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Stochastic Landau-Lifshitz-Gilbert equation with anisotropy energy driven by pure jump noise. (English) Zbl 1442.35566

Summary: In this work we study a stochastic three-dimensional Landau-Lifshitz-Gilbert equation with non-zero anisotropy energy, which is drive by pure jump noise. We show existence of weak martingale solutions taking values in a two-dimensional sphere \(\mathbb S^2\). The construction of the solution is based on the classical Faedo-Galerkin approximation, the compactness method and the Jakubowski version of the Skorokhod Theorem for nonmetric spaces.

MSC:

35R60 PDEs with randomness, stochastic partial differential equations
35Q60 PDEs in connection with optics and electromagnetic theory
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[1] Peszat, S.; Zabczyk, J., (Stochastic Partial Differential Equations with Lévy Noise: An Evolution Equation Approach. Stochastic Partial Differential Equations with Lévy Noise: An Evolution Equation Approach, Encyclopedia of Mathematics an (2007), Cambridge University Press) · Zbl 1205.60122
[2] Néel, L., Bases d’une nouvelle théorie générale du champ coercitif, Ann. Univ. Grenoble, 22, 299-343 (1946)
[3] Brown, W. F., Thermal fluctuations of a single-domain particle, Phys. Rev., 130, 5, 1677-1686 (1963)
[4] Kamppeter, T.; Mertens, F. G., Stochastic vortex dynamics in two-dimensional easy-plane ferromagnets: Multiplicative versus additive noise, Phys. Rev. B, 59, 17, 11349-11357 (1999)
[5] Brzeźniak, Z.; Goldys, B.; Jegaraj, T., Weak solutions of a stochastic Landau-Lifshitz-Gilbert equation, Appl. Math. Res. Express, 2013, 1-33 (2013) · Zbl 1272.60041
[6] Brzeźniak, Z.; Goldys, B.; Jegaraj, T., Large deviations and transitions between equilibria for stochastic Landau-Lifshitz-Gilbert equation, Arch. Ration. Mech. Anal., 226, 497-558 (2017) · Zbl 1373.35294
[7] Z. Brzeźniak, U. Manna, Weak Solutions of a Stochastic Landau-Lifshitz-Gilbert Equation Driven by Pure Jump Noise, submitted for publication.; Z. Brzeźniak, U. Manna, Weak Solutions of a Stochastic Landau-Lifshitz-Gilbert Equation Driven by Pure Jump Noise, submitted for publication.
[8] Blundell, S., Magnetism in Condensed Matter (2001), Oxford University Press
[9] Mayergoyz, I.; Bertotti, G.; Serpico, C., Magnetization dynamics driven by a jump-noise process, Phys. Rev. B, 83, 020402(R) (2011)
[10] Mayergoyz, I.; Bertotti, G.; Serpico, C., Landau-Lifshitz magnetization dynamics driven by a random jump-noise process, J. Appl. Phys., 109, 07D312 (2011)
[11] Fruchart, O.; Thiaville, A., Magnetism in reduced dimensions, C. R. Phys., 6, 9, 921-933 (2005)
[12] Marcus, S. L., Modelling and approximations of stochastic differential equations driven by semimartingales, Stochastics, 4, 223-245 (1981) · Zbl 0456.60064
[13] Applebaum, D., Lévy Processes and Stochastic Calculus (2009), Cambridge University Press · Zbl 1200.60001
[14] Kunita, H., Stochastic differential equations based on lévy processes and stochastic flows of diffeomorphisms, (Real and Stochastic Analysis (2004), Trends Math.: Trends Math. Boston, MA), 305-373 · Zbl 1082.60052
[15] I. Chevyrev, P.K. Friz, Canonical RDEs and general semimartingales as rough paths, preprint, arXiv:1704.08053; I. Chevyrev, P.K. Friz, Canonical RDEs and general semimartingales as rough paths, preprint, arXiv:1704.08053
[16] Brzeźniak, Z.; Li, L., Weak solutions of the stochastic Landau-Lifshitz-Gilbert equations with nonzero anisotrophy energy, Appl. Math. Res. Express (2016), available online http://dx.doi.org/10.1093/amrx/abw003 · Zbl 1398.35304
[17] Ikeda, N.; Watanabe, S., Stochastic Differential Equations and Diffusion Processes (1981), North-Holland Mathematical Library. North-Holland Publishing Company · Zbl 0495.60005
[18] Z. Brzeźniak, F. Hornung, L. Weis, Martingale solutions for the stochastic nonlinear Schrödinger equation in the energy space, preprint, arXiv:1707.05610; Z. Brzeźniak, F. Hornung, L. Weis, Martingale solutions for the stochastic nonlinear Schrödinger equation in the energy space, preprint, arXiv:1707.05610
[19] Brzeźniak, Z.; Hausenblas, E.; Razafimandimby, P., Stochastic Reaction-diffusion Equations Driven by Jump Processes, Potential Anal., 1-71 (2017)
[20] Gyöngy, I.; Krylov, N. V., On stochastics equations with respect to semimartingales. II. Itô formula in Banach spaces, Stochastics, 6, 3-4, 153-173 (1981/82) · Zbl 0481.60060
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