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Teoriya Veroyatnostei i ee Primeneniya, 1969, Volume 14, Issue 1, Pages 35–42
(Mi tvp1076)
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This article is cited in 1 scientific paper (total in 1 paper)
On nontransitivity and ergodicity of Gaussian Markov processes
M. B. Nevel'son Moscow
Abstract:
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
$$
\max_{1\le i\le n}\operatorname{Re}\lambda_i<0.
$$
}
Received: 12.06.1967
Citation:
M. B. Nevel'son, “On nontransitivity and ergodicity of Gaussian Markov processes”, Teor. Veroyatnost. i Primenen., 14:1 (1969), 35–42; Theory Probab. Appl., 14:1 (1969), 35–43
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https://www.mathnet.ru/eng/tvp1076 https://www.mathnet.ru/eng/tvp/v14/i1/p35
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