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

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Abstract

In this work we study a stochastic three-dimensional Landau–Lifshitz–Gilbert equation perturbed by pure jump noise in the Marcus canonical form. We show the existence of a weak martingale solution taking values in a two-dimensional sphere \({\mathbb{S}^2}\) and discuss certain regularity results. The construction of a solution is based on the classical Faedo–Galerkin approximation, the compactness methods and the Jakubowski version of the Skorokhod Theorem for nonmetric spaces.

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Acknowledgements

We would like to thank David Applebaum for the suggestion of using the Marcus canonical equations. We would also like to thank Ben Goldys for drawing our attention to Barkhausen effect. We dedicate this work to memory of our colleague, collaborator and friend, Terence Jegaraj, who recently tragically passed away and whose dedication and enthusiasm to the subject of stochastic ferromagnetism we will always remember.

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Authors and Affiliations

  1. Department of Mathematics, The University of York, Heslington, York, YO10 5DD, UK

    Zdzisław Brzeźniak

  2. School of Mathematics, Indian Institute of Science Education and Research Thiruvananthapuram, Trivandrum, 695551, India

    Utpal Manna

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  1. Zdzisław Brzeźniak
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  2. Utpal Manna
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Correspondence to Utpal Manna.

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Supported by a Royal Society Grant (No. IE140328) “Stochastic Landau–Lifshitz–Gilbert equation with Lévy noise and ferromagnetism”.

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Brzeźniak, Z., Manna, U. Weak Solutions of a Stochastic Landau–Lifshitz–Gilbert Equation Driven by Pure Jump Noise. Commun. Math. Phys. 371, 1071–1129 (2019). https://doi.org/10.1007/s00220-019-03359-x

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