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

Ľubomír Baňas; Zdzisław Brzeźniak; Mikhail Neklyudov; Martin Ondreját; Andreas Prohl

Czechoslovak Mathematical Journal (2015)

  • Volume: 65, Issue: 3, page 617-657
  • ISSN: 0011-4642

Abstract

top
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.

How to cite

top

Baňas, Ľubomír, et al. "Ergodicity for a stochastic geodesic equation in the tangent bundle of the 2D sphere." Czechoslovak Mathematical Journal 65.3 (2015): 617-657. <http://eudml.org/doc/271838>.

@article{Baňas2015,
abstract = {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.},
author = {Baňas, Ľubomír, Brzeźniak, Zdzisław, Neklyudov, Mikhail, Ondreját, Martin, Prohl, Andreas},
journal = {Czechoslovak Mathematical Journal},
keywords = {geometric stochastic wave equation; stochastic geodesic equation; ergodicity; attractivity; invariant measure; numerical approximation},
language = {eng},
number = {3},
pages = {617-657},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Ergodicity for a stochastic geodesic equation in the tangent bundle of the 2D sphere},
url = {http://eudml.org/doc/271838},
volume = {65},
year = {2015},
}

TY - JOUR
AU - Baňas, Ľubomír
AU - Brzeźniak, Zdzisław
AU - Neklyudov, Mikhail
AU - Ondreját, Martin
AU - Prohl, Andreas
TI - Ergodicity for a stochastic geodesic equation in the tangent bundle of the 2D sphere
JO - Czechoslovak Mathematical Journal
PY - 2015
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 65
IS - 3
SP - 617
EP - 657
AB - 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.
LA - eng
KW - geometric stochastic wave equation; stochastic geodesic equation; ergodicity; attractivity; invariant measure; numerical approximation
UR - http://eudml.org/doc/271838
ER -

References

top
  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. (1993) Zbl0790.60047MR1354161
  2. Arnold, L., Kliemann, W., 10.1080/17442508708833450, Stochastics 21 (1987), 41-61. (1987) Zbl0617.60076MR0899954DOI10.1080/17442508708833450
  3. Baňas, Ľ., Brzeźniak, Z., Neklyudov, M., Prohl, A., 10.1093/imanum/drt020, IMA J. Numer. Anal. 34 (2014), 502-549. (2014) Zbl1298.65012MR3194798DOI10.1093/imanum/drt020
  4. Baňas, Ľ., Brzeźniak, Z., Neklyudov, M., Prohl, A., Stochastic Ferromagnetism. Analysis and Numerics, De Gruyter Studies in Mathematics 58 De Gruyter, Berlin (2014). (2014) Zbl1288.82001MR3157451
  5. Baňas, Ľ., Prohl, A., Schätzle, R., 10.1007/s00211-009-0282-y, Numer. Math. 115 (2010), 395-432. (2010) Zbl1203.65174MR2640052DOI10.1007/s00211-009-0282-y
  6. Bartels, S., Lubich, C., Prohl, A., 10.1090/S0025-5718-09-02221-2, Math. Comput. 78 (2009), 1269-1292. (2009) Zbl1198.65178MR2501050DOI10.1090/S0025-5718-09-02221-2
  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. (1993) MR1205200
  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. (1994) MR1288358
  9. Brzeźniak, Z., Ondreját, M., 10.1214/11-AOP690, Ann. Probab. 41 (2013), 1938-1977. (2013) Zbl1286.60058MR3098063DOI10.1214/11-AOP690
  10. Brzeźniak, Z., Ondreját, M., 10.1080/03605302.2011.574243, Commun. Partial Differ. Equations 36 (2011), 1624-1653. (2011) Zbl1238.60073MR2825605DOI10.1080/03605302.2011.574243
  11. Brzeźniak, Z., Ondreját, M., 10.1016/j.jfa.2007.03.034, J. Funct. Anal. 253 (2007), 449-481. (2007) Zbl1141.58019MR2370085DOI10.1016/j.jfa.2007.03.034
  12. Cabaña, E. M., 10.1007/BF00538902, Z. Wahrscheinlichkeitstheor. Verw. Geb. 22 (1972), 13-24. (1972) Zbl0214.16801MR0322974DOI10.1007/BF00538902
  13. Carmona, R., Nualart, D., 10.1214/aop/1176991784, Ann. Probab. 16 (1988), 730-751. (1988) Zbl0643.60045MR0929075DOI10.1214/aop/1176991784
  14. Carmona, R., Nualart, D., 10.1007/BF00318783, Probab. Theory Relat. Fields 79 (1988), 469-508. (1988) Zbl0635.60073MR0966173DOI10.1007/BF00318783
  15. Chow, P.-L., 10.1214/aoap/1015961168, Ann. Appl. Probab. 12 (2002), 361-381. (2002) Zbl1017.60071MR1890069DOI10.1214/aoap/1015961168
  16. Prato, G. Da, Zabczyk, J., Ergodicity for Infinite-Dimensional Systems, London Mathematical Society Lecture Note Series 229 Cambridge Univ. Press, Cambridge (1996). (1996) Zbl0849.60052MR1417491
  17. Prato, G. Da, Zabczyk, J., Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and Its Applications 44 Cambridge University Press, Cambridge (1992). (1992) Zbl0761.60052MR1207136
  18. Dalang, R. C., Frangos, N. E., 10.1214/aop/1022855416, Ann. Probab. 26 (1998), 187-212. (1998) Zbl0938.60046MR1617046DOI10.1214/aop/1022855416
  19. Dalang, R. C., Lévêque, O., 10.1214/aop/1079021472, Ann. Probab. 32 (2004), 1068-1099. (2004) Zbl1046.60058MR2044674DOI10.1214/aop/1079021472
  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). (1997) 
  21. Ginibre, J., Velo, G., 10.1016/0003-4916(82)90077-X, Ann. Phys. 142 (1982), 393-415. (1982) MR0678488DOI10.1016/0003-4916(82)90077-X
  22. Gyöngy, I., Pröhle, T., 10.1016/0898-1221(90)90082-U, Comput. Math. Appl. 19 (1990), 65-70. (1990) Zbl0711.60051MR1026782DOI10.1016/0898-1221(90)90082-U
  23. Hörmander, L., 10.1007/BF02392081, Acta Math. 119 (1967), 147-171. (1967) Zbl0156.10701MR0222474DOI10.1007/BF02392081
  24. Ichihara, K., Kunita, H., 10.1007/BF00533476, Z. Wahrscheinlichkeitstheor. Verw. Geb. 30 (1974), 235-254. (1974) Zbl0326.60097MR0381007DOI10.1007/BF00533476
  25. Ichihara, K., Kunita, H., 10.1007/BF01844875, Z. Wahrscheinlichkeitstheor. Verw. Geb. 39 (1977), 81-84. (1977) Zbl0382.60069MR0488328DOI10.1007/BF01844875
  26. Ikeda, N., Watanabe, S., Stochastic Differential Equations and Diffusion Processes, North-Holland Mathematical Library 24 North-Holland; Publishing Co. Tokyo: Kodansha, Amsterdam (1989). (1989) Zbl0684.60040MR1011252
  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. (2000) Zbl0990.60065MR1832378
  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. (2001) Zbl0978.60066MR1818028
  29. Marcus, M., Mizel, V. J., 10.1080/17442509108833720, Stochastics Stochastics Rep. 36 (1991), 225-244. (1991) Zbl0739.60059MR1128496DOI10.1080/17442509108833720
  30. Maslowski, B., Seidler, J., Vrkoč, I., Integral continuity and stability for stochastic hyperbolic equations, Differ. Integral Equ. 6 (1993), 355-382. (1993) Zbl0777.35096MR1195388
  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). (1986) MR0847207
  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. (2002) Zbl1075.60072MR1931266
  33. Meyn, S., Tweedie, R. L., Markov Chains and Stochastic Stability, With a prologue by Peter W. Glynn Cambridge University Press Cambridge (2009). (2009) Zbl1165.60001MR2509253
  34. Millet, A., Morien, P.-L., 10.1214/aoap/1015345353, Ann. Appl. Probab. 11 (2001), 922-951. (2001) Zbl1017.60072MR1865028DOI10.1214/aoap/1015345353
  35. Millet, A., Sanz-Solé, M., 10.1214/aop/1022677387, Ann. Probab. 27 (1999), 803-844. (1999) MR1698971DOI10.1214/aop/1022677387
  36. Ondreját, M., 10.1142/S0219493706001633, Stoch. Dyn. 6 (2006), 23-52. (2006) Zbl1092.60024MR2210680DOI10.1142/S0219493706001633
  37. Ondreját, M., 10.1007/s00028-003-0130-y, J. Evol. Equ. 4 (2004), 169-191. (2004) Zbl1054.60068MR2059301DOI10.1007/s00028-003-0130-y
  38. Peszat, S., 10.1007/PL00013197, J. Evol. Equ. 2 (2002), 383-394. (2002) MR1930613DOI10.1007/PL00013197
  39. Peszat, S., Zabczyk, J., 10.1007/s004400050257, Probab. Theory Relat. Fields 116 (2000), 421-443. (2000) Zbl0959.60044MR1749283DOI10.1007/s004400050257
  40. Peszat, S., Zabczyk, J., 10.1016/S0304-4149(97)00089-6, Stochastic Processes Appl. 72 (1997), 187-204. (1997) Zbl0943.60048MR1486552DOI10.1016/S0304-4149(97)00089-6
  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). (1998) Zbl0993.35001MR1674843
  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. (1972) Zbl0255.60056MR0400425

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.