Homogeneous Markov Processes Without Discontinuities оf the Second Kind

A. V. Skorokhod

Teoriâ veroâtnostej i ee primeneniâ, Tome 12 (1967) no. 2, pp. 258-278

Résumé

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*}