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<p>The paper deals with the problem of large-time behaviour of trajectories for discrete-time dynamical systems driven by a random noise. Assuming that the phase space is finite-dimensional and compact, and the noise is a Markov process with a transition probability satisfying some regularity hypotheses, we prove that all the trajectories converge to a unique measure in the total variation metric. The proof is based on the Markovian reduction of the system in question and a result on mixing for Markov processes. Then we present an extension of this result to the case of systems driven by stationary noises. The results of this paper were announced in Kuksin-Shirikyan (Russian Math. Surveys 79:1098–1100, 2024).</p>

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Markovian Reduction and Exponential Mixing in Total Variation for Random Dynamical Systems

  • 期刊论文

  • 发表日期：

    2026年3月21日

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  • Sergei Kuksin,
  • Armen Shirikyan

摘要

The paper deals with the problem of large-time behaviour of trajectories for discrete-time dynamical systems driven by a random noise. Assuming that the phase space is finite-dimensional and compact, and the noise is a Markov process with a transition probability satisfying some regularity hypotheses, we prove that all the trajectories converge to a unique measure in the total variation metric. The proof is based on the Markovian reduction of the system in question and a result on mixing for Markov processes. Then we present an extension of this result to the case of systems driven by stationary noises. The results of this paper were announced in Kuksin-Shirikyan (Russian Math. Surveys 79:1098–1100, 2024).