Persistence Probabilities of a Smooth Self-Similar Anomalous Diffusion Process

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