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 Teor. Veroyatnost. i Primenen.: Year: Volume: Issue: Page: Find

 Teor. Veroyatnost. i Primenen., 1966, Volume 11, Issue 3, Pages 381–423 (Mi tvp638)

On local structure of continuous Markov processes

A. V. Skorokhod

Kiev

Abstract: Let $x_t$ be a continuous Markov process on a locally compact space $X$. In the article the following result is proved. There exists an additive positive functional $\varphi_t$ such that the process $y_t=x_{\tau_t}$ where $\tau_t$ is determined by the equality $\varphi_{\tau_t}=\tau$ posesses such a property: if $F(\xi_1,…,\xi_k)$ is a continuous bounded function which has derivatives of the first and the second orders and $\varphi_1,…,\varphi_k$ belong to the domain of the infinitesimal generator of the process $y_t$ then
\begin{gather*} \mathbf M_yF(\varphi_1(y_t),…,\varphi_k(y_t))-F(\varphi_1(y),…,\varphi_k(y))=\int_0^t\mathbf M\psi(y_s) ds,
\psi(y)=\sum a_i(y)\frac{\partial F}{\partial\xi_i}(\varphi_1(y),…,\varphi_k(y))+\frac12\sum b_{ij}(y)\frac{\partial^2F}{\partial\xi_i\partial\xi_j}(\varphi_1(y),…,\varphi_k(y)), \end{gather*}
where the coefficients $a_i(y)$, $b_{ij}(y)$ depend on the functions $\varphi_1,…,\varphi_k$.

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English version:
Theory of Probability and its Applications, 1966, 11:3, 336–372

Bibliographic databases:

Citation: A. V. Skorokhod, “On local structure of continuous Markov processes”, Teor. Veroyatnost. i Primenen., 11:3 (1966), 381–423; Theory Probab. Appl., 11:3 (1966), 336–372

Citation in format AMSBIB
\Bibitem{Sko66} \by A.~V.~Skorokhod \paper On local structure of continuous Markov processes \jour Teor. Veroyatnost. i Primenen. \yr 1966 \vol 11 \issue 3 \pages 381--423 \mathnet{http://mi.mathnet.ru/tvp638} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=203815} \zmath{https://zbmath.org/?q=an:0203.18004} \transl \jour Theory Probab. Appl. \yr 1966 \vol 11 \issue 3 \pages 336--372 \crossref{https://doi.org/10.1137/1111036}