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

Abstract

We propose a set-indexed family of capacities $\{\cap_G \}_{G \subseteq \R_+}$ on the classical Wiener space $C(\R_+)$. This family interpolates between the Wiener measure ($\cap_{\{0\}}$) on $C(\R_+)$ and the standard capacity ($\cap_{\R_+}$) on Wiener space. We then apply our capacities to characterize all quasi-sure lower functions in $C(\R_+)$. 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 $ε$-entropy of $G$, and $f^* := \sup_{[0,1]}|f|$.