Markov dynamics on the cone of discrete Radon measures

Abstract

Configuration spaces form an important and actively developing area in the infinite dimensional analysis. The spaces not only contain rich mathematical structures which require non-trivial combination of continuous and combinatoric analysis, they also provide a natural mathematical framework for the applications to mathematical physics, biology, ecology, and beyond. Spaces of discrete Radon measures (DRM) may be considered as generalizations of configuration spaces. Main peculiarity of a DRM is that its support is typically not a configuration (i.e. not a locally finite set). The latter changes drastically the techniques for the study of the spaces of DRM. Spaces of DRM have various motivations coming from mathematics and applications. In particular, random DRM appear in the context of the Skorokhod theorem [17] in the theory of processes with independent increments. Next, in the representation theory of current groups, the role of measures on spaces of DRM was clarified in fundamental works by Gelfand, Graev, and Vershik; see [15] for the development of this approach. Additionally, DRM gives a useful technical equipment in the study of several models in mathematical physics, biology, and ecology. In the present paper, we start with a brief overview of the known facts about the spaces of DRM (Section 2). In [10], the concept of Plato subspaces of the spaces of marked configurations was introduced. Using this, one can define topological, differential and functional structures on spaces of DRM, as well as transfer the harmonic analysis considered in [11] to the spaces of DRM. This allows us to extend the study of nonequilibrium dynamics, see e.g. [8, 12, 13], to the spaces of DRM.