Vol. 25, no. 1, 2019
Assume that $X$ is a topological space and $Y$ is a separable metric space. Let these spaces be equipped with Borel $\sigma$-algebras $\mathcal{B}_X$ and $\mathcal{B}_Y$, respectively. Suppose that $P(x,B)$ is a stochastic transition kernel; i.e., the mapping $x \mapsto P(x,B)$ is measurable for all $B \in \mathcal{B}_Y$ and the mapping $B\mapsto P(x, B)$ is a probability measure for any $x \in X$. Denote by supp$(P(x,\cdot))$ the topological support of the measure $B\mapsto P(x, B)$. If the transition kernel $P(x,B)$ satisfies the Feller property, i.e., the mapping $x \mapsto P(x,\cdot)$ is continuous in the weak topology on the space of probability measures, then the set-valued mapping $x\mapsto$supp$(P(x,\cdot))$ is lower semicontinuous. Conversely, consider a set-valued mapping $x\mapsto S(x)$, where $x\in X$ and $S(x)$ is a nonempty closed subset of a Polish space $Y$. If $x \mapsto S(x)$ is lower semicontinuous, then, under some general assumptions on the space $X$, there exists a Feller transition kernel such that supp$(P(x,\cdot))=S(x)$ for all $x\in X$.
Keywords: Feller property, transition kernel, topological support of a measure, lower semicontinuous set-valued mapping, continuous branch (selection)
Received July 13, 2018
Revised November 16, 2018
Accepted November 19, 2018
Funding Agency: This study was carried out at the Faculty of Computational Mathematics and Cybernetics of Moscow State University within the project “Optimization Methods in Control Problems for Complex Systems under Available Information” (state registration no. AAAA-A16-116021110324-8).
Sergei Nikolaevich Smirnov, Cand. Sci. (Phys.-Math.), Lomonosov Moscow State University, Moscow, 119991 Russia, e-mail: s.n.smirnov@gmail.com
Cite this article as: S.N. Smirnov. A Feller transition kernel with measure supports given by a set-valued mapping, Trudy Instituta Matematiki i Mekhaniki URO RAN, 2019, vol. 25, no. 1, pp. 219-228.
REFERENCES
1. Smirnov S.N. Feller Processes. In: Encyclopaedia of mathematics, vol. 10. N Y: Springer, 2012, pp. 483–485.
2. Kurzhanskii A.B. Upravlenie i nablyudenie v usloviyakh neopredelennosti [Control and Observation Under the Conditions of Uncertainty]. Moscow, Nauka Publ., 1977, 392 p.
3. Tulcea I.C. Mesures dans les espaces produits. Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat., 1949, vol. 8, no. 7, pp. 208–211.
4. Michael E.A. Continuous selections. I. Ann. Math., 1956, vol. 63, no. 2, pp. 361–382. doi: 10.2307/1969615
5. Lange K.L. Borel sets of probability measures. Pacific J. Math., 1973, vol. 48, no. 1, pp. 141–162.
6. Vakhania N.N., Tarieladze V.I., Chobanyan S.A. Probability distributions on Banach spaces. Dordrecht, D. Reidel Publishing Company, 1987, 482 p. ISBN: 90-277-2496-2 .
7. Hu S., Papageorgiou N. Handbook of multivalued analysis. Vol. I: Theory. Ser. Math. and Its Appl., vol. 419. Berlin: Springer, 1997, 968 p. ISBN: 0792346823 .
8. Billingsley P. Convergence of probability measures. N Y: Wiley, 1968, 253 p. ISBN: 0471072427. Translated to Russian under the title Skhodimost’ veroyatnostnykh mer. Moscow, Nauka Publ., 1977, 352 p.
9. Alexandroff A.D. Additive set-functions in abstract spaces. Rec. Math. [Mat. Sbornik] N.S., 1943, vol. 13 (55), no. 2-3, pp. 169–238.
10. Ash R.B. Real analysis and probability. N Y: Acad. Press., 1972. 494 p. ISBN: 978-0120652013 .
11. Bogachev V.B. Measure theory, Vol. II. N Y: Springer, 2007, 586 p. Original Russian text published in Bogachev V.I., Osnovy teorii mery, vol. 2. M.; Izhevsk: NITs “Regulyarnaya i Khaoticheskaya Dinamika” Publ., 2003, 576 p.
12. Shiryaev A.N. Probability. Ser. Graduate Texts in Math., vol. 95, N Y etc.: Springer-Verlag, 1995, 624 p. ISBN: 0387945490 . Original Russian text (3rd ed.) published in Shiryaev A.N. Veroyatnost’-1. Moscow, Nauka Publ., 2004, 408 p.
13. Weaver N. Lipschitz algebras. Singapur: World Scientic Publishing Co. Pte. Ltd., 1999, 223 p. doi: 10.1142/4100 .
14. Hille S.C., Worm D.T.H. Embedding of semigroups of Lipschitz maps into positive linear semigroups on ordered Banach spaces generated by measures. Integr. Equ. Oper. Theory, 2009, vol. 63, no. 3, pp. 351–371. doi: 10.1007/s00020-008-1652-z
15. Dudley R.M. Real analysis and probability. Cambridge: Cambridge University Press, 2004, 555 p. ISBN: 0-511-04208-6 .
16. Kolmogorov A.N., Fomin S.V. Elements of the theory of functions and functional analysis. Vol. 1, 2. Mineola, N Y, Dover Publ., 1999, 288 p. ISBN: 0486406830 . Original Russian text (4th ed.) published in Kolmogorov A.N., Fomin S.V. Elementy teorii funktsii i funktsional’nogo analiza. Moscow, Nauka Publ., 1976, 543 p.
17. Engelking R. General topology. Sigma series in pure mathematics, vol. 6, Berlin: Heldermann Verlag, 1989, 535 p. ISBN: 3885380064 . Translated to Russian inder the title Obshchaya topologiya. Moscow: Mir Publ., 1986, 752 p.
18. Jain P.K., Khalil Ahmad, Ahuja Om P. Functional analysis. New Delhi etc.: New Age International, 1995, 326 p. ISBN: 9788122408010 .
19. Prokhorov Yu.V. Convergence of random processes and limit theorems in probability theory. Theory Probab. Appl., 1956, vol. 1, no. 2, pp. 157–214. doi: 10.1137/1101016 .
20. Dudley R.M. Distances of probability measures and random variables. Ann. Math. Statis., 1968, vol. 39, no. 5, pp. 1563–1572. doi: 10.1214/aoms/1177698137 .
21. Denkowski Z., Migorski S., Papageorgiou N.S. An introduction to nonlinear analysis: Theory. N Y: Springer Science & Business Media, 2003, 823 p. ISBN: 0306474565 .
