In "An introduction to stochastic differential equations" of Evans, he define a SDE as follow : Consider a ODE $$\begin{cases}\dot x(t)=b(x(t))\\ x(0)=x_0\end{cases} $$
We call $x(t)$ the state of the system at time $t$. In many application however, the experimentally measured trajectories of system modeled by ODE do not in fact behave as predicted: the observed state seems to more or less follow the trajectory predicted by the ODE, but is apparently subject also to random perturbation. Hence, it seems reasonable to modify the ODE somehow to include the possibility of random effect disturbing the system. A formal way to do it is : $$\begin{cases}\dot x(t)=b(x(t))+B(x(t))\xi(t)\\ x(0)=x_0\end{cases},$$ with $B$ linear.
Take for example $b=0$, $B(t)=t$. Then the solution is setting to be the brownian motion, i.e. $\dot W(t)=\xi(t)$.
I really don't get this point. Why do we get the brownian motion ?