We explore Itô stochastic differential equations where the drift term possibly depends on the infinite past. Assuming the existence of a Lyapunov function, we prove the existence of a stationary solution assuming only minimal continuity of the coefficients. Uniqueness of the stationary solution is proven if the dependence on the past decays sufficiently fast. The results of this paper are then applied to stochastically forced dissipative partial differential equations such as the stochastic Navier-Stokes equation and stochastic Ginsburg-Landau equation.
- Lyapunov functions
- Stationary solutions
- Stochastic Ginsburg-Landau equation
- Stochastic Navier-Stokes equation
- Stochastic differential equations
ASJC Scopus subject areas
- Applied Mathematics