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In this paper, we prove the existence and uniqueness of random attractors for the FitzHugh-Nagumo system driven by colored noise with a nonlinear diffusion term. We demonstrate that the colored noise is much easier to deal with than the white noise for studying the pathwise dynamics of stochastic systems. In addition, we show the attractors of the random FitzHugh-Nagumo system driven by a linear multiplicative colored noise converge to that of the corresponding stochastic system driven by a linear multiplicative white noise.

In this paper, we study the limiting behavior of dynamics for stochastic reaction-diffusion equations driven by an additive noise and a deterministic non-autonomous forcing on an (*n*+1)-dimensional thin region when it collapses into an *n*-dimensional region. We first established the existence of attractors and their properties for these equations on (*n*+1)-dimensional thin domains. We then show that these attractors converge to the random attractor of the limit equation under the usual semi-distance as the thinness goes to zero.

This paper deals with the multivalued non-autonomous random dynamical system generated by the non-autonomous stochastic wave equations on unbounded domains, which has a non-Lipschitz nonlinearity with critical exponent in the three dimensional case. We introduce the concept of weak upper semicontinuity of multivalued functions and use such continuity to prove the measurability of multivalued functions from a metric space to a separable Banach space. By this approach, we show the measurability of pullback attractors of the multivalued random dynamical system of the wave equations regardless of the completeness of the underlying probability space. The asymptotic compactness of solutions is proved by the method of energy equations, and the difficulty caused by the non-compactness of Sobolev embeddings on $\mathbb{R}^n$ is overcome by the uniform estimates on the tails of solutions.

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