Atomic operators, random dynamical systems and invariant measures

A. Ponosov ; E. Stepanov

Algebra i analiz, Tome 26 (2014) no. 4, pp. 148-194.

Résumé

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.

Détail

Keywords: stochastic solution flow, invariant measure, atomic operator.

Bibliographie

[1] Appell J., Zabrejko P. P., Nonlinear superposition operators, Cambridge Tracts in Math., 95, Cambridge Univ. Press, Cambridge, 1990 | MR | Zbl

[2] Arnold L., Random dynamical systems, Springer Monogr. in Math., Springer-Verlag, Berlin, 1998 | DOI | MR

[3] Arnold L., Wihtutz V., “Stationary solutions of linear systems with additive and multiplicative noise”, Stochastics, 7:1–2 (1982), 133–156 | DOI | MR

[4] Buttazzo G., Semicontinuity, relaxation and integral representation in the calculus of variations, Pitman Research Notes in Math., 207, Longman Sci. and Tech., Harlow, 1989 | MR | Zbl

[5] Buttazzo G., Pratelli A., Solimini S., Stepanov E., Optimal urban networks via mass transportation, Lecture Notes in Math., 1961, Springer-Verlag, Berlin, 2009 | DOI | MR | Zbl

[6] Castaing C., Raynaud de Fitte P., Valadier M., Young measures on topological spaces: with applications in control theory and probability theory, Math. Appl., 571, Kluwer Acad. Publ., Dordrecht, 2004 | MR | Zbl

[7] Castaing C., Valadier M., Convex analysis and measurable multifunctions, Lecture Notes in Math., 580, Springer-Verlag, Berlin, 1977 | DOI | MR | Zbl

[8] Crauel H., Random probability measures on Polish spaces, Stochastics Monogr., 11, Taylor Francis, London, 2002 | MR | Zbl

[9] Da Prato G., Zabczyk J., Stochastic equations in infinite dimensions, Encyclopedia Math. Appl., 44, Cambridge Univ. Press, Cambridge, 1992 | MR | Zbl

[10] Drakhlin M. E., Ponosov A., Stepanov E., “On some classes of operators determined by the structure of their memory”, Proc. Edinb. Math. Soc. (2), 45:2 (2002), 467–490 | DOI | MR | Zbl

[11] Drakhlin M. E., Litsyn E., Ponosov A., Stepanov E., “Generalizing the property of locality: atomic/coatomic operators and applications”, J. Nonlinear Convex Anal., 7:2 (2006), 139–162 | MR | Zbl

[12] Gikhman I. I., Skorokhod A. V., Teoriya stokhasticheskikh protsessov, v. 3, Nauka, M., 1975 | MR

[13] Liberzon D., Switching in systems and control, Birkhäuser, Boston, MA, 2003 | MR | Zbl

[14] Mohammed S. E. A., Stochastic functional differential equations, Research Notes in Math., 99, Pitman, Boston, MA, 1984 | MR | Zbl

[15] Øksendal B., Stochastic differential equations. An introduction with applications, Universitext, 6th ed., Springer-Verlag, Berlin, 2003 | DOI | MR

[16] Ponosov A., “O gipoteze Nemytskogo”, Dokl. AN SSSR, 289:6 (1986), 1308–1311 | MR

[17] Ponosov A., “Metod nepodvizhnoi tochki v teorii stokhasticheskikh differentsialnykh uravnenii”, Dokl. AN SSSR, 299:3 (1988), 562–565 | MR | Zbl

[18] Protter P., “Semimartingales and measure preserving flows”, Ann. Inst. H. Poincare Probab. Statist., 22:2 (1986), 127–147 | MR | Zbl

[19] Scheutzow M., “Exponential growth rate for a singular linear stochastic delay differential equation”, Discrete Contin. Dyn. Syst. Ser. B, 18:6 (2013), 1683–1696 | DOI | MR | Zbl

[20] Shragin I. V., “Abstraktnye operatory Nemytskogo – lokalno opredelennye operatory”, Dokl. AN SSSR, 227:1 (1976), 47–49 | MR | Zbl

[21] Valadier M., “Young measures”, Methods of Nonconvex Analysis (Varenna, 1989), Lecture Notes in Math., 1446, Springer-Verlag, Berlin, 1990, 152–188 | DOI | MR