I'm interested in understanding the properties of the long term behaviour of a system of stochastic differential equations. For example, given a coupled system \begin{align} d x&=f(x,y)dt+\phi(x,y)dW_1\\ d y&=g(x,y)dt+\sigma(x,y)dW_2 \end{align} where $W_1$ and $W_2$ are Wiener processes, what can be said about the long term behaviour of the errors (or stability) \begin{align} \lim_{t\to\infty}|x(t+\Delta t)-x(t)|\\ \lim_{t\to\infty}|y(t+\Delta t)-y(t)| \end{align} I am not quite sure where to start, so I'd be glad to have some suggestions.
My approach: Assume $(x^*,y^*)$ represents the steady state solution of the deterministic system. In the mean square, $\Delta W= \Delta t^\frac12\xi$, where $\xi$ represents white noise. For large $t$ and without loss of generality, we take the infinitesimal difference of $x$ \begin{align*} x(t+\Delta t)-x(t)&=f(x^*+\delta x,y^*+\delta y)\Delta t+\phi(x^*+\delta x,y^*+\delta y)\Delta t^\frac12\xi_1 \end{align*} First-order Taylor expanding around $(\delta x,\delta y)$ leads to the vanishing of the terms involving $f$ and thus \begin{align*} x(t+\Delta t)-x(t)&=\phi(x^*,y^*)\Delta t^\frac12\xi_1+\frac{\partial}{\partial x}\phi(x^*,y^*)\delta x\Delta t^\frac12\xi_1+\frac{\partial}{\partial y}\phi(x^*,y^*)\delta y\Delta t^\frac12\xi_1 \end{align*} excluding higher order terms. Would it be possible to simplify or estimate this further? My ultimate goal is to get an upper bound for the error for large $t$.
Any way of justifying, for example, taking the large $t$ estimate $$ \frac{x(t+\Delta t)-x(t)}{\Delta t^\frac12}\approx \phi(x^*,y^*)\xi_1 $$