NEW PERSPECTIVE IN THE THEORY OF 2.ND-ORDER STOCHASTIC-PROCESSES

Starting from Langevin equations, we derive Fokker-Planck-like equatio ns (FPLEs) for the joint distribution of displacements and velocities, p(x, v, t), for a particle in a Gaussian random force field, firstly for the inertial process (i.e. in the absence of a frictional force) w ith a time correlated force, and secondly, for the Brownian motion wit h a white-noise force. From two different forms of the Langevin equati on as coupled or decoupled first-order equations, we obtain two differ ent forms of FPLEs for each one of these processes. In the inertial ca se one of the FPLEs reduces to an equation studied earlier by the auth or, while the other coincides with the equation obtained recently by D rory from an involved time discretization. In the Brownian motion case one of the FPLEs coincides with the free particle Kramers equation ob tained from the Fokker-Planck formalism for Markov processes. For each one of these processes the exactly determined initial value solutions of the two FPLEs are found to coincide. It follows, in particular, th at the Markovian character of p(x, v, t) for the Brownian motion is re spected, regardless of which FPLE is used for defining it. Furthermore , for each process the two FPLEs lead to the same diffusion-like equat ion for the marginal distribution of displacements. The latter have be en used elsewhere for studying first passage times, as well as surviva l probabilities in the presence of traps.