  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 $\mean{\phi(x)}=0$ and $\mean{\phi(x)\phi(x')}=g
\delta(x-x')$. 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 $\smean{P(x,t|x,0)}\sim{}t^{-\sqrt{2\rho/g}}$ (to be
compared with the usual Lifshits exponential decay $\exp{-(\rho^2t)^{1/3}}$ in
the absence of the random force field). The localisation properties of the
underlying quantum Hamiltonian are discussed as well.
