Sample paths of stochastic processes

Question

My teacher said, speaking about two Brownian motions, that if their finite dimensional distributions coincide does not mean that they are equal ${\omega}$ by ${\omega}$.
If so, what kind of "useful" information the finite dimensional distributions can give us, since, also when we have equality in distribution the paths can be really different?

I do not know if the question makes complete sense, maybe I have not well understood the meaning of "equality in distribution".

Answer 1

Forget Brownian motion for a moment. If $X_1,X_2$ are i.i.d. (say, standard Gaussian) then $X_1$ and $X_2$ are equal in distribution - that is, $\mathbb P(X_1 \in E) = \mathbb P(X_2 \in E)$ for every Borel set $E\subseteq\mathbb R$ - but they are certainly not equal at each sample point.

The same is true for stochastic processes. We can in fact have two independent Brownian motions, $\{B_1(t)\}_{t\ge0}$ and $\{B_2(t)\}_{t\ge0}$. Obviously, they cannot be equal at each sample point if they are independent (since, heuristically, information about one should not give you information about the other) but they are equal in distribution, which for stochastic processes is the same as all finite dimensional distributions being equal.