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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.

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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[1] S. Aida, S. Kusuoka, D. Stroock: On the support of Wiener functionals. Asymptotic Problems in Probability Theory: Wiener Functionals and Asymptotics. Proc. Conf., Sanda and Kyoto, Japan, 1990 (K. D. Elworthy et al., eds.). Pitman Res. Notes Math. Ser. 284, Longman Scientific & Technical, Harlow, Essex; John Wiley & Sons, New York, 1993, pp. 3–34. · Zbl 0790.60047
[2] L. Arnold, W. Kliemann: On unique ergodicity for degenerate diffusions. Stochastics 21 (1987), 41–61. · Zbl 0617.60076
[3] Ľ. Baňas, Z. Brzeźniak, M. Neklyudov, A. Prohl: A convergent finite-element-based discretization of the stochastic Landau-Lifshitz-Gilbert equation. IMA J. Numer. Anal. 34 (2014), 502–549. · Zbl 1298.65012
[4] Ľ. Baňas, Z. Brzeźniak, M. Neklyudov, A. Prohl: Stochastic Ferromagnetism. Analysis and Numerics. De Gruyter Studies in Mathematics 58, De Gruyter, Berlin, 2014. · Zbl 1288.82001
[5] Ľ. Baňas, A. Prohl, R. Schätzle: Finite element approximations of harmonic map heat flows and wave maps into spheres of nonconstant radii. Numer. Math. 115 (2010), 395–432. · Zbl 1203.65174
[6] S. Bartels, C. Lubich, A. Prohl: Convergent discretization of heat and wave map flows to spheres using approximate discrete Lagrange multipliers. Math. Comput. 78 (2009), 1269–1292. · Zbl 1198.65178
[7] G. Ben Arous, M. Grǎdinaru: Hölder norms and the support theorem for diffusions. C. R. Acad. Sci., Paris, Sér. I 316 (1993), 283–286. (In French.) · Zbl 0768.60067
[8] G. Ben Arous, M. Grǎdinaru, M. Ledoux: Hölder norms and the support theorem for diffusions. Ann. Inst. Henri Poincaré, Probab. Stat. 30 (1994), 415–436.
[9] Z. Brzeźniak, M. Ondreját: Stochastic geometric wave equations with values in compact Riemannian homogeneous spaces. Ann. Probab. 41 (2013), 1938–1977. · Zbl 1286.60058
[10] Z. Brzeźniak, M. Ondreját: Weak solutions to stochastic wave equations with values in Riemannian manifolds. Commun. Partial Differ. Equations 36 (2011), 1624–1653. · Zbl 1238.60073
[11] Z. Brzeźniak, M. Ondreját: Strong solutions to stochastic wave equations with values in Riemannian manifolds. J. Funct. Anal. 253 (2007), 449–481. · Zbl 1141.58019
[12] E. M. Cabaña: On barrier problems for the vibrating string. Z. Wahrscheinlichkeitstheor. Verw. Geb. 22 (1972), 13–24.
[13] R. Carmona, D. Nualart: Random nonlinear wave equations: propagation of singularities. Ann. Probab. 16 (1988), 730–751. · Zbl 0643.60045
[14] R. Carmona, D. Nualart: Random nonlinear wave equations: smoothness of the solutions. Probab. Theory Relat. Fields 79 (1988), 469–508. · Zbl 0635.60073
[15] P.-L. Chow: Stochastic wave equations with polynomial nonlinearity. Ann. Appl. Probab. 12 (2002), 361–381. · Zbl 1017.60071
[16] G. Da Prato, J. Zabczyk: Ergodicity for Infinite-Dimensional Systems. London Mathematical Society Lecture Note Series 229, Cambridge Univ. Press, Cambridge, 1996. · Zbl 0849.60052
[17] G. Da Prato, J. Zabczyk: Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and Its Applications 44, Cambridge University Press, Cambridge, 1992.
[18] R. C. Dalang, N. E. Frangos: The stochastic wave equation in two spatial dimensions. Ann. Probab. 26 (1998), 187–212. · Zbl 0938.60046
[19] R. C. Dalang, O. Lévêque: Second-order linear hyperbolic SPDEs driven by isotropic Gaussian noise on a sphere. Ann. Probab. 32 (2004), 1068–1099. · Zbl 1046.60058
[20] P. Diaconis, D. Freedman: On the hit and run process. University of California Berkeley, Statistics Technical Report no. 497, http://stat-reports.lib.berkeley.edu/accessPages/497.html (1997).
[21] J. Ginibre, G. Velo: The Cauchy problem for the O(N), \(\mathbb{C}\)P(N ), and G \(\mathbb{C}\)(N, p) models. Ann. Phys. 142 (1982), 393–415. · Zbl 0512.58018
[22] I. Gyöngy, T. Pröhle: On the approximation of stochastic differential equation and on Stroock-Varadhan’s support theorem. Comput. Math. Appl. 19 (1990), 65–70. · Zbl 0711.60051
[23] L. Hörmander: Hypoelliptic second order differential equations. Acta Math. 119 (1967), 147–171. · Zbl 0156.10701
[24] K. Ichihara, H. Kunita: A classification of the second order degenerate elliptic operators and its probabilistic characterization. Z. Wahrscheinlichkeitstheor. Verw. Geb. 30 (1974), 235–254. · Zbl 0326.60097
[25] K. Ichihara, H. Kunita: Supplements and corrections to the paper: ”A classification of the second order degenerate elliptic operators and its probabilistic characterization”. Z. Wahrscheinlichkeitstheor. Verw. Geb. 39 (1977), 81–84. · Zbl 0382.60069
[26] N. Ikeda, S. Watanabe: Stochastic Differential Equations and Diffusion Processes. North-Holland Mathematical Library 24, North-Holland; Publishing Co. Tokyo: Kodansha, Amsterdam, 1989. · Zbl 0684.60040
[27] A. Karczewska, J. Zabczyk: Stochastic PDE’s with function-valued solutions. Infinite Dimensional Stochastic Analysis. Proceedings of the Colloquium, Amsterdam, Netherlands, 1999 (P. Clément et al., eds.). Verh. Afd. Natuurkd. 1. Reeks. K. Ned. Akad. Wet. 52, Royal Netherlands Academy of Arts and Sciences, Amsterdam, 2000, pp. 197–216. · Zbl 0990.60065
[28] A. Karczewska, J. Zabczyk: A note on stochastic wave equations. Evolution Equations and Their Applications in Physical and Life Sciences. Proc. Conf., Germany, 1999 (G. Lumer et al., eds.). Lecture Notes in Pure and Appl. Math. 215, Marcel Dekker, New York, 2001, pp. 501–511.
[29] M. Marcus, V. J. Mizel: Stochastic hyperbolic systems and the wave equation. Stochastics Stochastics Rep. 36 (1991), 225–244. · Zbl 0739.60059
[30] B. Maslowski, J. Seidler, I. Vrkoč: Integral continuity and stability for stochastic hyperbolic equations. Differ. Integral Equ. 6 (1993), 355–382. · Zbl 0777.35096
[31] V. Matskyavichyus: The support of the solution of a stochastic differential equation. Lith. Math. J. 26 (1986), 91–98 (In Russian.); English translation, Lith. Math. J. 26 (1986), 57–62.
[32] J. C. Mattingly, A. M. Stuart, D. J. Higham: Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise. Stochastic Processes Appl. 101 (2002), 185–232. · Zbl 1075.60072
[33] S. Meyn, R. L. Tweedie: Markov Chains and Stochastic Stability. With a prologue by Peter W. Glynn, Cambridge University Press, Cambridge, 2009.
[34] A. Millet, P.-L. Morien: On a nonlinear stochastic wave equation in the plane: existence and uniqueness of the solution. Ann. Appl. Probab. 11 (2001), 922–951. · Zbl 1017.60072
[35] A. Millet, M. Sanz-Solé: A stochastic wave equation in two space dimension: smoothness of the law. Ann. Probab. 27 (1999), 803–844. · Zbl 0944.60067
[36] M. Ondreját: Existence of global martingale solutions to stochastic hyperbolic equations driven by a spatially homogeneous Wiener process. Stoch. Dyn. 6 (2006), 23–52. · Zbl 1092.60024
[37] M. Ondreját: Existence of global mild and strong solutions to stochastic hyperbolic evolution equations driven by a spatially homogeneous Wiener process. J. Evol. Equ. 4 (2004), 169–191. · Zbl 1054.60068
[38] S. Peszat: The Cauchy problem for a nonlinear stochastic wave equation in any dimension. J. Evol. Equ. 2 (2002), 383–394. · Zbl 1375.60109
[39] S. Peszat, J. Zabczyk: Nonlinear stochastic wave and heat equations. Probab. Theory Relat. Fields 116 (2000), 421–443. · Zbl 0959.60044
[40] S. Peszat, J. Zabczyk: Stochastic evolution equations with a spatially homogeneous Wiener process. Stochastic Processes Appl. 72 (1997), 187–204. · Zbl 0943.60048
[41] J. Shatah, M. Struwe: Geometric Wave Equations. Courant Lecture Notes in Mathematics 2, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, 1998.
[42] D. W. Stroock, S. R. S. Varadhan: On the support of diffusion processes with applications to the strong maximum principle. Proc. Conf. Berkeley, Calififornia, 1970/1971, Vol. III: Probability Theory (L. M. Le Cam et al., eds.). Univ. California Press, Berkeley, Calififornia, 1972, pp. 333–359.
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