Problem arising from martingale solutions to SPDE: $Law(u)=Law(v)$ on $C([0,T]; X)$, can $Law(u)=Law(v)$ on $C([0,t]; X)$ for $t<T$?

Question

I ask this question because I found in some papers of martingale solutions to SPDE, to prove the approximate solutions $u_n$ is a convergent sequence, one can use "stochastic compact" method to find another sequence $v_n$ (on another probability space), which has the same distribution as the original approximate solutions $u_n$, like (1) below. Moreover, $v_n$ is a convergent sequence. One can prove the new sequence $v_n$ also satisfies the approximate scheme and then take limit to get a martingale solution $v$ to the target SPDE. But how can we know the initial distribution of $v(0)$ is the same as the given initial distribution? For example, does (3) below hold true particularly for $t=0$???. Many papers on different models omit this, so it seems very easy. Maybe I am too stupid to know why. If necessary, I can give the references to focus on a specific example. I also want to know wether we can have (2).

Thanks in advance!!!

Let $(\Omega_i, \mathcal{F}^i,\mathbb{P}_i,\{\mathcal{F}^i_t\}_{t\geq0})$ $i=1,2$ be two probability space. Let $T>0$. Let $u:\Omega_1\times[0,T]\mapsto X$ and $v:\Omega_2\times[0,T]\mapsto X$ be two $X$-valued process such that $u\in L^2\left(\Omega_1;C([0,T];X)\right)$ and $v\in L^2\left(\Omega_2;C([0,T];X)\right)$.

Assume that $$\mathbb{P}_1(u\in A)=\mathbb{P}_2(v\in A)\ \ \ \forall A\in\mathcal{B}(C([0,T];X)),\ \ \ (1)$$ where $\mathcal{B}(C([0,T];X))$ denotes the Borel sets of $C([0,T];X)$.

My questions are:

(a) For any $t\in[0,T)$, can we have $$\mathbb{P}_2(u\in A)=\mathbb{P}_2(v\in A)\ \ \ \forall A\in\mathcal{B}(C([0,t];X))\ \ ???\ \ \ (2)$$

(b) For any $t\in[0,T]$, can we have $$\mathbb{P}_1(u(t)\in A)=\mathbb{P}_2(v(t)\in A)\ \ \ \forall A\in\mathcal{B}(X)\ \ ???\ \ \ (3)$$

Answer 1

Yes, both statements are true. For (a), note that for every Borel set $A$ of $C([0,t],X)$, the set $\{u \in C([0,T], X) : u|_{[0,t]} \in A\}$ is a Borel set in $C([0,T],X)$. This follows immediately from the fact that the restriction map from $C([0,T],X)$ to $C([0,t], X)$ is continuous and therefore Borel.

Likewise for (b), use the fact that the evaluation map $u \mapsto u(t)$ is a continuous map from $C([0,T],X)$ to $X$.