Modulation spaces, Wiener amalgam spaces, and Brownian motions

AbstractWe study the local-in-time regularity of the Brownian motion with respect to localized variants of modulation spaces Msp,q and Wiener amalgam spaces Wsp,q. We show that the periodic Brownian motion belongs locally in time to Msp,q(T) and Wsp,q(T) for (s−1)q<−1, and the condition on the indices is optimal. Moreover, with the Wiener measure μ on T, we show that (Msp,q(T),μ) and (Wsp,q(T),μ) form abstract Wiener spaces for the same range of indices, yielding large deviation estimates. We also establish the endpoint regularity of the periodic Brownian motion with respect to a Besov-type space bˆp,∞s(T). Specifically, we prove that the Brownian motion belongs to bˆp,∞s(T) for (s−1)p=−1, and it obeys a large deviation estimate. Finally, we revisit the regularity of Brownian motion on usual local Besov spaces Bp,qs, and indicate the endpoint large deviation estimates