A probabilityrepresentation of solutions of wave equations, and the function of Greene

Abstract

A probability representation of the solutions of the Cauchy problem for the equations of oscillation of a string, vibration of a membrane and oscillatory processes in continuous media (sound waves) is constructed. The main achievement is the use of a simple technique of uniformly distributed random variables. Solutions are presented in the form of mathematical expectations of functions from uniformly distributed random variables. With the help of the explicit Green function for the oscillation equation of a bounded string, probability representations of solutions of boundary value problems are obtained.