Calculus of Variations and Geometric Measure Theory

A. Ponosov - E. Stepanov

Atomic operators, random dynamical systems and invariant measures

created by stepanov on 22 Jun 2010
modified on 18 Jan 2015


Published Paper

Inserted: 22 jun 2010
Last Updated: 18 jan 2015

Journal: St. Petersburg Math. J
Volume: 26
Number: 4
Pages: 148-194
Year: 2014


We prove the existence of an invariant measure for (families of) generalized weighted shifts, which we call ``atomic operators''. Typically, such operators may come from infinite dimensional stochastic differential equations generating non-Carathéodory stochastic flows, for instance, from stochastic differential equation with time delay in the diffusion. The main result is applied then to such flows giving the existence of invariant measures for those.

Keywords: Young Measures, shift operator, local operator, random dynamical system, invariant measure, cocycle property, Wiener shift