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Ergodicity for a stochastic geodesic equation in the tangent bundle of the 2D sphere. (English) Zbl 1363.58015
Consider a stochastic wave equation, the solution \(u(t,x)\) of which takes its values in the sphere \(S^2\subset \mathbb R^3\), and consider only solutions \(u(t)\) which do not depend on the space variable \(x\). Then the wave equation becomes a Stratonovich equation \[ du'=-|u'|^2dt+(u\times u')\circ dW,\quad|u|=1,\quad u(0)\perp u'(0) \] for \(u(t)\) and its derivative \(u'(t)\). The solution \((u,u')\) is therefore in the tangent bundle \(TS^2\).
The authors verify the Feller property for this diffusion process, and the existence of many invariant measures. Actually, denoting by \(\lambda_r\) the normalized surface measure on \(M_r=\{(u,v)\in TS^2;\;|v|=r\}\), they prove that \(\lambda_r\) is the only invariant probability measure on \(M_r\) for \(r>0\); for \(r=0\), the Dirac masses are the invariant probability measures. Moreover, all these measures are exactly the ergodic probability measures.
A numerical analysis is also worked out, and results of simulations are given.

MSC:
58J65 Diffusion processes and stochastic analysis on manifolds
60J60 Diffusion processes
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
37A25 Ergodicity, mixing, rates of mixing
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
65C30 Numerical solutions to stochastic differential and integral equations
65C20 Probabilistic models, generic numerical methods in probability and statistics
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
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