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 {capG}G ⊆R on the classical Wiener space C(R+). This family interpolates between the Wiener measure (cap{0}) on C(R+) and the standard capacity (capR+) 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 (Theorem 2.3) which may be of independent interest: There exists a constant a>1 such that for all r>0,
a-1KG (r6) e-π2/(8r2) ≤ capG{ƒ* ≤ r} ≤ aKG (r6) e-π2/(8r2).
Here, KG denotes the Kolmogorov ε-entropy of G, and ƒ* := sup[0,1]| ƒ |.

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