The Structure of Almost Periodic Semigroups of Bebutov–Feller Operators

In this paper, we introduce classes of homogeneous Markov processes satisfying Bebutov–Feller property—specifically, weakly almost periodic and almost periodic processes. In terms of the topology defined on the space of Bebutov–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 which 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.