Invariant measures for Markov operators with application to function systems

Abstract

Abstract.A new sufficient condition for the existence of an invariant measure for Markov operators defined on Polish spaces is presented. This criterion is applied to iterated function systems.0. Introduction. The theory of Markov processes is a fast developing topic which has been extensively studied during the last few years. The reason for this study was the progress in the theory of fractals. Markov processes can be considered from two points of view. They can be investigated by purely probabilistic and purely analytic methods. In our paper we use the second approach. More precisely, let {Z n } be a homogeneous Markov chain taking values in a metric space (X, ) and let π be its transition kernel, i.e. Prob(Zfor all n ∈ N and Borel sets A. It is of great interest to give sufficient conditions for the existence of an invariant probability measure for the Markov chain {Z n }, i.e. a probability measure µ * satisfyingfor all Borel sets A (see [4,9,[13][14][15][16]).In our analytic approach we consider the Markov operator P defined on the space of all Borel measures given byConsequently, searching for invariant measures of Markov processes can be replaced by searching for fixed points of P . Using this method does not involve the probabilistic background of the transition kernels π.2000 Mathematics Subject Classification: Primary 47A35; Secondary 28D15.

Quantum Fractals on n–Spheres. Clifford Algebra Approach

“…Even if the spheres S n and balls B n+1 are compact, the Möbius transformations of these spheres are non-contractive. The question of existence and uniqueness of invariant measures for non-contractive iterated function systems has been discussed in the mathematical literature [34,35,36], yet none of the sufficient conditions seems to be easily applicable to our case. Apanasov has a whole book devoted to conformal maps, yet we find that his criteria, esp.…”

Criterion on stability for Markov processes applied to a model with jumps

“…, p N ). In [17] it was proved that under conditions (3.1)-(3.2) it is asymptotically stable. Denote by μ 0 its invariant distribution.…”

“…It was proved that the Markov operator P is nonexpansive with respect to some Wasserstein metric (see [17]). To be precise, we proved that there exists a metric ρ in X equivalent to the metric ρ (equivalence means that any sequence converges in ρ iff it is convergent inρ) such that From Lemma 3.1 in [17] it follows that we may find n 0 ≥ 1 such that P n 0 δ x B(z, 2ε/3) ≥ β/2 for any x in some open neighbourhood O of K.…”

“…Our proof is based on the asymptotic stability of the system. The proof of stability was given in [2] in the case of finitely dimensional spaces (see also [12]) and in [17] when the iterated system is defined on an arbitrary Polish space. There is a huge literature of models with jumps partly due to a large class of possible applications in physics or biology, partly due to their purely mathematical properties.…”

“…The proof is based on the lower bound technique introduced by A. Lasota and J. Yorke in [12], where the authors showed the existence of an absolutely continuous invariant measure for the Frobenius-Perron operator corresponding to piecewise monotonic transformations. Since then, the technique has been generalized first for Markov semigroups acting on densities (see [10]), subsequently for general Markov semigroups defined on arbitrary Borel measures in finite dimensions (see [13]) and finally it has been extended to infinite dimensional spaces (see [17]). Generally speaking the method relies on an easy observation that two regular trajectories starting at different measures which visit some small set with positive, bounded from below, probability converge in the weak topology.…”