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

A Markov process for an infinite age-structured population