Definitions of Path Integrals for General Diffusion Processes

Abstract

Stochastic processes with Gaussian white noise can be described by the Langevin equation [1] or by the stochastic equivalent [1,2] Fokker-Planck equation (11), which is a non-linear multidimensional generalization of a diffusion equation. The solution of this equation is the probability distribution f(x,t) to find a certain value xfor the stochastic variable at a given time t. In this description nothing can be said about the path x(t) which leads to this value. ONSAGER and MACHLUP [3] were the first dealing with the probability for paths of linear diffusion processes. There the solution f(x,t) of the Fokker-Planck equation is expressed by an integral (1) over all possible paths. The generalization of such path integrals on general non-linear, multidimensional Gaussian processes has become of increasing interest in the last years [4–21].