On the existence of continuous processes with given one-dimensional distributions

Let $mathcal{P}$ be the collection of Borel probability measures on $mathbb{R}$, equipped with the weak topology, and let $mu:[0,1] ightarrowmathcal{P}$ be a continuous map. Say that $mu$ is presentable if $X_tsimmu_t$, $tin [0,1]$, for some real process $X$ with continuous paths. It may be that $mu$ fails to be presentable. Hence, firstly, conditions for presentability are given. For instance, $mu$ is presentable if $mu_t$ is supported by an interval (possibly, by a singleton) for all but countably many $t$. Secondly, assuming $mu$ presentable, we investigate whether the quantile process $Q$ induced by $mu$ has continuous paths. The latter is defined, on the probability space $((0,1),mathcal{B}(0,1),,$Lebesgue measure$)$, by egin{gather*} Q_t(alpha)=inf,igl{xinmathbb{R}:mu_t(-infty,x]gealphaigl}quadquad ext{for all }tin [0,1] ext{ and }alphain (0,1). end{gather*} A few open problems are discussed as well.