| Russian Mathematical Surveys |
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Brief communications Mixing in stochastic dynamical systems with stationary noise S. B. Kuksinabc, A. R. Shirikyandb a Université Paris Cité & Sorbonne Université CNRS, IMJ-PRG, Paris, France b RUDN University c Steklov Mathematical Institute of Russian Academy of Sciences, Moscow d Department of Mathematics, CY Cergy Paris University, CNRS UMR 8088, Cergy–Pontoise, France
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    The asymptotic behaviour of trajectories of stochastic dynamical systems with white noise has been studied in considerable depth both in the finite- and infinite-dimensional cases. As is well known, in this situation the system has a unique globally stable state, provided that the transition function of the Markov process generated by the system has some properties of regularity and recurrence: see [1], [5], and [2]. The aim of this note is to announce some recent results in the case when, in place of white noise, a stochastic dynamical system is driven by Markovian or stationary noise. The reader can find the proofs of theorems below in [3] and [4]. For the simplicity of presentation we limit ourselves to the case of finite-dimensional phase space. A stochastic dynamical system with Markovian noiseLet $H$ and $E$ be finite-dimensional Euclidean spaces, ${\mathcal K}\,{\subset}\, E$ be a compact subset, and $S\colon H\times E\to H$ be a $C^2$-smooth map. Consider the random dynamical system where $u_k\in H$, and $\{\eta_k\}_{k\in\mathbb{Z}_+}$ is a stationary stochastic process in $E$ such that for each $k\in\mathbb{Z}_+$ the distribution ${\mathcal D}(\eta_k)$ has support in ${\mathcal K}$. We assume that the following conditions are satisfied:
For each vector $u\in H$ the dynamical system (1) defines a trajectory $\{u_k\}_{k\geqslant 0}$ with initial condition $u_0=u$. The following theorem describes the large-time asymptotic behaviour of the trajectory. Theorem 1. Assume that the above four conditions are satisfied. Then there exists a translation-invariant probability measure ${\boldsymbol\mu}$ on the space $H^{\mathbb{Z}}$ and $\gamma>0$ such that for each initial condition $u\in H$ the corresponding trajectory $\{u_k\}_{k\geqslant 0}$ of (1) satisfies the inequality where ${\boldsymbol\mu}_m$ denotes the projection of ${\boldsymbol\mu}$ onto $H^{m+1}$, $\|{\,\cdot\,}\|_{\mathrm{var}}$ is the total variation norm of measures, and $C_m>0$ is a constant independent of $u$. Note that the result on the convergence of trajectories in the total variation norm also holds in the case when the noise is a stationary stochastic process. In this case we must impose certain conditions on the conditional probabilities of the process, when the whole past is fixed. For a precise statement the reader can consult [3], Theorem 3.1. Applications to ordinary differential equations with random perturbationIn Euclidean space $H=\mathbb{R}^d$ with scalar product $\langle\,\cdot\,{,}\,\cdot\,\rangle$ and the corresponding norm $|{\,\cdot\,}|$ consider the following ordinary differential equation with random perturbation: Here $V\colon \mathbb{R}^d\to\mathbb{R}^d$ is a $C^2$-vector field that for some $c>0$ satisfies $\langle V(x),x\rangle\leqslant -c|x|^2$ for each $x\in H$, and $\eta$ is a stochastic process of the form where $\delta(t)$ is the Dirac mass at the origin and $\{\eta_k\}$ is a homogeneous Markov process in $H$. Trajectories of (3) make jumps at integer points of the time axis, and we assume that they are right continuous. Letting $x_k$ denote the values of the solution $x(t)$ at the time $t=k$, it is easy to see that the sequence $\{x_k\}$ satisfies (1) for $S(x,\eta)=\varphi_1(x)+\eta$, where $\varphi_t\colon\mathbb{R}^d\to\mathbb{R}^d$ is the phase flow associated with the unperturbed equation (3). Thus, to each initial condition $x\in H$ there corresponds a trajectory $\{x_k\}_{k\geqslant 0}$ in $H$. The following result is a consequence of Theorem 1.Theorem 2. Assume that the transition function of the process $\{\eta_k\}$ satisfies conditions (MR) and (SR). Then there exist a translation-invariant probability measure ${\boldsymbol\mu}$ on the space $H^\mathbb{Z}$ and $\gamma>0$ such that inequality (2) with $u_k=x_k$ and $u=x$ holds for each initial condition $x\in H$ and each integer $m\geqslant 0$, for a sufficiently large constant $C_m$ independent on $x$. 
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\Bibitem{KukShi24} |
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