Some Prevalent Sets in Multifractal Analysis: How Smooth is Almost Every Function in $$T_p^\alpha (x)$$

Abstract

We present prevalent results concerning generalized versions of the T α p spaces, initially introduced by Calderón and Zygmund. We notably show that the logarithmic correction appearing in the quasi-characterization of such spaces is mandatory for almost every function; it is in particular true for the Hölder spaces, for which the existence of the correction was showed necessary for a specific function. We also show that almost every function from T α p (x 0 ) has α as generalized Hölder exponent at x 0 .

Multifractional Hermite processes: definition and first properties

“…Finally, from (48), equality (50), Proposition 2.2 and Condition 2.8 (a), we deduce the existence of a deterministic constant c 1 > 0, only depending on H and I, such that, for all 0 < ε < ξ and t, u ∈…”

“…It is particularly true when we consider stochastic processes, see for instance [31,27,30]. Note that one can define generalized Hölder spaces associated with modulus of continuity [39,40,46] and that these spaces lead to specific multifractal formalisms [47,48].…”