The stochastic evolution of an infinite population with logistic-type interaction
Yuri Kozitsky, Michael Röckner
TL;DR
The paper analyzes an infinite spatial population with logistic-type interaction, introducing a Kolmogorov generator $L=L^{+}+L^{-}$ that includes a challenging quadratic death term. The authors establish a unique evolution $t\mapsto \mu_t$ within the class of sub-Poissonian measures $\mathcal{P}_{\rm exp}$ by solving a Fokker-Planck equation $(L,\mathcal{F},\mu_0)$ and then construct a cadlag Markov process on tempered configurations via restricted martingale problems, using auxiliary models and McKean-type correlation-function analysis. The approach ensures absence of clustering (no heavy tails) and provides a robust framework that connects to Bolker-Pacala-type models and prior stochastic-geometry results. The results yield both analytic control over the state evolution and a probabilistic pathwise representation, enabling broader applicability to spatially structured populations with density-dependent regulation. Overall, the work advances the rigorous construction of infinite-population Markov dynamics with logistic-type interactions and lays groundwork for extensions to related spatial-lattice and dispersal-competition models.
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 $Γ$ is equipped with the vague topology and thus with the corresponding Borel $σ$-field. The population evolution is described at two levels. At the first level, we deal with the Fokker-Planck equation for $(L,\mathcal{F},μ_0)$ where $\mathcal{F}$ is an appropriate set of bounded test functions $F:Γ\to \mathds{R}$ (domain of $L$) and $μ_0$ is an initial state, which is supposed to belong to the set $\mathcal{P}_{\rm exp}$ of sub-Poissonian probability measures on $Γ$. We prove that the Fokker-Planck equation has a unique solution $t\mapstoμ_t$ 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 $μ_t$. The process is obtained as the unique solution of the corresponding martingale problem.