Long time behavior of diffusions with Markov switching

2009/12/16 by Jean-Baptiste Bardet, Bardet, Jean-Baptiste, Hélène Guérin +3 · 2 citations
Mathematics · #60H25 #60J60 #60J75 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.0912.3231

openalex publication_date 2009/12/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let Y be an Ornstein-Uhlenbeck diffusion governed by an ergodic finite state Markov process X: dY_t=-λ(X_t)Y_tdt+σ(X_t)dB_t, Y₀ given. Under ergodicity condition, we get quantitative estimates for the long time behavior of Y. We also establish a trichotomy for the tail of the stationary distribution of Y: it can be heavy (only some moments are finite), exponential-like (only some exponential moments are finite) or Gaussian-like (its Laplace transform is bounded below and above by Gaussian ones). The critical moments are characterized by the parameters of the model.

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