We establish the existence of an ergodic invariant measure on \(H^1(D,\mathbb {R}^3)\cap L^2(D,\mathbb {S}^2)\) for the stochastic Landau-Lifschitz-Gilbert equation on a bounded one-dimensional interval D. The conclusion follows from the classical Krylov-Bogoliubov theorem. Unlike for many other equations, verifying the hypotheses of the Krylov-Bogoliubov theorem is not a standard procedure. We use rough paths theory to show that the semigroup associated with the equation has the Feller property in \(H^1(D,\mathbb {R}^3)\cap L^2(D,\mathbb {S}^2)\) . Using only classical Stratonovich calculus does not appear to allow for the same conclusion. On the other hand, we employ the classical Stratonovich calculus to prove the tightness hypothesis. The Krein-Milman theorem implies the existence of an ergodic invariant measure. In case of spatially constant noise, we show that there exists a unique Gibbs invariant measure, and we establish the qualitative behaviour of the unique stationary solution. In the absence of anisotropy and for a spatially constant noise, we can provide a pathwise long-time behaviour result: in particular, every solution is recurrent for large times.
