2021/12/09 by Jasinska, Dominika, Kozitsky, Yuri
#60J25 #60J85 #92D25 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)
paper · doi:10.48550/arxiv.2112.04992
For an infinite system of particles arriving in and departing from a habitat X -- a locally compact Polish space with a positive Radon measure χ -- a Markov process is constructed in an explicit way. Along with its location x∈ X, each particle is characterized by age α≥ 0 -- time since arriving. As the state space one takes the set of marked configurations \widehatΓ, equipped with a metric that makes it a complete and separable metric space. The stochastic evolution of the system is described by a Kolmogorov operator L, expressed through the measure χ and a departure rate m(x,α)≥ 0, and acting on bounded continuous functions F:\widehatΓ→ \mathdsR. For this operator, we pose the martingale problem and show that it has a unique solution, explicitly constructed in the paper. We also prove that the corresponding process has a unique stationary state and is temporarily egrodic if the rate of departure is separated away from zero.