I have an ODE system with one random parameter, $\alpha$. $X \in \mathbb{R^2}$ and is governed by $$\dot{X}=f(X,\alpha (t))$$ where $\alpha$ is random parameter the follows a Ornstein–Uhlenbeck process. That is $$d\alpha(t)=-\theta \alpha_t dt+\sigma dW(t)$$. Now OU process has a stationary distribution https://en.wikipedia.org/wiki/Ornstein%E2%80%93Uhlenbeck_process
I assume that that the ODE part of the system has a stable node. My question is , can i analyse the long run behavior of the system by looking at the behavior of the stable node under the stationary distribution. I know that I can easily write this system as a proper Ito stochastic differential equation and analyse it numerically, but will the other method be valid? Am I missing something here? Thanks in advance
