Modulation spaces, Wiener amalgam spaces, and Brownian motions

2010/07/12 by Árpád Bényi, Tadahiro Oh, Bényi, Árpád +1 · 3 citations
Mathematics · #60G51 #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Primary 42B35 #Probability (math.PR) #Secondary 42A61 #math.AP #math.FA #math.PR #msc:42A61 #msc:42B35 #msc:60G51

paper · pdf · doi:10.48550/arxiv.1007.1957

35 pages. The introduction is expanded. Appendices are added (A: derivation of Fourier-Wiener series, B: passing estimates from T to bounded intervals on R.) To appear in Adv. Math

arxiv created 2011/08/18 · arxiv updated 2011/08/19

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 μon T, we show that (M^{p, q}_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 \ftb^s_{p, ∞} (T). Specifically, we prove that the Brownian motion belongs to \ftb^s_{p, ∞} (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.

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