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