# Indistinguishability processes vs finite dimensional distributions

I have been wondering about two concepts of "sameness" of two stochastic processes. Suppose that both stochastic processes $X= \{X(t), t \ge 0 \}$ and $Y= \{Y(t), t \ge 0 \}$ are defined on same probability space.

We say that $X$ and $Y$ Indistinguishable if $P(X(t)=Y(t), \forall t \ge 0) = 1.$ I know that Indistinguishability imples that they have same finite dimensional distributions.

My questions:

• Does the other implication also hold under some assumptions? For example for continuous processes?
• If I have two wienner processes $W_1$ and $W_2$ on same probability space, are they indistinguishable?

Let $W_1$ be a Wiener process and define $W_2(t):= -W_1(t)$. Then $W_2$ is again a Wiener process on the same space, but the two are certainly not indistinguishable! Note that $W_2$ also has the same finite-dimensional distributions as $W_1$.

The natural notion that is weaker than indistinguishability is one process being a modification of the other. This means that the two processes agree a.s. at any finite set of deterministic times. As a consequence they also agree a.s. at any countable set of deterministic times. Thus if they enjoy some pathwise regularity such as continuity or the cadlag property then they will also be indistinguishable.

Merely having the same finite dimensional distributions is a total non-starter; consider even just the case where the time set is a single point (so you just have one random variable) to see why.

• Dear Ian, could you explain or give a reference on why if two process have the same finite dimensional distributions then they agree on a countable set of deterministic times as well? Thank you Commented Apr 28, 2022 at 13:03
• @Sarah It is just $P \left ( \bigcap_{i=1}^n \{ \omega : X_{t_i}=Y_{t_i} \} \right )=1 \: \forall n \Rightarrow P \left ( \bigcap_{i=1}^\infty \{ \omega : X_{t_i}=Y_{t_i} \} \right )=1$ from continuity of measure.
– Ian
Commented Apr 28, 2022 at 13:17
• Oh... Thanks a lot! Commented Apr 28, 2022 at 13:34
• @Sarah Sorry, to be clear, what I aid is that if two processes are equal a.s. at any finite set of deterministic times, then they are also equal a.s. at any countable set of deterministic times. Having the same finite dimensional distributions is much weaker than these hypotheses.
– Ian
Commented Apr 28, 2022 at 15:04
• Dear Ian, if two processes are equal a.s. at any finite set of deterministic sets then they have the same finite dimensional distributions, or not? Commented Apr 28, 2022 at 15:14