In this thesis, we study the existence of stationary solutions for two cases. One
is for random difference equations. For this, we prove the existence and uniqueness
of the stationary solutions in a finite-dimensional Euclidean space Rd by applying the
coupling method. The other one is for semi linear stochastic evolution equations. For
this case, we follows Mohammed, Zhang and Zhao 's work. In an infinite-dimensional
Hilbert space H, we release the Lipschitz constant restriction by using Arzela-Ascoli
compactness argument. And we also weaken the globally bounded condition for F by
applying forward and backward Gronwall inequality and coupling method.
A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University.