Random-search efficiency in a bounded interval with spatially heterogeneous diffusion coefficient

Abstract

We consider random walkers searching for a target in a bounded 1D heterogeneous environment, in the interval , where diffusion is described by a space-dependent diffusion coefficient D(x). Boundary conditions are absorbing at the position of the target (set at x = 0) and reflecting at the border x = L. We calculate and compare the estimates of efficiency and . In the Stratonovich framework of the multiplicative random process, both measures are analytically calculated for arbitrary D(x). For other interpretations of the stochastic integrals (e.g. Itô and anti-Itô), we obtain general results for ɛ 2, while ɛ 1 is obtained for particular forms of D(x). The impact of the diffusivity profile on these measures of efficiency is discussed. Symmetries and peculiar properties arise when the search starts at the border (). In particular, heterogeneity spoils the efficiency of the search within the Stratonovich framework, while for other interpretations the searcher can perform better for certain heterogeneous diffusivity profiles.