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

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

2004-10-09

arxiv.org/abs/math/0410236v1

Mathematics

Probability

13 pages

Scientific paper

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 $r > 0$, \[ \frac {1}{a} \K_G(r^6) e^{-\pi^2/(8r^2)} \le \cap_G \{f^* \le r\} \le a \K_G(r^6) e^{-\pi^2/(8r^2)}. \] Here, $\K_G$ denotes the Kolmogorov $\epsilon$-entropy of $G$, and $f^* := \sup_{[0,1]}|f|$.

Affiliated with

Also associated with

No associations

Rating

Capacities in Wiener Space, Quasi-Sure Lower Functions, and Kolmogorov's Epsilon-Entropy does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Capacities in Wiener Space, Quasi-Sure Lower Functions, and Kolmogorov's Epsilon-Entropy, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Capacities in Wiener Space, Quasi-Sure Lower Functions, and Kolmogorov's Epsilon-Entropy will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-217727