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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



Atomic operators, random dynamical systems and invariant measures

Authors: A. Ponosov and E. Stepanov
Original publication: Algebra i Analiz, tom 26 (2014), nomer 4.
Journal: St. Petersburg Math. J. 26 (2015), 607-642
MSC (2010): Primary 37H10
Published electronically: May 6, 2015
MathSciNet review: 3289188
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Abstract: It is proved that the existence of invariant measures for families of the so-called atomic operators (nonlinear generalized weighted shifts) defined over spaces of measurable functions follows from the existence of appropriate invariant bounded sets. Typically, such operators come from infinite-dimensional stochastic differential equations generating not necessarily regular solution flows, for instance, from stochastic differential equations with time delay in the diffusion term (regular solution flows called also Carathéodory flows are those almost surely continuous with respect to the initial data). Thus, it is proved that to ensure the existence of an invariant measure for a stochastic solution flow it suffices to find a bounded invariant subset, and no regularity requirement for the flow is necessary. This result is based on the possibility to extend atomic operators by continuity to a suitable set of Young measures, which is proved in the paper. A motivating example giving a new result on the existence of an invariant measure for a possibly nonregular solution flow of some model stochastic differential equation is also provided.

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  • 1. J. Appell and P. P. Zabrejko, Nonlinear superposition operators, Cambridge Tracts in Math., vol. 95, Cambridge Univ. Press, Cambridge, 1990. MR 1066204 (91k:47168)
  • 2. L. Arnold, Random dynamical systems, Springer Monogr. in Math., Springer-Verlag, Berlin, 1998. MR 1723992 (2000m:37087)
  • 3. L. Arnold and V. Wihtutz, Stationary solutions of linear systems with additive and multiplicative noise, Stochastics 7 (1982), no. 1-2, 133-156. MR 665411 (83j:60058)
  • 4. G. Buttazzo, Semicontinuity, relaxation and integral representation in the calculus of variations, Pitman Research Notes in Math., vol. 207, Longman Sci. and Tech., Harlow, 1989. MR 1020296 (91c:49002)
  • 5. G. Buttazzo, A. Pratelli, S. Solimini, and E. Stepanov, Optimal urban networks via mass transportation, Lecture Notes in Math., vol. 1961, Springer-Verlag, Berlin, 2009. MR 2469110 (2010c:49002)
  • 6. C. Castaing, P. Raynaud de Fitte, and M. Valadier, Young measures on topological spaces: with applications in control theory and probability theory, Math. Appl., vol. 571, Kluwer Acad. Publ., Dordrecht, 2004. MR 2102261 (2005k:28001)
  • 7. C. Castaing and M. Valadier, Convex analysis and measurable multifunctions, Lecture Notes in Math., vol. 580, Springer-Verlag, Berlin, 1977. MR 0467310 (57:7169)
  • 8. H. Crauel, Random probability measures on Polish spaces, Stochastics Monogr., vol. 11, Taylor & Francis, London, 2002. MR 1993844 (2004e:60005)
  • 9. G. Da Prato and J. Zabczyk, Stochastic equations in infinite dimensions, Encyclopedia Math. Appl., vol. 44, Cambridge Univ. Press, Cambridge, 1992. MR 1207136 (95g:60073)
  • 10. M. E. Drakhlin, A. Ponosov, and E. Stepanov, On some classes of operators determined by the structure of their memory, Proc. Edinb. Math. Soc. (2) 45 (2002), no. 2, 467-490. MR 1912652 (2003g:47114)
  • 11. M. E. Drakhlin, E. Litsyn, A. Ponosov, and E. Stepanov, Generalizing the property of locality: atomic/coatomic operators and applications, J. Nonlinear Convex Anal. 7 (2006), no. 2, 139-162. MR 2254115 (2008b:47058)
  • 12. I. I. Gikhman and A. V. Skorokhod, Theory of random processes. Vol. III, Nauka, Moscow, 1975; English transl.; Grundlehren Math. Wiss., vol. 232, Springer-Verlag, Berlin, 1979. MR 0651014 (58:31323a)
  • 13. D. Liberzon, Switching in systems and control, Birkhäuser, Boston, MA, 2003. MR 1987806 (2004i:93003)
  • 14. S. E. A. Mohammed, Stochastic functional differential equations, Research Notes in Math., vol. 99, Pitman, Boston, MA, 1984. MR 0754561 (86j:60151)
  • 15. B. Øksendal, Stochastic differential equations. An introduction with applications, 6th ed., Universitext, Springer-Verlag, Berlin, 2003. MR 2001996 (2004e:60102)
  • 16. A. Ponosov, The Nemytskiĭ conjecture, Dokl. Akad. Nauk SSSR 289 (1986), no. 6, 1308-1311; English transl., Soviet Math. Dokl. 34 (1987), no. 1, 231-233. MR 856838 (88b:47092)
  • 17. -, A fixed-point method in the theory of stochastic differential equations, Dokl. Akad. Nauk SSSR 299 (1988), no. 3, 562-565; English transl., Soviet Math. Dokl. 37 (1988), no. 2, 426-429. MR 936492 (89g:60189)
  • 18. P. Protter, Semimartingales and measure preserving flows, Ann. Inst. H. Poincare Probab. Statist. 22 (1986), no. 2, 127-147. MR 850752 (88m:60138)
  • 19. M. Scheutzow, Exponential growth rate for a singular linear stochastic delay differential equation, Discrete Contin. Dyn. Syst. Ser. B 18 (2013), no. 6, 1683-1696. MR 3038775
  • 20. I. V. Shragin, Abstract Nemetskiĭ operators are locally defined operators, Dokl. Akad. Nauk SSSR 227 (1976), no. 1, 47-49; English transl., Soviet Math. Dokl. 17 (1976), no. 2, 354-357. MR 0412918 (54:1039)
  • 21. M. Valadier, Young measures, Methods of Nonconvex Analysis (Varenna, 1989), Lecture Notes in Math., vol. 1446, Springer-Verlag, Berlin, 1990, pp. 152-188. MR 1079763 (91j:28006)

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Additional Information

A. Ponosov
Affiliation: Department of Mathematical Sciences and Technology, Norwegian University of Life Sciences, P.O. Box 5003, -1432 Ås, Norway

E. Stepanov
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia; Division of Mathematical Physics, Department of Mathematics and Mechanics, St. Petersburg State University, Universitetskiĭ pr. 28, Old Peterhof, St. Petersburg 198504, Russia; St. Petersburg National Research University of Information Technologies, Mechanics, and Optics, Kronverkskiĭ pr. 49, St. Petersburg 197101, Russia

Keywords: Stochastic solution flow, invariant measure, atomic operator
Received by editor(s): October 10, 2013
Published electronically: May 6, 2015
Additional Notes: The work has been sponsored by the St. Petersburg State University grants #6.38.670.2013 and # The work of the second author was also partially financed by GNAMPA, by RFBR grant #14-01-00534, by the project 2010A2TFX2 “Calcolo delle variazioni” of the Italian Ministry of Research and by the Russian government grant NSh-1771.2014.1
Dedicated: Dedicated to the memory of Professor M. E. Drakhlin
Article copyright: © Copyright 2015 American Mathematical Society

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