Persistence Probabilities of a Smooth Self-Similar Anomalous Diffusion Process

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↓ 0 and H↑ 1 , respectively, is studied. Finally, for H→ 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.