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

Let $mathcalP$ be the collection of Borel probability measures on $mathbbR$, equipped with the weak topology, and let $mu:[0,1] ightarrowmathcalP$ 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 $((0,1),mathcalB(0,1),,$Lebesgue measure$)$, by egingather* Q_t(alpha)=inf,iglxinmathbbR:mu_t(-infty,x]gealphaiglquadquad extfor all tin [0,1] ext and alphain (0,1). endgather* A few open problems are discussed as well.