Ergodicity for Functional Stochastic Equations Without Dissipativity

Abstract

Dealing with diffusions with “pure delay” in which both the drift and the diffusion coefficients depend only on the arguments with delays, most of the existing results are not applicable. This chapter uses variation-of-constants formulae to overcome the difficulties due to the lack of the information at the current time, and establishes existence and uniqueness of invariant measures for functional stochastic equations that need not satisfy dissipative conditions.