Diffusion approximation for the advection of particles in a strongly turbulent random environment

Abstract

In this paper we prove several theorems concerning the motion of a particle in a random environment. The trajectory of a particle is the solution of the differential equation $dx(t)/dt=V(x(t))$, where $V(x) = v + \varepsilon^{1-\alpha} F(x), \; 0\leq\alpha\leq 1,$ $v$ is a constant vector, $F$ is a mean-zero fluctuation field and $\varepsilon^{1-\alpha}$ is a parameter measuring the size of the fluctuations. We show that both in case of a motion of a single particle and of a particle system considered in the macroscopic coordinate system moving along with velocity $v$ [i.e., $x \sim (x-vt)/\varepsilon^{\alpha}, \; t\sim t/\varepsilon^2$] the diffusion approximation holds provided that $F$ is divergence free. Moreover we show how to renormalize trajectories to obtain a similar result for non-divergence-free fields. These results generalize theorems due to Khasminskii and to Kesten and Papanicolaou.