Rate of convergence to equilibrium for discrete-time stochastic dynamics with memory. (English) Zbl 1431.62422

Varvenne, Maylis

Bernoulli 25, No. 4B, 3234-3275 (2019).

In a continuous time framework, stochastic differential equations (SDEs) driven by Gaussian processes with stationary increments have been introduced to model random evolution phenomena with long random dependence properties. In the main result of this paper, the existence and uniqueness of the invariant distribution is shown, and some upper bounds on the rate of convergence to equilibrium in terms of the asymptotic behavior of the covariance function of the Gaussian noise and fractional Brownian motion \( H \in (0,1/2) \) is outlined.
This paper is well-organized with remarkable results.

Reviewer: Rózsa Horváth-Bokor (Budakalász)

MSC:

62M10	Time series, auto-correlation, regression, etc. in statistics (GARCH)
60G22	Fractional processes, including fractional Brownian motion
60G15	Gaussian processes
60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)

Keywords:

discrete stochastic dynamics; Lyapunov function; rate of convergence to equilibrium; stationary Gaussian noise; Toeplitz operator; total variation distance

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