For any \(\alpha\in(0,1)\), the \(\alpha\)-quantile of a measure \(\mu\) on \(\mathbb{R}\) is the infimum number \(x\) such that \(\mu((-\infty,x])\geq \alpha\). One associates to a family \((\mu_t)_{t\in\mathbb{R}}\) of measures on \(\mathbb{R}\) the process of their quantiles, defined on the interval \((0,1)\) with the Lebesgue measure, with chance element \(\alpha\). (One talks of the quantile process associated with the family \((\mu_t)_{t\in\mathbb{R}}\).) The authors prove that there is a unique Markovian process on the probability space \((0,1)\) that has \(\mu_t\) as its one dimensional marge and whose finite dimensional marges are given by a twicked version of the quantile process whose past and future at a finite number of times are made independent of one another. This process is called ‘Markov quantile’. It is proved to be almost surely increasing if the family \((\mu_t)_{t\in\mathbb{R}}\) is increasing for the stochastic order.