One-dimensional diffusion in a semi-infinite Poisson random force
  We consider the one-dimensional diffusion of a particle on a semi-infinite line and in a piecewise linear random potential. We first present a new formalism which yields an analytical expression for the Green function of the Fokker-Planck equation, valid for any deterministic construction of the potential profile. The force is then taken to be an asymmetric dichotomic process. Solving the corresponding energy-dependent stochastic Riccati equation in the space-asymptotic regime, we give an exact probabilistic description of returns to the origin. This method allows for a time-asymptotic characterization of the underlying dynamical phases. When the two values taken by the dichotomic force are of different signs, there occur trapping potential wells with a broad distribution of trapping times and dynamical phases may appear, depending on the mean force. If both values are negative, the time-asymptotic mean value of the probability density at the origin is proportional to the absolute value of the mean force. If they are both positive, traps no more exist and the dynamics is always normal. Problems with a shot-noise force and with a Gaussian white-noise force are solved as appropriate limiting cases. 
Petr Chvosta and Noelle Pottier