What does a generic Markov operator look like?

A. M. Vershik
2006 St. Petersburg Mathematical Journal
Generic (i.e., forming an everywhere dense massive subset) classes of Markov operators in the space L 2 (X, µ) with a finite continuous measure are considered. In a canonical way, each Markov operator is associated with a multivalued measure-preserving transformation (i.e., a polymorphism), and also with a stationary Markov chain; therefore, one can also talk of generic polymorphisms and generic Markov chains. Not only had the generic nature of the properties discussed in the paper been unclear
more » ... before this research, but even the very existence of Markov operators that enjoy these properties in full or partly was known. The most important result is that the class of totally nondeterministic nonmixing operators is generic. A number of problems is posed; there is some hope that generic Markov operators will find applications in various fields, including statistical hydrodynamics. I was fortunate to be friends with O. A., especially in the 1970s, and some time I will write about this. In the late 1960s and 1970s, she was very interested in dynamical systems, and this was an additional motive for our contacts. Working on the Hopf equation, she arrived at the necessity of considering multivalued (Markov) mappings and suggested that we start a joint research on multivalued solutions of equations. Our work resulted in a series of papers; see [4, 5] . We also had grandiose projects for further research, for example, working on metric hydrodynamics, but these were never realized. Approximately at the same time, I started to develop general (multivalued) dynamics ([1]), and recently, after a long interval, I have returned to this subject. In this paper, dedicated to the unforgettable O. A., I continue this topic. §1. Markov operators 1.1. Definitions. Definition 1. A Markov operator in the Hilbert space L 2 (X, µ) of complex-valued square-integrable functions on a Lebesgue-Rokhlin space (X, µ) with a continuous normalized measure µ is a continuous linear operator V satisfying the following conditions: 1) V is a contraction: V ≤ 1 (in the operator norm); 2) V 1 = V * 1 = 1, where 1 is the function identically equal to one; 3) V preserves the nonnegativity of functions: V f is nonnegative whenever f ∈ L 2 (X, µ) is nonnegative. Note that condition 1) follows from 2) and 3), and the second condition in 2) follows from the others. In short, a Markov operator is a unity-preserving positive contraction.