If a particle follows a Wiener process $W(t)$, in physics, we often write that the infinitesimal displacement is: $$dx\sim dW\sim\sqrt{dt}\mathcal{N}(0,1)$$
It makes sense if we treat $dt$ and $dx$ as infinitesimals, but can it be made rigourous when $dx$ is treated as a differential form? Does the square root of a differential form can be defined?