Stochastic partial differential equations in continuum physics—on the foundations of the stochastic interpolation method for ITO's type equations

This paper deals with the mathematical analysis of mathematical models in continuum physics which are described by partial differential equations, in one space dimension, with additional random noise. The first part of the paper provides the framework for the mathematical modelling of a large class of physical systems. Then, the so-called ‘stochastic interpolation method’ is applied to obtain quantitative solutions by transforming the original equation into a system of stochastic ordinary differential equations, and an analysis of the error estimates is developed in order to provide a bound estimate of the distance between the solution of the true original problem and the one obtained by solving the system of ordinary differential equations. As an application, a linear random model for the heat equation is finally considered following the analysis proposed here.