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

TL;DR

This paper analyzes one-dimensional stochastic energy balance climate models with Legendre weighted diffusion, transforming them into pathwise random PDEs to study solution stability and long-term behavior.

Contribution

It introduces a method to convert stochastic PDEs with Wiener process forcing into deterministic-like equations depending on random parameters, enabling new stability and asymptotic analyses.

Findings

Solutions are stable as Wiener process noise diminishes.

Solutions exhibit specific asymptotic behavior over large time scales.

The approach simplifies the analysis of complex stochastic climate models.

Abstract

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.