In this paper we investigate the long-time behavior of stochastic reaction-diffusion equations of the type du = (Au + f (u))dt + σ(u)dW (t), where A is an elliptic operator, f and σ are nonlinear maps and W is an infinite dimensional nuclear Wiener process. The emphasis is on unbounded domains. Under the assumption that the nonlinear function f possesses certain dissipative properties, this equation is known to have a solution with an expectation value which is uniformly bounded in time. Together with some compactness property, the existence of such a solution implies the existence of an invariant measure which is an important step in establishing the ergodic behavior of the underlying physical system. In this paper we expand the existing classes of nonlinear functions f and σ and elliptic operators A for which the invariant measure exists, in particular, in unbounded domains. We also show the uniqueness of the invariant measure for an equation defined on the upper half space if A is the Shrödinger-type operator A = 1 ρ (divρ∇u) where ρ = e −|x| 2 is the Gaussian weight.

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The bidomain equations driven by stochastic forces and subject to nonlinearities of FitzHugh-Nagumo or Allen-Cahn type are considered for the first time. It is shown that this set of equations admits a global weak solution as well as a stationary solution, which generates a uniquely determined invariant measure.

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