I need to prove or disprove that for a stochastic process $(X_t)_{t \in [0,1]}$ with marginals $(\mu_t)_{t \in [0,1]}$, 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, 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 $t$?
Dongwei
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