Stochastic diffusive energy balance climate model with a multiplicative noise modeling the Solar variability

We prove the existence, uniqueness, and comparison of solutions for a nonlinear stochastic parabolic partial differential equation that includes the Solar variability in terms of a multiplicative Wiener cylindrical noise in the term of the absorbed radiative energy in a simplified diffusive one-dimensional Energy Balance Model. We introduce a hybrid co-albedo nonlinear term, which has the advantages of both the Sellers model, as it is a continuous function, and the Budyko model, as it has an infinite derivative at u=-10C (the temperature at which ice is white), allowing the location of the polar ice caps to be easily detected. We show that, despite the lack of differentiability of this function, the method of successive approximations can be satisfactorily applied.