The stochastic evolution of an infinite population with logistic-type interaction

Abstract

An infinite population of point entities dwelling in the habitat $X=\mathds{R}^d$ is studied. Its members arrive at and depart from $X$ at random. The departure rate has a term corresponding to a logistic-type interaction between the entities. Thereby, the corresponding Kolmogorov operator $L$ has an additive quadratic part, which usually produces essential difficulties in its study. The population's pure states are locally finite counting measures defined on $X$. The set of such states $\Gamma$ is equipped with the vague topology and thus with the corresponding Borel $\sigma$-field. The population evolution is described at two levels. At the first level, we deal with the Fokker-Planck equation for $(L,\mathcal{F},\mu_0)$ where $\mathcal{F}$ is an appropriate set of bounded test functions $F:\Gamma\to \mathds{R}$ (domain of $L$) and $\mu_0$ is an initial state, which is supposed to belong to the set $\mathcal{P}_{\rm exp}$ of sub-Poissonian probability measures on $\Gamma$. We prove that the Fokker-Planck equation has a unique solution $t\mapsto\mu_t$ which also belongs to $\mathcal{P}_{\rm exp}$. Some of the properties of this solution are also obtained. The second level description yields a Markov process such that its one dimensional marginals coincide with the mentioned states $\mu_t$. The process is obtained as the unique solution of the corresponding martingale problem.