Random shifts which preserve measure

Abstract:

Given a flow ${\theta _g},g \in G$ a group, over a probability space $(\Omega ,\mathfrak {F},P)$ and a $G$-valued random variable $Z$, we exhibit the Lebesgue decomposition of the measure $P \circ \theta _Z^{ - 1}$ relative to $P$, and give necessary and sufficient conditions for equality $(P \circ \theta _Z^{ - 1} = P)$, absolute continuity $(P \circ \theta _Z^{ - 1} \ll P)$, and singularity $(P \circ \theta _Z^{ - 1} \bot P)$ in terms of the Haar measure. The proof rests on the theory of “Palm measures” as developed by Mecke and the authors. Specializing the group $G$, we retrieve some known results for the integers and real line, and compute the Radon-Nikodým derivatives in various cases.