Bifurcations of stationary measures of random diffeomorphisms

Abstract:

Random diffeomorphisms with bounded absolutely continuous noise are known
to possess a finite number of stationary measures. We discuss the
dependence of stationary measures on an auxiliary parameter, thus
describing bifurcations of families of random diffeomorphisms. A
bifurcation theory is developed under mild regularity assumptions on the
diffeomorphisms and the noise distribution (e.g. smooth diffeomorphisms
with uniformly distributed additive noise are included). We distinguish
bifurcations where the density function of a stationary measure varies
discontinuously or where the support of a stationary measure varies
discontinuously. We establish that generic random diffeomorphisms are
stable. The densities of stable stationary measures are shown to be smooth
and to depend smoothly on an auxiliary parameter, except at bifurcation
values. The bifurcation theory explains the occurrence of transients and
intermittency as the main bifurcation phenomena in random diffeomorphisms.
Quantitative descriptions by means of average escape times from sets as
functions of the parameter are provided. Further quantitative properties
are described through the speed of decay of correlations as a function of
the parameter. Random differentiable maps which are not necessarily
injective are studied in one dimension; we show that stable
one-dimensional random maps occur open and dense and that in one-parameter
families bifurcations are typically isolated. We classify codimension-one
bifurcations for one-dimensional random maps; we distinguish three
possible kinds, the random saddle node, the random homoclinic and the
random boundary bifurcation. The theory is illustrated on families of
random circle diffeomorphisms and random unimodal maps.