A Markov process for an infinite age-structured population

For an infinite system of particles arriving in and departing from a habitat $X$ -- locally compact Polish space with positive Radon measure $\chi$ Markov process is constructed explicit way. Along its location $x\in X$, each particle characterized by age $\alpha\geq 0$ time since arriving. As the state one takes set marked configurations $\widehat{\Gamma}$, equipped metric that makes it complete separable space. The stochastic evolution described Kolmogorov operator $L$, expressed through departure rate $m(x,\alpha)\geq 0$, acting on bounded continuous functions $F:\widehat{\Gamma}\to \mathds{R}$. this operator, we pose martingale problem show has unique solution, explicitly paper. We also prove corresponding stationary temporarily egrodic if separated away zero.