This paper is inspired by artides of Chow [Ch] and Nualart-Zakai [NZ], in which certain (linear) stochastic heat equations are treated within the framework of generalized Brownian functionals. In particular, in [Ch] a stochastic heat equation with a gradient-coupled noise (namely, the noise associated with a Wiener integral with respect to Brownian motion) is proposed as a model for the transport of a substance in a turbulent medium. The present artide extends the work in [DP,P2] (and also [CLP]) in several ways: most notably, to the non-linear case and to very general noise terms which may depend Ort space and time. [...] Our article is organized as follows. Section 2 provides some mathematical background from white noise analysis, a precise formulation of the Cauchy problem and the notion of solution to be used. In Section 3 the necessary Banach spaces of U-functionals are introduced. In Sections 4 and 5 we prove existence and uniqueness of solutions of the Cauchy problem under certain global (Section 4), resp. local (Section 5) Lipschitz conditions on the coefficients F and G. Moreover, a number of examples are considered there. Finally, as a byproduct of our method we treat in Section 6 non-linear anticipating stochastic differential equations and stochastic Volterra equations, and give again several examples.
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