I have a question regarding the characterization of the differences between the deterministic ODE system versus its stochastic counterpart. For the sake of concreteness, suppose i have a deterministic autonomous system. Let $x\in \mathbb{R^d}$ $$\dot{x(t)}=f(x)$$
Now lets us suppose that we introduce a little bit of noise in the system, and say that the stochastic counter part is $$dx(t)=f(x)dt+\sigma dW(t)$$ where $W(t)$ is d-dimensional independent Wiener processes. I am new to this subject and I would like to find some vocabulary to describe the difference in behavior of the two systems.
What i know till now: The second system can be solved by Euler-Muruyama discretization, and I could perhaps compare the trajectories for different values of $\sigma$. Also for some linear systems, we can expect to find the stationary distribution from the Fokker-Plank equation. (I have not found any results for nonlinear drift function) . Is there any other approach to studying the properties?