Existence And Uniqueness Of Stationary Solution Of Nonlinear Stochastic Differential Equation With Memory

In this paper a stochastic differential equation (SDE) with infinite memory is considered. The drift coefficient of the equation is a nonlinear functional of the past history of the solution. Sufficient conditions for existence and uniqueness of stationary solution are given. This work is motivated by recent papers [1] and [2] where stochastically forced nonlinear equations of hydrodynamics were considered and it was shown how the infinitedimensional stochastic Markovian dynamics related to these equations can be reduced to finite-dimensional stochastic dynamics. The corresponding finite-dimensional systems are however essentially non-Markovian. So, the important problem of existence and uniqueness of stationary solutions for stochastic hydrodynamical equations is tightly related to existence and uniqueness of stationary solutions of SDEs with infinite memory. Some results in the area were established in [3]. In the first part of this paper some necessary notions are introduced and the main result is stated. A proof of the main result is given in the second part. We combine the approach of [3] with an interesting method for establishing the desired uniqueness suggested in [1] and [2] for the problems considered therein. The equation under consideration is