The Generalized Multifractional Field: A nice tool for the study of the generalized Multifractional Brownian Motion

The Generalized Multifractional Brownian Motion (GMBM) is a continuous Ganssian process {X (t)}(tis an element of[0,1]) that extends the classical Fractional Brownian Motion (FBM) and the Multifractional Brownian Motion (MBM) [ 15, 4, 1, 2]. Its main interest is that, its Holder regularity can change widely from point to point. In this article we introduce the Generalized Multifractional Field (GMF), a continuous Gaussian field {Y(x, Y)}((x,y)is an element of[0.1])2 that satisfies for every t, X(t) = Y(t. t). Then, we give a wavelet decomposition of Y and using this nice decomposition, we show that Y is beta-Holder in Y, uniformly in x. Generally speaking this result seems to be quite important for the study of the GMBM. In this article, it will allow us to determine, without any restriction, its pointwise, almost sure, Holder exponent and to prove that two GMBM's with the same Holder regularity differ by a "smoother" process.