One-dimensional classical diffusion in a random force field with weakly concentrated absorbers

A one-dimensional model of classical diffusion in a random force field with a weak concentration rho of absorbers is studied. The force field is taken as a Gaussian white noise with <phi(x)> = 0 and <phi(x)phi(x')> = g delta(x - x'). Our analysis relies on the relation between the Fokker-Planck operator and a quantum Hamiltonian in which absorption leads to breaking of supersymmetry. Using a Lifshits argument, it is shown that the average return probability is a power law < P(x, t vertical bar x, 0)> similar to t(-)root 2 rho/g (to be compared with the usual Lifshits exponential decay exp - (rho(2)t)(1/3) in the absence of the random force field). The localisation properties of the underlying quantum Hamiltonian are discussed as well. Copyright (C) EPLA, 2009