Ergodicity for a Stochastic Geodesic Equation in the Tangent Bundle of the 2D Sphere

Lubomír  Banas; Zdzislaw Brzezniak; Mikhail Neklyudov; Martin Ondreját; Andreas Prohl

doi:10.1007/s10587-015-0200-7

Ergodicity for a Stochastic Geodesic Equation in the Tangent Bundle of the 2D Sphere

Lubomír Banas, Zdzislaw Brzezniak, Mikhail Neklyudov, Martin Ondreját, Andreas Prohl

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

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 motivatea and illustrate the theoretical results.

Original language	English
Pages (from-to)	617-657
Number of pages	41
Journal	Czechoslovak Mathematical Journal
Volume	65
Issue number	3
DOIs	https://doi.org/10.1007/s10587-015-0200-7
Publication status	Published - 2015

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10.1007/s10587-015-0200-7

Cite this

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abstract = "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 motivatea and illustrate the theoretical results.",

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AU - Neklyudov, Mikhail

AU - Ondreját, Martin

AU - Prohl, Andreas

PY - 2015

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AB - 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 motivatea and illustrate the theoretical results.

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