Stationary distribution of stochastic population dynamics with infinite delay
In this paper, the authors consider the stationary distribution of a stochastic Lotka-Volterra model with infinite delay. The authors use the variable substituting technique to obtain a higher-dimensional stochastic differential equation without delay and then show that the new equation satisfies the strong Feller property and strong Markov property, by which the invariant measure follows from the Krylov-Bogoliubov Theorem. Since distribution of the original model is actually the marginal distribution of the new equation, these results also show that there exists the stationary distribution for the original equation. Furthermore, when the noise intensity is strongly dependent on the population size, the authors also show that the stationary distribution of this stochastic Lotka-Volterra system with infinite delay holds a smooth density by using Hörmander's theorem.
stochastic functional differential equation
scientific article; zbMATH DE number 7604814
scientific article