An asymptotic analysis of a generalized Langevin equation

The motion of a particle in a nonhomogeneous turbulent flow can be described on two levels: either as a Markov displacement process or as a joint velocity, displacement Markov process. This paper investigates a method of deriving the first process from the second by a direct expansion of the stochastic integral equation. The generalized Langevin equation (defined in the Ito sense), 38dV[small n with long right leg]=-[small n with long right leg]2[alpha][small n with long right leg](Z[small n with long right leg],[small n with long right leg]s)V[small n with long right leg] ds+dW[small n with long right leg], V[small n with long right leg](s0=v0;36dZ[small n with long right leg]=[small n with long right leg]V[small n with long right leg]ds, Z[small n with long right leg](s0)=z0 is often used as a model of the velocity, V[eta], of a particle suspended in a turbulent medium. Here, dW[small n with long right leg] = [small n with long right leg][lambda][small n with long right leg](Z[small n with long right leg], [small n with long right leg]s) d[omega] + [small n with long right leg][beta][small n with long right leg](Z[small n with long right leg], [eta]s) ds and [omega](s) is a Wiener process with zero that as [small n with long right leg]å[infinity], the displacement, Z[small n with long right leg], converges uniformly in mean square to the process Z that satisfies 69dZ=[sigma](Z,s) d[Omega]+[gamma](Z,s) ds, Z(s0)=z0; where [sigma] = [lambda]/[alpha], [gamma] = [beta]/[alpha] - 1-([lambda]2/[alpha]3)[not partial differential][alpha]/[not partial differential]z and [alpha][small n with long right leg], [not partial differential][alpha][small n with long right leg]/[not partial differential]z, [lambda][small n with long right leg] and [beta][small n with long right leg] converge [not partial differential][alpha]/[not partial differential]z, [lambda] and [beta], respectively. That is, the displacement process Z[small n with long right leg] approximates a diffusion as [small n with long right leg] tends to infinity.