This is a continuation of the authors’ paper [Stochastic Processes Appl. 72, No. 2, 187-204 (1997; Zbl 0943.60048)]. Let be a homogeneous Wiener process valued in with a positive symmetric tempered measure as a space correlation . The stochastic wave equation with initial conditions and the heat equation with initial condition are considered on with Lipschitz and . Comparing the paper cited above, in the present paper is extended to a generalised function and the Fourier transform of which is not necessarily absolutely continuous with respect to the Lebegue measure . Let Condition (H) be: there exists such that . Condition (G) is defined by Condition (H) together with
The main results are: (1) For , (G) ensures the existence and uniqueness of the stochastic wave equation on and when (H) is true and and there exist solutions of the stochastic wave equation of some and for , then (G) holds conversely. (2) The same conclusions as in (1) hold for the heat equation for any dimension .