Large deviation principle for the stationary solutions of stochastic functional differential equations with infinite delay

We investigate the large deviation principle (LDP) of the stationary solutions of stochastic functional differential equations (SFDEs) with infinite delay under small random perturbation. First, we demonstrate the existence and uniqueness of the corresponding stationary solutions. Second, by the weak convergence approach, we show the uniform large deviation principle for the solution maps, and then prove the LDP for stationary solutions. Furthermore, we obtain the LDP for invariant measures of SFDEs through the LDP for stationary solutions and the contraction principle.