Well-posedness and limit behavior of stochastic fractional Boussinesq equation driven by nonlinear noise

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This paper primarily explores the well-posedness and limit behavior of the stochastic fractional Boussinesq equation driven by periodic force and nonlinear noise. Firstly, we establish the existence and uniqueness of the strong solution in probabilistic sense. Subsequently, the existence of weak mean random attractor and evolution systems of measures will be proved. Furthermore, we investigate the asymptotic behavior of the invariant measure as the noise intensity approaches zero. To this end, we utilize the Wentzell-Freidlin large deviation principle by demonstrating the compactness of level sets and the weak convergence between stochastic and controlled equations. Finally, we establish the ergodicity and exponential mixing by employing the Harris-like theorem under periodic force.