This article is cited in
6 papers
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
$F(\xi_1,\dots,\xi_k)$ is a continuous bounded function which has derivatives of the first and the second orders and
$\varphi_1,\dots,\varphi_k$ belong to the domain of the infinitesimal generator of the process
$y_t$ then
\begin{gather*}
\mathbf M_yF(\varphi_1(y_t),\dots,\varphi_k(y_t))-F(\varphi_1(y),\dots,\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),\dots,\varphi_k(y))+\frac12\sum b_{ij}(y)\frac{\partial^2F}{\partial\xi_i\partial\xi_j}(\varphi_1(y),\dots,\varphi_k(y)),
\end{gather*}
where the coefficients
$a_i(y)$,
$b_{ij}(y)$ depend on the functions
$\varphi_1,\dots,\varphi_k$.
Received: 09.01.1966
Fulltext:
PDF file (2200 kB)