In 1961, Ciesielski established a remarkable isomorphism of spaces
of Holder continuous functions and Banach spaces of real valued se-
quences. This isomorphism leads to wavelet decompositions of Gaus-
sian processes giving access for instance to a precise study of their large
deviations, as shown by Baldi and Roynette. We will use Schauder rep-
resentations for a pathwise approach of integration, along Ciesielski's
isomorphism. It can be formulated in terms of dyadic martingales and
Rademacher functions. In a more general and analytical setting, this
pathwise approach of rough path analysis can be understood in terms
of Paley-Littlewood decompositions of distributions, and Bony para-
products in Besov spaces. This talk is based on joint work with M.
Gubinelli and N. Perkowski (U Paris-Dauphine).
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