Ergodicity for a stochastic geodesic equation in the tangent bundle of the 2D sphere

被引：0
作者：
Ľubomír Baňas
Zdzisłlaw Brzeźniak
Mikhail Neklyudov
Martin Ondreját
Andreas Prohl
机构：
[1] Universität Bielefeld,Fakultät für Mathematik
[2] The University of York,Department of Mathematics
[3] University of Pisa,Department of Mathematics
[4] The Institute of Information Theory and Automation of the Czech Academy of Sciences,Mathematisches Institut
[5] Universität Tübingen,undefined
来源：
Czechoslovak Mathematical Journal | 2015年 / 65卷
关键词：
geometric stochastic wave equation; stochastic geodesic equation; ergodicity; attractivity; invariant measure; numerical approximation; 58J65; 60H10; 60H35; 65C30; 60J60; 65C20; 37A25; 60H15;
D O I：
暂无
中图分类号：
学科分类号：
摘要：
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.
引用
收藏
页码：617 / 657
页数：40
相关论文
共 62 条
  • [1] Arnold L.(1987)On unique ergodicity for degenerate diffusions Stochastics 21 41-61
  • [2] Kliemann W.(2014)A convergent finite-element-based discretization of the stochastic Landau-Lifshitz-Gilbert equation IMA J. Numer. Anal. 34 502-549
  • [3] Baňas Z.(2010)Finite element approximations of harmonic map heat flows and wave maps into spheres of nonconstant radii Numer. Math. 115 395-432
  • [4] Brzeźniak M.(2009)Convergent discretization of heat and wave map flows to spheres using approximate discrete Lagrange multipliers Math. Comput. 78 1269-1292
  • [5] Neklyudov A.(1993)Hölder norms and the support theorem for diffusions C. R. Acad. Sci., Paris, Sér. I 316 283-286
  • [6] Prohl A.(1994)Hölder norms and the support theorem for diffusions Ann. Inst. Henri Poincaré, Probab. Stat. 30 415-436
  • [7] Baňas R.(2013)Stochastic geometric wave equations with values in compact Riemannian homogeneous spaces Ann. Probab. 41 1938-1977
  • [8] Prohl S.(2011)Weak solutions to stochastic wave equations with values in Riemannian manifolds Commun. Partial Differ. Equations 36 1624-1653
  • [9] Schätzle C.(2007)Strong solutions to stochastic wave equations with values in Riemannian manifolds J. Funct. Anal. 253 449-481
  • [10] Bartels A.(1972)On barrier problems for the vibrating string Z. Wahrscheinlichkeitstheor. Verw. Geb. 22 13-24
1234567