Test-Particle Propagator and Its Application to the Decay of Plasma Wave Echoes

The conditional probability density W (x, v, t | x0, v0, t0) describing the motion of a test particle satisfies a kinetic-type equation, which was first systematically discussed by Rostoker for stable plasmas and was generalized by Rogister and Oberman to describe weakly tubulent plasmas. A systematic expansion for the solution of this equation is presented which is valid for short times on the kinetic time scale. However, these times are of sufficient length to describe the decay of plasma wave echoes due to collisions or weak turbulence.