Ergodicity of stochastic Burgers equation in unbounded domains with space-time white noise

In this paper, we investigate the stochastic Burgers equation with multiplicative noise defined on the entire real line. The equation can be expressed as follows: ∂ u∂ t=∂^2 u∂ x^2-ku +12∂ u^2∂ x+σ(u)∂^2 W∂ t∂ x. The noise is characterized as space-time white noise. We establish that the solution remains uniformly bounded in time. Furthermore, by employing the Krylov-Bogolioubov theorem, we establish the existence of invariant measures for this equation. Additionally, we demonstrate the uniqueness of invariant measures by utilizing the strong Feller and irreducible properties.