大阪大学　数理・データ科学教育研究センター
Center for Mathematical Modeling and Data Science,Osaka University

Speed of Convergence to Equilibrium of Random Dynamical Systems - With Applications

Rabi Bhattacharya (Department of Mathematics, University of Arizona)

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4. Speed of Convergence to Equilibrium of Random Dynamical Systems - With Applications

大阪大学 金融・保険セミナーシリーズ 第40回(CSFI-CRESTジョイントセミナー)

Speed of Convergence to Equilibrium of Random Dynamical Systems - With Applications

Rabi Bhattacharya (Department of Mathematics, University of Arizona)

Markov processes in discrete time on a standard state space S are of the form Xn = αn…α1X0, where αn (n=1,2,…) are i.i.d. random maps on S, independent of the initial state X0. For processes of interest here, S is a partially ordered metric space and αn are monotone maps on S into S. We find conditions for the existence of a unique equilibrium π and compute the speed of convergence of Xn to π in an appropriate distance. Applications are given to some growth and ruin problems in economics and insurance, and to the 2D Ising model on finite lattices. In continuous time we consider processes Xt (t≥0) on S= R^k (k≥1) governed by Ito’s stochastic differential equation dXt = b(Xt)dt + σ(Xt)dBt, with an initial X0 independent of the k-dimensional Brownian motion Bt (t≥0). Of particular interest are slow (i.e., polynomial) convergence rates to equilibrium signifying long range dependence.