interpolates between the Wiener measure (${\cap }_{\{0\}}$) on $C({\mathbb{R}}_{+})$ and the standard capacity (${\cap }_{{\mathbb{R}}_{+}}$) on Wiener space. We then apply our capacities to characterize all quasi-sure lower functions in $C({\mathbb{R}}_{+})$. In order to do this we derive the following capacity estimate which may be of independent interest: There exists a constant $a>1$ such that for all $r>0$, \[ \frac {1}{a} \K_G(r^6) e^{-π^2/(8r^2)} \le \cap_G \{f^* \le r\} \le a \K_G(r^6) e^{-π^2/(8r^2)}. \] Here, $\K_G$ denotes the Kolmogorov $\epsilon$-entropy of $G$, and $f^{\ast }:={\sup }_{[0,1]}|f|$.

13 pages