On local structure of continuous Markov processes

A. V. Skorokhod

Teoriâ veroâtnostej i ee primeneniâ, Tome 11 (1966) no. 3, pp. 381-423

Résumé

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 $\varphi_1,\dots,\varphi_k$.