Continuous dependence on initial data for solutions of nonlinear stochastic evolution equations

We consider stochastic evolution equations in the framework of white noise analysis. Contraction operators on inductive limits of Banach spares arise naturally in this context and we first extend Banach's fixed point theorem to this type of spaces. In order to apply the fixed point theorem to evolution equations, we construct a topological isomorphism between spaces of generalized random fields and the corresponding spaces of U-functionals. As an application we show that the solutions of some nonlinear stochastic heat equations depend continuously on their initial data. This method also applies to stochastic Volterra equations, stochastic reaction-diffusion equations and to anticipating stochastic differential equations.