The stochastic evolution of a general contact model

The evolution of states of a self-propelled population is studied. The population dwells in a locally compact Polish space X, and its pure states are locally finite counting measures on X. The set of all such states $$\varGamma $$ is equipped with the vague topology that makes it a Polish space as well; mixed states are the probability measures $$\mu $$ defined thereon. Let the integer-valued random variable $$Z_{\mu ,\varLambda }$$ , $$\varLambda \subset X$$ , be the number of population members contained in $$\varLambda $$ in state $$\mu $$ . The evolution consists in random appearance of new members attracted by the existing population and in their independent disappearance. It is specified by the Fokker–Planck equation in which the model is represented by the Kolmogorov operator L and its domain $${\mathcal {F}}$$ . Let $${\mathcal {P}}_*$$ be the set of all measures $$\mu $$ for which the factorial moments of $$Z_{\mu ,\varLambda }$$ satisfy $$\phi _k (Z_{\mu ,\varLambda }) \le C_\mu k! b_{\mu ,\varLambda }^k$$ with $$b_{\mu ,\varLambda }<+\infty $$ for compact $$\varLambda $$ . The result of the paper is the statement that the Fokker–Planck equation with $$\mu _0\in {\mathcal {P}}_*$$ and the domain $${\mathcal {F}}$$ presented in the paper has a unique global solution $$\mu _t \in {\mathcal {P}}_*$$ . This statement is proved, discussed and compared with the results known for similar models.