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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 $ρ$ of absorbers is studied. The force field is taken as a Gaussian white noise with $\mean{ϕ(x)}=0$ and $\mean{ϕ(x)ϕ(x')}=g δ(x-x')$. 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 $\smean{P(x,t|x,0)}\sim{}t^{-\sqrt{2ρ/g}}$ (to be compared with the usual Lifshits exponential decay $\exp{-(ρ^2t)^{1/3}}$ in the absence of the random force field). The localisation properties of the underlying quantum Hamiltonian are discussed as well.