Hitting Probability for Anomalous Diffusion Processes

We present the universal features of the hitting probability Q(x, L), the probability that a generic stochastic process starting at x and evolving in a box [0, L] hits the upper boundary L before hitting the lower boundary at 0. For a generic self-affine process, we show that Q(x, L) Q(z = x/L) has a scaling Q(z) similar to z(phi) as z -> 0, where phi = phi/H, H, and theta being the Hurst and persistence exponent of the process, respectively. This result is verified in several exact calculations, including when the process represents the position of a particle diffusing in a disordered potential. We also provide numerical support for our analytical results.