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 I α ( ⋅ ) and D α ( ⋅ ) of the Liouville fractional integral and derivative operators, respectively, where α ( u ) , u ∈ R , is a general function taking values in ( 0 , 1 ) and satisfying some regularity conditions. The proof of D α ( ⋅ ) I α ( ⋅ ) f = f relies on a surprising integral identity. We also discuss properties of multifractional generalizations of fractional Brownian motion defined as white noise integrals X t = ∫ 0 t ( I α ( ⋅ ) B ̇ ) ( s ) d s and Y t = ∫ 0 t ( D − α ( ⋅ ) B ̇ ) ( s ) d s.