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Homogeneous Markov Processes Without Discontinuities îf the Second Kind
A. V. Skorokhod Kiev
Abstract:
Let
$x_t$ be a homogeneous Markov process in a compact subset
$U$ of a linear space
$X$. Suppose that for all
$t\ge0$ both
$x_{t-0}$,
$x_{t+0}$ exist and
$x_t=x_{t+0}$. Let further the transition probability
$P(t,x,E)$ of
$x_t$ satisfy the following conditions:
I. $\lim\limits_{t\downarrow0}\sup\limits_{x\in U}P(t,x,\{y\colon|x-y|>\varepsilon\})=0$ for all
$\varepsilon>0$,
II. If
$\varphi(x)$ is a continuous function on
$U$ then
$\int\varphi(y)P(t,x,dy)$ is also a continuous function of
$x$ on
$U$.
Under these assumptions there exists a positive homogeneous additive functional
$\delta_t$ such that the process
$y_t=x_{\tau_t}$ where
$\delta_{\tau_t}=t$ possesses the following property: if
$\varphi_1,\dots,\varphi_n\in D_A$ (
$A$ is the infinitesimal operator of the process
$y_t$) and
$F(\xi_1,\dots,\xi_n)$ is a function with continuous derivatives
$\frac{\partial^2F}{\partial\xi_i\partial\xi_j}$ $(i,j=1,\dots,n)$ then $\Phi(x)=F(\varphi_1,\dots,\varphi_n)\in D_{\widetilde A}$ where
$\widetilde A$ is the quasiinfinitesimal operator of
$y_t$ and
\begin{gather*}
\widetilde A\Phi(x)=\sum a_i(x)\frac{\partial\Phi}{\partial\varphi_i}(x)+\sum b_{ij}(x)\frac{\partial^2\Phi}{\partial\varphi_i\partial\varphi_j}(x)+
\\
+\int\biggl\{\Phi(x+y)-\Phi(x)-\sum\frac{\partial\Phi}{\partial\varphi_i}(x)[\varphi_i(x+y)-\varphi_i(x)]\biggr\}\Lambda(x,dy).
\end{gather*}
Received: 17.11.1966
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