Stochastic functional differential equations with infinite delay under non-Lipschitz coefficients: Existence and uniqueness, Markov property, ergodicity, and asymptotic log-Harnack inequality

Abstract

This paper focuses on a class of stochastic functional differential equations with infinite delay and non-Lipschitz coefficients. Under one-sided super-linear growth and non-Lipschitz conditions, this paper establishes the existence and uniqueness of strong solutions and strong Markov properties of the segment processes. Under additional assumption on non-degeneracy of the diffusion coefficient, exponential ergodicity for the segment process is derived by using asymptotic coupling method. In addition, the asymptotic log-Harnack inequality is established for the associated Markovian semigroup by using coupling and change of measures, which implies the asymptotically strong Feller property. Finally, an example is given to demonstrate these results.