I need to prove or disprove that for a stochastic process $(X\_t)\_{t \in [0,1]}$ with marginals $(\mu\_t)\_{t \in [0,1]}$ on $\mathbb{R}$, if the sample paths of $(X\_t)\_{t \in [0,1]}$ are continuous, then $(\mu\_t)\_{t \in [0,1]}$ is weakly continuous in $t$. I see a similar question [Continuity of the densities of a stochastic process](https://mathoverflow.net/questions/370159/continuity-of-the-densities-of-a-stochastic-process?noredirect=1&lq=1&newreg=56800865f773464bb9d45240cac0863a), but it's for the discontinuity of densities of $(X\_t)\_{t \in [0,1]}$. Any help is greatly appreciated.



For stochastic process $X_t$ with marginals $\mu_t$, is it true that the sample-path continuity of $X_t$ implies $\mu_t$ is weakly continuous in $t$?
Yes, under mild assumptions.


If the state space $E$ is Polish (including $E = \mathbb{R}^n$ in particular), then the space $\mathcal{P}(E)$ of Borel probability measures on $E$, with the weak topology, is metrizable, and so it suffices to show weak sequential continuity. That is, for every sequence of times $t\_n \to t$, that we have $\mu\_{t\_n} \to \mu\_t$ weakly.


Let $f : E \to \mathbb{R}$ be continuous and bounded. We must show $\int\_E f\,d\mu\_{t\_n} \to \int\_E f\,d\mu\_t$, which is to say that $\mathbb{E}[f(X\_{t\_n})] \to \mathbb{E}[f(X\_t)]$. Now we have $X\_{t\_n} \to X\_t$ a.s., by sample path continuity. Hence $f(X\_{t\_n}) \to f(X\_t)$ a.s., because $f$ is continuous. Hence
$\mathbb{E}[f(X\_{t\_n})] \to \mathbb{E}[f(X\_t)]$ by dominated convergence, because $f$ is bounded.


