Persistence Probabilities of a Smooth Self-Similar Anomalous Diffusion Process

Abstract

AbstractWe 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 ↓ 0 and $$H\uparrow 1$$ H ↑ 1 , respectively, is studied. Finally, for $$H\rightarrow 1/2$$ H → 1 / 2 , the suitably renormalized process converges to a non-trivial limit with non-vanishing persistence exponent, contrary to the fact that $$M^{1/2}$$ M 1 / 2 vanishes.