A White Noise Approach to a Class of Non-linear Stochastic Heat Equations

Abstract

Cauchy problems for a class of non-linear stochastic evolution equations are studied. They are formulated as integral equations for generalized random fields. By methods of white noise analysis (S-transformation, characterization theorem, etc.) these problems are reduced to fixed point problems in appropriatly constructed Banach spaces. This technique provides a systematic treatment of existence and uniqueness questions for a variety of equations. The method is applied to non-linear heat equations, non-linear Volterra equations and non-linear ordinary differential equations, which may all be anticipating. For each case examples are given, such as the stochastic Burgers equation and stochastic reaction–diffusion equations.