Wong–Zakai Approximation for Landau–Lifshitz–Gilbert Equation Driven by Geometric Rough Paths

We adapt Lyon’s rough path theory to study Landau–Lifshitz–Gilbert equations (LLGEs) driven by geometric rough paths in one dimension, with non-zero exchange energy only. We convert the LLGEs to a fully nonlinear time-dependent partial differential equation without rough paths term by a suitable transformation. Our point of interest is the regular approximation of the geometric rough path. We investigate the limit equation, the form of the correction term, and its convergence rate in controlled rough path spaces. The key ingredients for constructing the solution and its corresponding convergence results are the Doss–Sussmann transformation, maximal regularity property, and the geometric rough path theory.

Transport and continuity equations with (very) rough noise

Existence and uniqueness for rough flows, transport and continuity equations driven by general geometric rough paths are established.

Random attractors for dissipative systems with rough noises

<p style='text-indent:20px;'>We provide an analytic approach to study the asymptotic dynamics of rough differential equations, with the driving noises of Hölder continuity. Such systems can be solved with Lyons' theory of rough paths, in particular the rough integrals are understood in the Gubinelli sense for controlled rough paths. Using the framework of random dynamical systems and random attractors, we prove the existence and upper semi-continuity of the global pullback attractor for dissipative systems perturbed by bounded noises. Moreover, if the unperturbed system is strictly dissipative then the random attractor is a singleton for sufficiently small noise intensity.</p>