Skorohod A. V. Markov processes with homogeneous second component. I.

I. I. Ezhov   ; A. V. Skorokhod

Teoriâ veroâtnostej i ee primeneniâ, Tome 14 (1969) no. 1, pp. 3-14

Résumé

We consider Markov processes $z_t=\{x_t,y_t\}$ in a product space $X\times Y$ ($x_t\in X$, $y_t\in Y$), $Y$ being a finite-dimensional Euclidean space. Such a process is called a process with homogeneous second component if its transition probability function $P(t,x,y,s,A,B)$, $x\in X$, $y\in Y$, $A\subset X$, $B\subset Y$, $t<s$, satisfies the condition $$ P(t,x,y,s,A,B)=P(t,x,0,s,A,B_{-y}), $$ where $B_{-y}$ is the set of $y'$'s such that $y+y'\in B$. In §1 we study general properties of such processes. In §2 the case is considered when $x_t$ is a process with denumerable set of states. §3 deals with time-homogeneous processes.