In this paper, we introduce classes of homogeneous Markov processes satisfying the Bebutov–Feller property, specifically weakly almost periodic and almost periodic processes. In terms of the topology defined on the space of the Bebutоv–Feller operators, this means that the closure of the semigroup of transition operators for a Markov process forms a compact semitopological (or, in the case of almost periodic processes, a topological) semigroup. We establish criteria for processes to belong to these classes and provide a result concerning the asymptotic behavior of their transition operators. In terms of the topology introduced on the space of Bebutov–Feller operators, this result describes an approximation (in a specific sense) using the kernel of the aforementioned compact semigroup. Notably, the kernel of this semigroup is identified as a compact Abelian group. To derive an approximating element of this group, the transition operator is multiplied (either from the right or the left) by the group identity. The structure of the idempotent operator (or projector) is also characterized, which is particularly useful given that the identity of the group is itself idempotent. The structure of the kernel can be further detailed as follows: for the group identity, the conservative part of the phase space is partitioned into “elementary” sets that are both closed and invariant. By considering these “elementary” sets as “enlarged” states, we construct a transition operator that acts on the redefined phase space consisting of these enlarged states. This operator is deterministic and corresponds to a specific spatial transformation belonging to the group of autohomeomorphisms of the new phase space.
Keywords: homogeneous Markov processes, Feller property, semigroup of transition operators, Radon measures, semi-topological semigroup, topological semigroup, kernel of semigroup, closed invariant sets, conservative part of phase space
Received February 10, 2025
Revised March 21, 2025
Accepted March 24, 2025
Funding Agency: The paper was published with the financial support of the Ministry of Education and Science of the Russian Federation as part of the program of the Moscow Center for Fundamental and Applied Mathematics under the agreement no. 075-15-2022-284.
Sergey Nikolayevich Smirnov, Dr. Phys.-Math. Sci., Prof., Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, Moscow, 119991 Russia, e-mail: s.n.smirnov@gmail.com
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Cite this article as: Sergey N. Smirnov. The structure of almost periodic semigroups of Bebutov–Feller operators. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2025, vol. 31, no. 2, pp. 215–228.
