Random composing of mappings, small stochastic perturbations and attractors

  • Paweł Góra


Let T and H r (0<rr0) be continuous mappings of a compact metric space (M,d), d](x,Hr(x))≦r for any x∈M. We consider Markov processes \(\tilde T_r \) with transition functions
$$\tilde P^r (x,A) = p\chi _A (T(x)) + q\chi _A (H_r (x))$$
. They are random compositions of T and H r . We study the existence, uniqueness and asymptotic (r→0) behaviour of \(\tilde T_r \)-invariant measures \(\tilde \mu _r \). We do this by converting the problem into the problem of small stochastic perturbations of the mapping T. The main result is that the weak limit points (for r→0) of the set \(\{ \tilde \mu _r :{\text{ 0 < }}r \leqq r_0 \} \) are measures concentrated on “attractors” of the mapping T.

Our definition of “attractors” is based on ideas similar to those of Ruelle [5].

The perturbations we deal with are nonlocal and singular, whereas so far most authors considered local and “absolutely continuous” stochastic perturbations (e.g. Ruelle [5]).


Continuous Mapping Stochastic Process Probability Theory Markov Process Invariant Measure 


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Copyright information

© Springer-Verlag 1985

Authors and Affiliations

  • Paweł Góra
    • 1
  1. 1.Institute of MathematicsWarsaw UniversityWarszawaPoland

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