Continuity in expectation of odd random attractors for stochastic Kuramoto-Sivashinsky equations

We consider the stochastic non-autonomous Kuramoto-Sivashinsky equation with multiplicative white noise and colored coefficients. Due to anti-dissipative term in equation, we restrict state space on Lebesgue of odd functions then obtain a pullback random attractor $ \mathcal{A} space, where special bridge function plays an crucial role priori estimate. Moreover, prove continuity expectation for set-valued mapping (t ,s) \to \mathcal{A}(t,\theta_s\omega ) respect Hausdorff metric establish residual dense pathwise sense. In application, verify four conditions including joint dynamical system, union closedness tempered universe, local compactness boundedness attractor.