Generic properties of Markov operators.(English)Zbl 1118.37011

Schneider, Rolf (ed.) et al., IV international conference on “Stochastic geometry, convex bodies, empirical measures and applications to engineering science”, Tropea, Italy, September 24–29, 2001. Vol. II. Palermo: Circolo Matematico di Palermo. Suppl. Rend. Circ. Mat. Palermo, II. Ser. 70, 191-200 (2002).
The authors show that most, in the Baire category sense (a subset of a metric space is called of the Baire category, if it can be presented as a countable union of nowhere dense sets), Markov operators $$P$$ acting on the space of finite Borel measures $$\mu$$ supported on a compact subset of $$\mathbb R^d$$ are asymptotically stable (in the Fortet-Mourier norm) and have singular stationary (i.e., $$P\mu=\mu$$) measure.
For the entire collection see [Zbl 1003.00019].

MSC:

 37A30 Ergodic theorems, spectral theory, Markov operators 60J35 Transition functions, generators and resolvents 47A35 Ergodic theory of linear operators 37C20 Generic properties, structural stability of dynamical systems 28A33 Spaces of measures, convergence of measures

Keywords:

invariant measure