Convergence rates and central limit theorem for 3-D stochastic fractional Boussinesq equations with transport noise

Abstract

This work investigates the 3-D stochastic modified fractional Boussinesq equations driven by transport noise on the torus. The existence of weak solutions is initially established through Galerkin approximation and compactness techniques. Subsequently, we demonstrate that the weak solutions of stochastic modified fractional Boussinesq equations converge to the unique solution of a deterministic Boussinesq model with fractional dissipation, subject to an appropriate noise scaling, along with providing the quantitative estimates with explicit convergent rates. Finally, the central limit theorem with an explicit convergence rate is established based on the aforementioned scaling limit. It is noteworthy that the scaling limit result of stochastic modified fractional Boussinesq equations to deterministic Boussinesq models with additional fractional dissipation implies that the transport noise regularizes the modified fractional Boussinesq equations, thus leading to “approximate weak uniqueness”.