2024/11/25 by Yuri Kozitsky, Michael Röckner, Kozitsky, Yuri +1
Economics, Econometrics and Finance · Mathematics · #35Q84 #60G55 #60J25 #60J75 #Complex Systems and Time Series Analysis #FOS: Mathematics #Mathematical Biology Tumor Growth #Probability (math.PR) #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2411.16647
openalex publication_date 2024/11/25 · openalex created_date 2024/12/04 · openalex updated_date 2026/07/28
An infinite population of point entities dwelling in the habitat X=\mathdsRd 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,F,μ0) where F is an appropriate set of bounded test functions F:Γ→ \mathdsR (domain of L) and μ0 is an initial state, which is supposed to belong to the set P\rm exp of sub-Poissonian probability measures on Γ. We prove that the Fokker-Planck equation has a unique solution t↦μt which also belongs to 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 μt. The process is obtained as the unique solution of the corresponding martingale problem.