Taiwanese Journal of Mathematics

ERGODIC PROPERTIES OF CONTINUOUS PARAMETER ADDITIVE PROCESSES

Ryotaro Sato

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Abstract

Let $\{T_{t}:t\in{\bf R}\}$ be a measure preserving flow in a probability measure space $(\Omega, {\cal A},\mu)$, and $\{F_{t}:t\in {\bf R}\}$ be a family of real-valued measurable functions on $(\Omega,{\cal A},\mu)$ such that $F_{t+s}=F_{t}+F_{s}\circ T_{t}$ (mod $\mu$) for all $t,\, s \in {\bf R}$. In this paper we deduce necessary and sufficient conditions for the existence of a real-valued measurable function $f$ on $\Omega$, with $f\in L_{p}(\Omega,\mu)$ where $0\leq p\leq \infty$, such that $F_{t}=f\circ T_{t}-f$ (mod $\mu$) for all $t\in {\bf R}$. Related results are also obtained. These may be considered to be continuous parameter refinements of the recent discrete parameter results of Alonso, Hong and Obaya concerning additive real coboundary cocycles.

Article information

Source
Taiwanese J. Math., Volume 7, Number 3 (2003), 347-390.

Dates
First available in Project Euclid: 20 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500558393

Digital Object Identifier
doi:10.11650/twjm/1500558393

Mathematical Reviews number (MathSciNet)
MR1998757

Zentralblatt MATH identifier
1044.28013

Subjects
Primary: 28D10: One-parameter continuous families of measure-preserving transformations 37A10: One-parameter continuous families of measure-preserving transformations 47A35: Ergodic theory [See also 28Dxx, 37Axx]

Keywords
additive process measure preserving flow measure preserving and nonsingular transformations skew-product transformation invariant measure pointwise and mean ergodic theorems additive real coboundary cocycle

Citation

Sato, Ryotaro. ERGODIC PROPERTIES OF CONTINUOUS PARAMETER ADDITIVE PROCESSES. Taiwanese J. Math. 7 (2003), no. 3, 347--390. doi:10.11650/twjm/1500558393. https://projecteuclid.org/euclid.twjm/1500558393


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