On the wavelet transform of fractional Brownian motion

The wavelet transform of a function f(t) is deened by the formula: Wf(t; a) = W a f(t) = 1 p a Z f(s)g(t ? s a) ds where g(t) is a xed function, t 2 R and a 2 R +. This transform yields a joint timescale representation the original input function that has been of great recent interest. (See for example D1] D2] and HW]). In a recent correspondence, Flandrin F] proposed the use of the wavelet transform to analyze the behavior of fractional Brownian motion, a highly nonstationary random process. (For a background on fractional Brownian motion and some of it's applications, see M1] and MV]). The wavelet transform of a stochastic process, X(t), is a random eld WX(t; a) on the upper half plane. The process t 7 ! W a X(t) can be thought of as the component of the original process at scale a. A consequence of Flandrin's computation is that fractional Brownian motion is stationary at each xed scale. In particular, when X(t) is a fractional Brownian motion, the covariance of the process t 7 ! W a X(t) is of the form (1) EW a X(t) W a X(s)] = a (t ? s a) where is a positive deenite function determined in an explicit manner by the order or the fractional Brownian motion and the deening function g(t) of the wavelet transform. This fact is used by Flandrin to make rigorous sense of the spectral content of fractional Brownian motion. It is natural to ask whether there are other Gaussian processes whose wavelet transforms have such a natural covariance structure. In addition, are there any Gaussian processes whose wavelet transform is stationary with respect to the aane group (i.e. the statistics of the wavelet transform do not depend on translations and