A Variation Embedding Theorem and Applications

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 Hölder/variation regularity of rough paths. 1. Fractional Sobolev Spaces For a real valued measurable path h : [0, 1] → R and δ ∈ (0, 1) and p ∈ (1,∞) we define the fractional Sobolev (semi-)norm