Wanderings of a Markov process

E. B. Dynkin

Teoriâ veroâtnostej i ee primeneniâ, Tome 16 (1971) no. 3, pp. 409-436

Résumé

Let $X=(x_t,\zeta,M_t,\mathbf P_x)$ be a standard Markov process in a semi-compact $E$ and let $D$ be an open subset of the space $E$. The random set $\{t\colon x_t\in D\}$ consists of intervals $(\gamma,\delta)$ with the beginnings $\gamma$ of some of them. Wanderings of $X$ are the paths $\omega^\gamma$ in the space $D$ defined by the formula $w_t^\gamma=x_{\gamma+t}$ ($0<t<\delta-\gamma$). For any left-continuous nonanticipating functional $F_t(\omega,w)$ ($t>0$, $\omega\in\Omega$, $w\in W$), we consider the sum of its values $F_\gamma(\omega,w^\gamma)$ over all wanderings of $X$ and we calculate the expectation of this sum in terms of an additive functional $\Phi$ of $X$ (the fundamental functional) and a kernel $b(x,\Gamma)$ (the entrance kernel). The main result is the formula of wanderings (1.8).