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 and is 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 $H\to 1\mathrm{/}2$, the suitably renormalized process converges to a non-trivial limit with non-vanishing persistence exponent, contrary to the fact that $M^{1\mathrm{/}2}$ vanishes.