Persistence Probabilities of a Smooth Self-Similar Anomalous Diffusion Process

Abstract

We consider the persistence probability of a certain fractional Gaussian process \(M^H\) that appears in the Mandelbrot-van Ness representation of fractional Brownian motion. This process is self-similar and smooth. We show that the persistence exponent of \(M^H\) exists, is positive and continuous in the Hurst parameter H. Further, the asymptotic behaviour of the persistence exponent for \(H\downarrow 0\) and \(H\uparrow 1\), respectively, is studied. Finally, for \(H\rightarrow 1/2\), the suitably renormalized process converges to a non-trivial limit with non-vanishing persistence exponent, contrary to the fact that \(M^{1/2}\) vanishes.