# Preprint: Capacities in Wiener Space, Quasi-sure Lower Functions, and Kolmogorov's ε-Entropy

## D. Khoshnevisan, D. A. Levin, and P. J. Méndez-Hernández

Abstract. We propose a set-indexed family of capacities $$\{\text{cap}_G\}_{G\subset{\bf R}}$$ on the classical Wiener space $$C({\bf R}_+)$$. This family interpolates between the Wiener measure ($$\text{cap}_{\{0\}}$$) on $$C({\bf R}_+)$$ and the standard capacity ($$\text{cap}_{{\bf R}_+}$$) on Wiener space. We then apply our capacities to characterize all quasi-sure lower functions in $$C({\bf R}_+)$$. In order to do this we derive the following capacity estimate (Theorem 2.3) which may be of independent interest: There exists a constant $$a>1$$ such that for all $$r>0$$,
$$\displaystyle a^{-1}{\rm K}_G(r^6) \exp\left(-\frac{\pi^2}{8r^2}\right) \le \text{cap}_G \left\{ f^*\le r\right\}\le a{\rm K}_G(r^6) \exp\left(-\frac{\pi^2}{8r^2}\right),$$
where $${\rm K}_G$$ denotes the Kolmogorov $$\epsilon$$-entropy of $$G$$, and $$f^*:=\sup_{[0,1]}|f|$$.

Keywords. Capacity in Wiener space, lower functions, Kolmogorov entropy.

AMS Classification (2000) 60J45, 60J65, 60Fxx, 28C20.

Support. The research of D. Kh. was supported in part by a grant from the National Science Foundation.

Pre/E-Prints. This paper is available in

 Davar Khoshnevisan Department of Mathematics University of Utah 155 S, 1400 E JWB 233 Salt Lake City, UT 84112-0090, U.S.A. davar@math.utah.edu David Asher Levin Department of Mathematics University of Utah 155 S, 1400 E JWB 233 Salt Lake City, UT 84112-0090, U.S.A. levin@math.utah.edu Pedro J. Méndez-Hernández Department of Mathematics University of Utah 155 S, 1400 E JWB 233 Salt Lake City, UT 84112-0090, U.S.A. mendez@math.utah.edu Current Address: Escuela de Matemática Universidad de Costa Rica San Pedro de Montes de Oca, Costa Rica pmendez@emate.ucr.ac.cr

Last Update: October 9, 2004
© 2004 - Davar Khoshnevisan, David Asher Levin, and Pedro J. Méndez-Hernández