Hitting probability for anomalous diffusion processes

Abstract

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 (describing, for instance, the polymer translocation through a nanopore) we show that $Q(x,L)=Q(x/L)$ and the scaling function $Q(z)\sim z^{\varphi }$ as $z\to 0$ with $\varphi =\theta /H$ where $H$ and $\theta$ are respectively the Hurst exponent and the persistence exponent of the process. 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 supports for our analytical results.

5 pages, 4 figures

References in corpus (3)

• Fractional Laplacian in Bounded Domains
• Anomalous Dynamics of Unbiased Polymer Translocation through a Narrow Pore
• Asymptotic behavior of self-affine processes in semi-infinite domains

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