Ergodicity for a stochastic geodesic equation in the tangent bundle of   the 2D sphere
  We study ergodic properties of stochastic geometric wave equations on a particular model with the target being the 2D sphere while considering the space variable-independent solutions only. 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 we obtain also results on attractivity towards an invariant measure. We also present a suitable numerical scheme for approximating the solutions subject to a sphere constraint. 
L'ubom\'ir Banas, Zdzis{\l}aw Brze\'zniak, Misha Neklyudov, Martin   Ondrej\'at and Andreas Prohl
math.PR