Fractional Brownian motion in presence of two fixed adsorbing boundaries

TL;DR

This paper analyzes the long-time survival probability of fractional Brownian motion within fixed boundaries, revealing how it decays over time for different Hurst indices and trap densities, with implications for subdiffusive and superdiffusive regimes.

Contribution

It provides a detailed asymptotic analysis of survival probabilities for fractional Brownian motion with fixed boundaries, extending understanding across all Hurst indices.

Findings

Survival probability decays as  t^{2H}/L^2 in log scale.

Decay rate varies with Hurst index, indicating different diffusion regimes.

Survival probability in trap systems follows  n^{2/3} t^{2H/3} asymptotics.

Abstract

We study the long-time asymptotics of the probability P_t that the Riemann-Liouville fractional Brownian motion with Hurst index H does not escape from a fixed interval [-L,L] up to time t. We show that for any H \in ]0,1], for both subdiffusion and superdiffusion regimes, this probability obeys \ln(P_t) \sim - t^{2 H}/L^2, i.e. may decay slower than exponential (subdiffusion) or faster than exponential (superdiffusion). This implies that survival probability S_t of particles undergoing fractional Brownian motion in a one-dimensional system with randomly placed traps follows \ln(S_t) \sim - n^{2/3} t^{2H/3} as t \to \infty, where n is the mean density of traps.