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 in 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 term, which usually produces essential difficulties in its study. The population pure states are locally finite counting measures defined on $X$. The set of such states $\Gamma$ is equipped with the vague topology, which allows one to use probability measures defined thereon. The population evolution is described at two levels. At the first level, one deals 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 belongs to $\mathcal{P}_{\rm exp}$. Some of the properties of this solution are also described. The second-level description yields a Markov process with cadlag paths 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. The results obtained are discussed and compared with those known for similar models with logistic-type interactions.