On Dobrushin Ergodicity Coefficient and weak ergodicity of Markov Chains on Jordan Algebras

Abstract

Abstract. In this paper we study certain properties of Dobrushin's ergodicity coefficient for stochastic operators defined on non-associative L 1 -spaces associated with semi-finite JBWalgebras. Such results extends the well-known classical ones to a non-associative setting. This allows us to investigate the weak ergodicity of nonhomogeneous discrete Markov processes (NDMP) by means of the ergodicity coefficient. We provide a necessary and sufficient conditions for such processes to satisfy the weak ergodicity. IntroductionIt is known (see [23]) that the investigations of asymptotical behavior of iterations of Markov operators on commutative L 1 -spaces are very important. On the other hand, these investigations are related with several notions of ergodicity of L 1 -contractions of measure spaces. To the investigation of such ergodic properties of Markov operators were devoted lots of papers (see for example, [7,23]). On the other hand, such kind of operators were studied in noncommutative settings. Since, the study of quantum dynamical systems has had an impetuous growth in the last years, in view of natural applications to various field of mathematics and physics. It is then of interest to understand among the various ergodic properties, which ones survive and are meaningful by passing from the classical to the quantum case. Due to noncommutativity, the latter situation is much more complicated than the former. The reader is referred e.g. to [2,15,16,18,29] for further details relative to some differences between the classical and the quantum situations. It is therefore natural to study the possible generalizations to quantum case of the various ergodic properties known for classical dynamical systems. One of the generalizations of noncommutative algebras is Jordan algebras. Note that Jordan Banach algebras [3,6] are a non-associative real analogue of von Neumann algebras. The existence of exceptional JBW -algebras does not allow one to use the ideas and methods from von Neumann algebras. The motivation of these investigations arose in quantum statistical mechanics and quantum field theory (see, [9]). Mostly, in those investigations homogeneous Markov processes were considered. Many ergodic type theorems have been proved for Markov operators acting in nonassociative and noncommutative L p -spaces (see for example, [4, 5, 20, 21, 8, 18]) On the other hand, nonhomogeneous Markov processes with general state space have become a subject of interest due to their applications in many branches of mathematics and natural sciences. In many papers (see for example, [24,17,31]) the weak ergodicity of nonhomogeneous Markov process are given in terms of Dobrushin's ergodicity coefficient [12]. In [33] some

Uniformization stable Markov models and their Jordan algebraic structure

“…Jordan algebras in particular have been of discussion in the context of genetics for the theory of DNA recombination [19]. More generally, Jordan algebras have been used to consider Markov processes in the context of non-associative state spaces [12].…”