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We establish 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 is achieved by employing the classical Krylov-Bogoliubov
theorem. In contrast to other equations, verifying the hypothesis of the
Krylov-Bogoliubov theorem is not a standard procedure. We employ rough paths
theory to show that the semigroup associated to the equation has the Feller
property in $H^1(D,\mathbb{R}^3)\cap L^2(D,\mathbb{S}^2)$. It does not seem
possible to achieve the same conclusion by the classical Stratonovich calculus.
On the other hand, we employ the classical Stratonovich calculus to prove the
tightness hypothesis. The Krein-Milman theorem implies 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 absence of the anisotropic
energy and for a spatially constant noise, we are able to provide a path-wise
long time behaviour result: in particular, every solution synchronises with a
spherical Brownian motion and it is recurrent for large times

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