On nontransitivity and ergodicity of Gaussian Markov processes

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
$$ \max_{1\le i\le n}\operatorname{Re}\lambda_i<0. $$
}