Stochastic nonlinear differential equations. I

Abstract

A solution method is developed for nonlinear differential equations having the following two properties. Their coefficients are stochastic through their dependence on a Markov process. The magnitude of the fluctuations, multiplied with their auto-correlation time, is a small quantity. Under these conditions, the solution is also approximately a Markov process. Its probability distribution obeys a master equation, whose kernel is found as an expansion in that small quantity. The general formula is derived. Applications will be given in the second part of this work.