2004/10/09 by Davar Khoshnevisan, David Levin, David A. Levin +5
Mathematics · #Markov Chains and Monte Carlo Methods #Mathematical Approximation and Integration #Mathematical Dynamics and Fractals #math.PR #msc:28C20 #msc:60J45 #msc:60J65
paper · pdf · doi:10.48550/arxiv.math/0410236
13 pages
arxiv created 2004/10/09 · arxiv updated 2009/12/01
We propose a set-indexed family of capacities \∩G \G ⊆ \R+ on the classical Wiener space C(\R+). This family interpolates between the Wiener measure (∩_\0\) on C(\R+) and the standard capacity (∩\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 1a \KG(r6) e-π2/(8r2) ≤ ∩G \f^* ≤ r\ ≤ a \KG(r6) e-π2/(8r2). Here, \KG denotes the Kolmogorov ε-entropy of G, and f^* := sup[0,1]|f|.