Nonhomogeneous fractional integration and multifractional processes

Extending the recent work of Philippe et al. [A. Philippe, D. Surgailis, M.-C. Viano, Invariance principle for a class of non stationary processes with long memory, C. R. Acad. Sci. Paris, Ser. 1. 342 (2006) 269–274; A. Philippe, D. Surgailis, M.-C. Viano, Time varying fractionally integrated processes with nonstationary long memory, Theory Probab. Appl. (2007) (in press)] on time-varying fractionally integrated operators and processes with discrete argument, we introduce nonhomogeneous generalizations Iα(⋅)Iα(⋅) and Dα(⋅)Dα(⋅) of the Liouville fractional integral and derivative operators, respectively, where α(u),u∈Rα(u),u∈R, is a general function taking values in (0,1)(0,1) and satisfying some regularity conditions. The proof of Dα(⋅)Iα(⋅)f=fDα(⋅)Iα(⋅)f=f relies on a surprising integral identity. We also discuss properties of multifractional generalizations of fractional Brownian motion defined as white noise integrals Xt=∫0t(Iα(⋅)Ḃ)(s)ds and Yt=∫0t(D−α(⋅)Ḃ)(s)ds.