Let $\Big(X_n(t)\Big)_{n\in \mathbb N}$ be a sequence of stochastic process in $[0,1]$ with probability density $u_n(x, t)$ with $(x, t)\in [0,1]\times \mathbb R^+$ i.e.
$$P(X_n(t)\in A)=\int_Au(x, t)dx\qquad\forall A\subset [0,1].$$
I can prove that
$$u_n\xrightarrow[n\to+\infty]{w}u,$$ where with $\xrightarrow[]{w}$ we mean that for every $f\in L^2([0,1]\times \mathbb R^+)$ and for every $t\in \mathbb R^+$ it holds that $$\int_0^t\int_0^1u_n(x, t)f(x, t)dxdt\xrightarrow[n\to+\infty]{}\int_0^t\int_0^1u(x, t)f(x, t)dxdt.$$
I also know (by a tightness argument) that the sequence of processes $X_n$ converges in distribution to a process $X_*\in D$ where $D:=D(\mathbb R^+, [0,1])$ is the space of all trajectories cadlag defined in $\mathbb R^+$ and taking value in $[0,1]$.
My question is the following. There is a way to conclude, only by these informations, that the limit function $u$ is the probability density of the limit process $X_*$?
Thank you very much to anyone might help me!