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 and Wiener amalgam spaces . We show that the periodic Brownian motion belongs locally in time to and for (s−1)q<−1, and the condition on the indices is optimal. Moreover, with the Wiener measure μ on T, we show that and 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 . Specifically, we prove that the Brownian motion belongs to 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 , and indicate the endpoint large deviation estimates.