Modulation Spaces, Wiener Amalgam Spaces, and Brownian Motions

We study the local-in-time regularity of the Brownian motion with respect to localized variants of modulation spaces M s and Wiener amalgam spaces W p,q s . We show that the periodic Brownian motion belongs locally in time to M s (T) and W p,q s (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 (M s (T), μ) and (W p,q s (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 b̂p,∞(T). Specifically, we prove that the Brownian motion belongs to b̂p,∞(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 B p,q, and indicate the endpoint large deviation estimates.