2005/11/21 by Friz, Peter, Victoir, Nicolas · 2 citations
#60G17 #60H99 #FOS: Mathematics #Functional Analysis (math.FA) #Probability (math.PR)
paper · doi:10.48550/arxiv.math/0511520
Fractional Sobolev spaces, also known as Besov or Slobodetzki spaces, arise in many areas of analysis, stochastic analysis in particular. We prove an embedding into certain q-variation spaces and discuss a few applications. First we show q-variation regularity of Cameron-Martin paths associated to fractional Brownian motion and other Volterra processes. This is useful, for instance, to establish large deviations for enhanced fractional Brownian motion. Second, the q-variation embedding, combined with results of rough path theory, provides a different route to a regularity result for stochastic differential equations by Kusuoka. Third, the embedding theorem works in a non-commutative setting and can be used to establish Hoelder/variation regularity of rough paths.