Stochastic Acceleration in an Inhomogenous Time Random Force Field

This paper studies the asymptotic behavior of a particle with large initial velocity and subject to a force field which is randomly time dependent and inhomogenous in space. We analyze the diffusive limit ǫ → 0 of the position–velocity pair under the appropriate space-time rescaling: ( ǫY (s/ǫ), ǫẎ (s/ǫ) ) . Two alternative approaches are proposed. The first one is based on hydrodynamic limits and homogenization techniques for the underlying kinetic equation; the second one on homogenization of the random distribution of trajectories. Time randomness is embodied into an underlying Markov process. Space inhomogeneity is modeled by a periodic structure in the first approach, and by a random field in the second one. In the first case, the analysis relies on the dissipation properties of the Markov process, whereas in the second one, the mixing properties of the random field are used. We point out more analogies and differences of the two obtained results.