On nontransitivity and ergodicity of Gaussian Markov processes

M. B. Nevel'son

Teoriâ veroâtnostej i ee primeneniâ, Tome 14 (1969) no. 1, pp. 35-42

Résumé

A stochastic process $X=(X^{s,x}(t),P)$ in an $n$-dimensional Euclidean space $E_n$ is studied which satisfies the stochastic differential equation $(1)$, where $B(t)$ and $C(t)$ are periodic matrices and $\eta(t)$ is an $n$-dimensional white noise process. Provided that, for any $t>s>0$, open set $U\subset E_n$ and $x\in E_n$, $$ P\{X^{s,x}(t)\in U\}>0, $$ the following results are obtained. Theorem 1. The $X$ process is recurrent if and only if the characteristic numbers $\alpha_1,\dots,\alpha_n$ of the system $(1)$ satisfy at least one of conditions 4–7. Theorem 2. {\em Let matrices $B$ and $C$ be constant. Then the $X$ process is ergodic if and only if the eigenvalues $\lambda_1,\dots,\lambda_n$ of $B$ satisfy the condition $$ \max_{1\le i\le n}\operatorname{Re}\lambda_i<0. $$ }