  We consider a class of one-dimensional nonlinear stochastic parabolic
problems associated with Sellers and Budyko diffusive energy balance climate
models with a Legendre weighted diffusion and an additive cylindrical Wiener
processes forcing. Our results use in an important way that, under suitable
assumptions on the Wiener processes, a suitable change of variables leads the
problem to a pathwise random PDE, hence an essentially "deterministic"
formulation depending on a random parameter. Two applications are also given:
the stability of solutions when the Wiener process converges to zero and the
asymptotic behaviour of solutions for large time.
