Weak Solutions of a Stochastic Landau–Lifshitz–Gilbert Equation Driven by Pure Jump Noise

Stochastic Landau–Lifshitz–Gilbert equation with anisotropy energy driven by pure jump noise

“…Hence the equation (4.13) has a unique global strong solution in H n (see [9] and [8] for more details). Moreover, we have the following a-priori estimates for all n ∈ N. Lemma 4.5.…”

“…However, to the best of our knowledge, there is no concrete work on stochastic partial differential equations driven by Lévy noise in the "Marcus" canonical form. Our recent work [9] and the current paper are motivated by this question and we believe similar questions are yet unanswered for many other constrained PDEs (e.g. harmonic map flow, nonlinear Schrödinger equation on a compact Riemannian manifold, nematic liquid crystal model etc.)…”

“…A simple way to incorporate the noise into the deterministic LLGE is to perturb the effective field by a Lévy noise in the so called "Marcus" form. It is worth to mention here that the program to analyze noise induced transitions was initiated by Néel [21] and further developed in [3], [14] and others, and most recently by the first named author, Goldys and Jegaraj in [4] and [5], and by both the authors in [9].…”

“…In a later paper Brzeźniak and Li [8], the result was generalised in the presence of anisotropy energy. In a very recent work [9], we have addressed existence of weak martingale solution of the stochastic LLGE in the absence of anisotropy energy. In this work we extend this result for a more general LLGE.…”

“…In this work we extend this result for a more general LLGE. The presence of anisotropy energy creates additional technical difficulty at various levels as compared to [9], e.g. in order to prove existence of solution we need to consider a special approximation involving projection π n applied to ∇φ, see approximate equation (4.12).…”