Geodesic
Ergodicity for a stochastic geodesic equation in the tangent bundle of the 2D sphere
Czechoslovak Mathematical Journal, Tome 65 (2015) no. 3, pp. 617-657.

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We study ergodic properties of stochastic geometric wave equations on a particular model with the target being the 2D sphere while considering only solutions which are independent of the space variable. This simplification leads to a degenerate stochastic equation in the tangent bundle of the 2D sphere. Studying this equation, we prove existence and non-uniqueness of invariant probability measures for the original problem and obtain also results on attractivity towards an invariant measure. We also present a structure-preserving numerical scheme to approximate solutions and provide computational experiments to motivate and illustrate the theoretical results.
DOI : 10.1007/s10587-015-0200-7
Classification : 37A25, 58J65, 60H10, 60H15, 60H35, 60J60, 65C20, 65C30
Keywords: geometric stochastic wave equation; stochastic geodesic equation; ergodicity; attractivity; invariant measure; numerical approximation 
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     title = {Ergodicity for a stochastic geodesic equation in the tangent bundle of the {2D} sphere},
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Baňas, Ľubomír; Brzeźniak, Zdzisław; Neklyudov, Mikhail; Ondreját, Martin; Prohl, Andreas. Ergodicity for a stochastic geodesic equation in the tangent bundle of the 2D sphere. Czechoslovak Mathematical Journal, Tome 65 (2015) no. 3, pp. 617-657. doi : 10.1007/s10587-015-0200-7. http://geodesic.mathdoc.fr/articles/10.1007/s10587-015-0200-7/

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