Persistence Probabilities of a Smooth Self-Similar Anomalous Diffusion Process

We consider the persistence probability of a certain fractional Gaussian process MH 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 MH M H exists, is positive and continuous in the Hurst parameter H . Further, the asymptotic behaviour of the persistence exponent for H↓ 0 H ↓ 0 and H↑ 1 H ↑ 1 , respectively, is studied. Finally, for H→ 1/2 H → 1 / 2 , the suitably renormalized process converges to a non-trivial limit with non-vanishing persistence exponent, contrary to the fact that M1/2 M 1 / 2 vanishes.