Random additive perturbation of a $k$ term recurrence relation

We are interested in stochastic processes satisfying a nonlinear recurrence relation of the form $$X_{n+k}={\mathrm{\Phi }}_0(X_n,\mathrm{.}\mathrm{.}\mathrm{.},X_{n+k-1})+{\mathrm{\Theta }}_n$$ where $\mathrm{\Theta }$ is a noise term. We establish the existence of an invariant measure for this process under given sufficient conditions on ${\mathrm{\Phi }}_0\mathrm{.}$