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We consider a class of one-dimensional nonlinear stochastic parabolicproblems associated with Sellers and Budyko diffusive energy balance climatemodels with a Legendre weighted diffusion and an additive cylindrical Wienerprocesses forcing. Our results use in an important way that, under suitableassumptions on the Wiener processes, a suitable change of variables leads theproblem 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 theasymptotic behaviour of solutions for large time.

Stochastic energy balance climate models with Legendre weighted diffusion and a cylindrical Wiener process forcing