How to describe stochastic steady states?

Question

Imagine I have the following system of ODEs, with $x\equiv x(t),y\equiv y(t)$, $$ \dot{x}=f(x,y)\\ \dot{y}=g(x,y) $$ If $f$ and $g$ are not stochastic, having the system converge into a steady state solution satisfies $\dot{x}=\dot{y}=0$. In other words, $$ \lim_{t\to\infty}|x(t)-x(t+\Delta t)|=0\\ \lim_{t\to\infty}|y(t)-y(t+\Delta t)|=0 $$ With $f$ and $g$ stochastic in some way, however, I would need to write something like $$ \lim_{t\to\infty}|x(t)-x(t+\Delta t)|<\xi_x\\ \lim_{t\to\infty}|y(t)-y(t+\Delta t)|<\xi_y $$ where $\xi_x,\xi_y$ are such that no stochastic phenomena would change the final state of the system to some "error". But this seems poorly written and I would like to present it more formally. I'm fairly new to dynamic systems, so do you have any ideas or suggestions?