  Using a rough path formulation, we investigate existence, uniqueness and
regularity for the stochastic Landau-Lifshitz-Gilbert equation with
Stratonovich noise on the one dimensional torus. As a main result we show the
continuity of the so-called It\^o-Lyons map in the energy spaces
$L^\infty(0,T;H^k)\cap L^2(0,T;H^{k+1})$ for any $k\ge1$. The proof proceeds in
two steps. First, based on an energy estimate in the aforementioned space
together with a compactness argument we prove existence of a unique solution,
implying the continuous dependence in a weaker norm. This is then strengthened
in the second step where the continuity in the optimal norm is established
through an application of the rough Gronwall lemma. Our approach is direct and
does not rely on any transformation formula, which permits to treat
multidimensional noise. As an easy consequence we then deduce a Wong-Zakai type
result, a large deviation principle for the solution and a support theorem.
