Shift-invariant measures on infinite-dimensional spaces: Integrable functions and random walks

Abstract

Averaging of random shift operators on a space of the square integrable by shift-invariant measure complex-valued functions on linear topological spaces has been studied. The case of the l∞ space has been considered as an example. A shift-invariant measure on the l∞ space, which was constructed by Caratheodory's scheme, is σ-additive, but not σ-finite. Furthermore, various approximations of measurable sets have been investigated. One-parameter groups of shifts along constant vector fields in the l∞ space and semigroups of shifts to a random vector, the distribution of which is given by a collection of the Gaussian measures, have been discussed. A criterion of strong continuity for a semigroup of shifts along a constant vector field has been established. Conditions for collection of the Gaussian measures, which guarantee the semigroup property and strong continuity of averaged one-parameter collection of linear operators, have been defined.

Keywords

• strongly continuous semigroups
• averaging of operator semigroups
• shift-invariant measures
• square integrable functions