Abstract
Let be a random process starting at with absorbing boundary conditions at both ends of the interval. Denote by the probability to first exit at the upper boundary. For Brownian motion, , which is equivalent to . For fractional Brownian motion with Hurst exponent , we establish that , where . The function is analytic and well approximated by its Taylor expansion , where is the Catalan constant. A similar result holds for moments of the exit time starting at . We then consider the span of , i.e., the size of the (compact) domain visited up to time . For Brownian motion, we derive an analytic expression for the probability that the span reaches 1 for the first time and then generalize it to fractional Brownian motion. Using large-scale numerical simulations with system sizes up to and a broad range of , we confirm our analytic results. There are important finite-discretization corrections which we quantify. They are most severe for small , necessitating going to the large systems mentioned above.
12 More- Received 2 August 2018
DOI:https://doi.org/10.1103/PhysRevE.99.032106
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