On the Fourier structure of the zero set of fractional Brownian motion

In this paper, we consider a one-dimensional fractional Brownian motion X and the Fourier transform of its associated Dirac measure δ(X). It is a measure, canonically associated with X (in the sense of Schwartz’s theory of generalized functions). As was shown by Kahane, this measure reflects the fractal geometry of the zero set of X to a remarkable degree. One can also think of this measure as being given by the local time of X at 0. Kahane initiated the problem of understanding the Fourier structure of the zero sets of fractional Brownian motion. In this paper, we address this problem by analysing the even moments of the Fourier transform of the Dirac measure of a one-dimensional fractional Brownian motion restricted to compact intervals. We shall represent these moments by Fresnel-type oscillatory integrals. In the case of Brownian motion, the second-order moment reveals close connections with Bessel functions and Fresnel integrals.