Calculus of Variations and Geometric Measure Theory

M. Drakhlin - E. Litsyn - A. Ponosov - E. Stepanov

Generalizing the property of locality: atomic/coatomic operators and applications

created by stepanov on 12 May 2005
modified on 30 Sep 2006


Published Paper

Inserted: 12 may 2005
Last Updated: 30 sep 2006

Journal: J. Nonlinear & Convex Anal.
Volume: 7
Number: 2
Pages: 139-162
Year: 2006


In this paper we strudy the so-called atomic and coatomic operators introduce in (M. Drakhlin, A. Ponosov, E. Stepanov. On some classes of operators determined by the structure of their memory, Proc. Edinburgh Math. Soc., 45(2002), no. 2, 467-490) and generalizing the classical notion of a local operator between ideal function spaces. In particular, we discover characteristic properties of such operators, which can serve as their new definitions. These properties are intrinsic in the sense that they are independent of a particular $\sigma$-homomorphism of the underlying $\sigma$-algebrae and are based on purely measure-theoretic notions of memory and comemory of an operator, which re also studied in details in the paper. We also prove some new results on analytic properties of atomic, coatomic and local operators. For the reader's convenience some of the known results regarding such operators that are exploited in the present paper are also provided without proofs. In the last section, we show that the study of strong periodic in law solutions to a stochastic functional differential equation can be put, under rather general assumptions, into the framework of atomic operators. This result can serve as an additional strong motivation for introducing and studying atomic and coatomic operators in their most general form.

Keywords: local operator, atomic operator, Carathéodory function, random dynamical system, invariant measures, coatomic operator, Nemytskiì operator