| Title:
|
Ergodicity for a stochastic geodesic equation in the tangent bundle of the 2D sphere (English) |
| Author:
|
Baňas, Ľubomír |
| Author:
|
Brzeźniak, Zdzisław |
| Author:
|
Neklyudov, Mikhail |
| Author:
|
Ondreját, Martin |
| Author:
|
Prohl, Andreas |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
65 |
| Issue:
|
3 |
| Year:
|
2015 |
| Pages:
|
617-657 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
geometric stochastic wave equation |
| Keyword:
|
stochastic geodesic equation |
| Keyword:
|
ergodicity |
| Keyword:
|
attractivity |
| Keyword:
|
invariant measure |
| Keyword:
|
numerical approximation |
| MSC:
|
37A25 |
| MSC:
|
58J65 |
| MSC:
|
60H10 |
| MSC:
|
60H15 |
| MSC:
|
60H35 |
| MSC:
|
60J60 |
| MSC:
|
65C20 |
| MSC:
|
65C30 |
| idZBL:
|
Zbl 06537684 |
| idMR:
|
MR3407597 |
| DOI:
|
10.1007/s10587-015-0200-7 |
| . |
| Date available:
|
2015-10-04T18:05:30Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144435 |
| . |
| Reference:
|
[1] Aida, S., Kusuoka, S., Stroock, D.: 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. Pitman Res. Notes Math. Ser. 284 Longman Scientific & Technical, Harlow, Essex; John Wiley & Sons, New York (1993), 3-34. Zbl 0790.60047, MR 1354161 |
| Reference:
|
[2] Arnold, L., Kliemann, W.: On unique ergodicity for degenerate diffusions.Stochastics 21 (1987), 41-61. Zbl 0617.60076, MR 0899954, 10.1080/17442508708833450 |
| Reference:
|
[3] Baňas, Ľ., Brzeźniak, Z., Neklyudov, M., Prohl, A.: A convergent finite-element-based discretization of the stochastic Landau-{L}ifshitz-{G}ilbert equation.IMA J. Numer. Anal. 34 (2014), 502-549. Zbl 1298.65012, MR 3194798, 10.1093/imanum/drt020 |
| Reference:
|
[4] Baňas, Ľ., Brzeźniak, Z., Neklyudov, M., Prohl, A.: Stochastic Ferromagnetism. Analysis and Numerics.De Gruyter Studies in Mathematics 58 De Gruyter, Berlin (2014). Zbl 1288.82001, MR 3157451 |
| Reference:
|
[5] Baňas, Ľ., Prohl, A., Schätzle, R.: 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, MR 2640052, 10.1007/s00211-009-0282-y |
| Reference:
|
[6] Bartels, S., Lubich, C., Prohl, A.: Convergent discretization of heat and wave map flows to spheres using approximate discrete Lagrange multipliers.Math. Comput. 78 (2009), 1269-1292. Zbl 1198.65178, MR 2501050, 10.1090/S0025-5718-09-02221-2 |
| Reference:
|
[7] Arous, G. Ben, Grădinaru, M.: Hölder norms and the support theorem for diffusions.C. R. Acad. Sci., Paris, Sér. I 316 French (1993), 283-286. MR 1205200 |
| Reference:
|
[8] Arous, G. Ben, Grădinaru, M., Ledoux, M.: Hölder norms and the support theorem for diffusions.Ann. Inst. Henri Poincaré, Probab. Stat. 30 (1994), 415-436. MR 1288358 |
| Reference:
|
[9] Brzeźniak, Z., Ondreját, M.: Stochastic geometric wave equations with values in compact Riemannian homogeneous spaces.Ann. Probab. 41 (2013), 1938-1977. Zbl 1286.60058, MR 3098063, 10.1214/11-AOP690 |
| Reference:
|
[10] Brzeźniak, Z., Ondreját, M.: Weak solutions to stochastic wave equations with values in Riemannian manifolds.Commun. Partial Differ. Equations 36 (2011), 1624-1653. Zbl 1238.60073, MR 2825605, 10.1080/03605302.2011.574243 |
| Reference:
|
[11] Brzeźniak, Z., Ondreját, M.: Strong solutions to stochastic wave equations with values in Riemannian manifolds.J. Funct. Anal. 253 (2007), 449-481. Zbl 1141.58019, MR 2370085, 10.1016/j.jfa.2007.03.034 |
| Reference:
|
[12] Cabaña, E. M.: On barrier problems for the vibrating string.Z. Wahrscheinlichkeitstheor. Verw. Geb. 22 (1972), 13-24. Zbl 0214.16801, MR 0322974, 10.1007/BF00538902 |
| Reference:
|
[13] Carmona, R., Nualart, D.: Random nonlinear wave equations: propagation of singularities.Ann. Probab. 16 (1988), 730-751. Zbl 0643.60045, MR 0929075, 10.1214/aop/1176991784 |
| Reference:
|
[14] Carmona, R., Nualart, D.: Random nonlinear wave equations: smoothness of the solutions.Probab. Theory Relat. Fields 79 (1988), 469-508. Zbl 0635.60073, MR 0966173, 10.1007/BF00318783 |
| Reference:
|
[15] Chow, P.-L.: Stochastic wave equations with polynomial nonlinearity.Ann. Appl. Probab. 12 (2002), 361-381. Zbl 1017.60071, MR 1890069, 10.1214/aoap/1015961168 |
| Reference:
|
[16] Prato, G. Da, Zabczyk, J.: Ergodicity for Infinite-Dimensional Systems.London Mathematical Society Lecture Note Series 229 Cambridge Univ. Press, Cambridge (1996). Zbl 0849.60052, MR 1417491 |
| Reference:
|
[17] Prato, G. Da, Zabczyk, J.: Stochastic Equations in Infinite Dimensions.Encyclopedia of Mathematics and Its Applications 44 Cambridge University Press, Cambridge (1992). Zbl 0761.60052, MR 1207136 |
| Reference:
|
[18] Dalang, R. C., Frangos, N. E.: The stochastic wave equation in two spatial dimensions.Ann. Probab. 26 (1998), 187-212. Zbl 0938.60046, MR 1617046, 10.1214/aop/1022855416 |
| Reference:
|
[19] Dalang, R. C., Lévêque, O.: Second-order linear hyperbolic {SPDE}s driven by isotropic Gaussian noise on a sphere.Ann. Probab. 32 (2004), 1068-1099. Zbl 1046.60058, MR 2044674, 10.1214/aop/1079021472 |
| Reference:
|
[20] Diaconis, P., Freedman, D.: 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). |
| Reference:
|
[21] Ginibre, J., Velo, G.: The Cauchy problem for the $ O(N), \Bbb C P(N-1),$ and $G_{\Bbb C}(N, p)$ models.Ann. Phys. 142 (1982), 393-415. MR 0678488, 10.1016/0003-4916(82)90077-X |
| Reference:
|
[22] Gyöngy, I., Pröhle, T.: On the approximation of stochastic differential equation and on Stroock-{V}aradhan's support theorem.Comput. Math. Appl. 19 (1990), 65-70. Zbl 0711.60051, MR 1026782, 10.1016/0898-1221(90)90082-U |
| Reference:
|
[23] Hörmander, L.: Hypoelliptic second order differential equations.Acta Math. 119 (1967), 147-171. Zbl 0156.10701, MR 0222474, 10.1007/BF02392081 |
| Reference:
|
[24] Ichihara, K., Kunita, H.: A classification of the second order degenerate elliptic operators and its probabilistic characterization.Z. Wahrscheinlichkeitstheor. Verw. Geb. 30 (1974), 235-254. Zbl 0326.60097, MR 0381007, 10.1007/BF00533476 |
| Reference:
|
[25] Ichihara, K., Kunita, H.: 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, MR 0488328, 10.1007/BF01844875 |
| Reference:
|
[26] Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes.North-Holland Mathematical Library 24 North-Holland; Publishing Co. Tokyo: Kodansha, Amsterdam (1989). Zbl 0684.60040, MR 1011252 |
| Reference:
|
[27] Karczewska, A., Zabczyk, J.: Stochastic {PDE}'s with function-valued solutions.Infinite Dimensional Stochastic Analysis. Proceedings of the Colloquium, Amsterdam, Netherlands, 1999 P. Clément et al. Verh. Afd. Natuurkd. 1. Reeks. K. Ned. Akad. Wet. 52 Royal Netherlands Academy of Arts and Sciences, Amsterdam (2000), 197-216. Zbl 0990.60065, MR 1832378 |
| Reference:
|
[28] Karczewska, A., Zabczyk, J.: A note on stochastic wave equations.Evolution Equations and Their Applications in Physical and Life Sciences. Proc. Conf., Germany, 1999 G. Lumer et al. Lecture Notes in Pure and Appl. Math. 215 Marcel Dekker, New York (2001), 501-511. Zbl 0978.60066, MR 1818028 |
| Reference:
|
[29] Marcus, M., Mizel, V. J.: Stochastic hyperbolic systems and the wave equation.Stochastics Stochastics Rep. 36 (1991), 225-244. Zbl 0739.60059, MR 1128496, 10.1080/17442509108833720 |
| Reference:
|
[30] Maslowski, B., Seidler, J., Vrkoč, I.: Integral continuity and stability for stochastic hyperbolic equations.Differ. Integral Equ. 6 (1993), 355-382. Zbl 0777.35096, MR 1195388 |
| Reference:
|
[31] Matskyavichyus, V.: The support of the solution of a stochastic differential equation.Lith. Math. J. 26 (1986), 91-98 Russian English translation Lith. Math. J. 26 57-62 (1986). MR 0847207 |
| Reference:
|
[32] Mattingly, J. C., Stuart, A. M., Higham, D. J.: Ergodicity for {SDE}s and approximations: locally Lipschitz vector fields and degenerate noise.Stochastic Processes Appl. 101 (2002), 185-232. Zbl 1075.60072, MR 1931266 |
| Reference:
|
[33] Meyn, S., Tweedie, R. L.: Markov Chains and Stochastic Stability.With a prologue by Peter W. Glynn Cambridge University Press Cambridge (2009). Zbl 1165.60001, MR 2509253 |
| Reference:
|
[34] Millet, A., Morien, P.-L.: 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, MR 1865028, 10.1214/aoap/1015345353 |
| Reference:
|
[35] Millet, A., Sanz-Solé, M.: A stochastic wave equation in two space dimension: smoothness of the law.Ann. Probab. 27 (1999), 803-844. MR 1698971, 10.1214/aop/1022677387 |
| Reference:
|
[36] Ondreját, M.: 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, MR 2210680, 10.1142/S0219493706001633 |
| Reference:
|
[37] Ondreját, M.: 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, MR 2059301, 10.1007/s00028-003-0130-y |
| Reference:
|
[38] Peszat, S.: The Cauchy problem for a nonlinear stochastic wave equation in any dimension.J. Evol. Equ. 2 (2002), 383-394. MR 1930613, 10.1007/PL00013197 |
| Reference:
|
[39] Peszat, S., Zabczyk, J.: Nonlinear stochastic wave and heat equations.Probab. Theory Relat. Fields 116 (2000), 421-443. Zbl 0959.60044, MR 1749283, 10.1007/s004400050257 |
| Reference:
|
[40] Peszat, S., Zabczyk, J.: Stochastic evolution equations with a spatially homogeneous Wiener process.Stochastic Processes Appl. 72 (1997), 187-204. Zbl 0943.60048, MR 1486552, 10.1016/S0304-4149(97)00089-6 |
| Reference:
|
[41] Shatah, J., Struwe, M.: Geometric Wave Equations.Courant Lecture Notes in Mathematics 2 New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence (1998). Zbl 0993.35001, MR 1674843 |
| Reference:
|
[42] Stroock, D. W., Varadhan, S. R. S.: 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. Univ. California Press Berkeley, Calififornia (1972), 333-359. Zbl 0255.60056, MR 0400425 |
| . |
