How to prove two stochastic processes have the same distribution

Question

Let $C([0,\infty), R)$ be the canonical space of continuous functions. Assume $(\Omega, \mathcal{F}, \{\mathcal{F}_{t}\}_{t\geq 0})$ is a measurable space with a filtration. Let $P, Q$ be two probability measures on $(\Omega, \mathcal{F})$. Assume $X_{t}$ and $Y_{t}$ are two stochastic processes adapted to $\{\mathcal{F}_{t}\}_{t\geq 0}$. If for any borel set $A$ and $t$, $$ P(X_{t}\in A)= Q(Y_{t}\in A) $$ Can we conclude that the law on $C([0,\infty), R)$ induced by $(X_{t}, P)$ is the same as that of $(Y_{t}, Q)$? Any references are very appreciated.

Answer 1

If only the marginals match it is not true, here is a nice counter example the fake Brownian Motion

Best regards

Answer 2

No. Try $(X_t)$ standard Brownian motion and $Y_t=\sqrt{t}\cdot Y_1$ for every $t$, where $Y_1$ is standard normal.