# On finite dimensional distributions

I have a question on distributions for some stoachastic processes.

Let $$(\Omega_i,\mathcal{F_i},P_i)$$, $$i=1,2$$, be probability spaces. Let $$X=(X_t)_{t \in [0,\infty)}$$ be an $$\mathbb{R}^d$$-valued stochastictic process defined on the probability space $$(\Omega_1,\mathcal{F_1},P_1)$$. The sample path of $$X$$ is continuous. Let $$B=(B_t)_{t \in [0,\infty)}$$ be a $$d$$-dimensional Brownian motion on $$(\Omega_2,\mathcal{F_2},P_2)$$.

We assume that for every $$t \ge 0$$ and bounded continuous function $$f \colon \mathbb{R}^d \to \mathbb{R}$$, \begin{align*} E_1[f(X_t)]=E_{2}[f(B_t)] \end{align*} Here, $$E_1$$ and $$E_2$$ denote the expectations under $$P_1$$ and $$P_2$$, respectively.

Then, can we show that the finite dimensional distributions of $$X$$ and $$B$$ are the same ? That is, for any $$m \in \mathbb{N}$$, $$0 \le t_1 \le t_2\le \cdots \le t_m$$, and any Borel subsets $$A_1,A_2,\ldots, A_m$$, \begin{align*} &P_1[X_{t_1} \in A_1,X_{t_2} \in A_2,\ldots, X_{t_m} \in A_m] \\ &=P_2[B_{t_1} \in A_1,B_{t_2} \in A_2,\ldots, B_{t_m} \in A_m]. \end{align*}

Take $$X_t=\sqrt{t}B_1$$. Then at any fixed point in time $$t$$, $$X_t=B_t$$ in distribution (normal distribution with mean 0 and variance $$t$$), but as processes $$X\neq B$$ in distribution since $$X$$ has no independent increments.
• You should take $X_t=\sqrt t B_1$. Commented Sep 23, 2021 at 23:13