The stochastic evolution of an infinite population with logistic-type interaction.
A talk in the Bielefeld Stochastic Afternoon series by
Jurij Kozicki from Uniwersytet Marii Curie-Sklodowskiej
| Abstract: | The Markov evolution of an infinite population of point entities dwelling in X=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. 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 is equipped with the vague topology, which allows one to use probability measures as population states. The population evolution is described at two levels. First we prove that the Fokker-Planck equation with a specially selected domain of L and the initial state taken from the set of sub-Poissonian measures
has a unique solution which lies in the mentioned set of measures. Some of the properties of this solution are also obtained. The second level description yields a (measure valued) Markov process such that its one-dimensional marginals coincide with the corresponding values of the mentioned solution. The process is obtained as the unique solution of the martingale problem for L, and its construction and uniqueness are essentially based on the results from the first part.
Within the CRC this talk is associated to the project(s): A5, B1 |