Modulation spaces, Wiener amalgam spaces, and Brownian motions

TL;DR

This paper investigates the local-in-time regularity of Brownian motion within various modulation, Wiener amalgam, and Besov-type spaces, establishing optimal conditions and large deviation estimates for these function spaces.

Contribution

It characterizes the regularity of Brownian motion in modulation and Wiener amalgam spaces, identifies optimal index conditions, and constructs associated abstract Wiener spaces with large deviation estimates.

Findings

Brownian motion belongs locally in time to M^{p, q}_s and W^{p, q}_s for (s-1)q < -1

Brownian motion belongs to t{b}^s_{p, } for (s-1) p = -1

Established large deviation estimates for Brownian motion in these spaces

Abstract

We study the local-in-time regularity of the Brownian motion with respect to localized variants of modulation spaces M^{p, q}_s and Wiener amalgam spaces W^{p, q}_s. We show that the periodic Brownian motion belongs locally in time to M^{p, q}_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 \mu on T, we show that (M^{p, q}_s (T), \mu) and (W^{p, q}_s (T), \mu) 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 \ft{b}^s_{p, \infty} (T). Specifically, we prove that the Brownian motion belongs to \ft{b}^s_{p, \infty} (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}^s, and indicate the endpoint large deviation estimates.