Comparison methods for a class of function valued stochastic partial differential equations

Abstract

A comparison theorem is derived for a class of function valued stochastic partial differential equations (SPDE's) with Lipschitz coefficients driven by cylindrical and regular Hilbert space valued Brownian motions. Moreover, we obtain necessary and sufficient conditions for the positivity of the mild solutions of the SPDE's where the sufficiency follows from the comparison theorem. Thereby it is, e.g., possible to identify a class of SPDE's, which can serve as stochastic space-time models for the density of particles. As a consequence we can construct unique mild solutions of SPDE's on the cone of positive functions with non-Lipschitz drift parts including the case of arbitrary polynomials R(x) with R(O)≧O and leading negative coefficient. © 1992 Springer-Verlag.