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The Markov-quantile process attached to a family of marginals

Let µ = (µt)t∈R be any 1-parameter family of probability measures on R. Its quantile process (Gt)t∈R : ]0, 1[ → RR, given by Gt(α) = inf{x ∈ R : µt(]−∞, x]) > α}, is not Markov in general. We modify it to build the Markov process we call “Markov-quantile”. 
 
We ﬁrst describe the discrete analogue: if (µn)n∈Z is a family of probability measures on R, a Markov process Y = (Yn)n∈Z such that Law(Yn) = µn is given by the data of its couplings from n to n + 1, i.e. Law((Yn, Yn+1)), and the process Y is the inhomogeneous Markov chain having those couplings as transitions. Therefore, there is a canonical Markov process with marginals µn and as similar as possible to the quantile process: the chain whose transitions are the quantile couplings. We show that an analogous process exists for a continuous parameter t: there is a unique Markov process X with the measures µt as marginals, and being a limit for the ﬁnite dimensional topology of quantile processes where the past is made independent of the future at ﬁnitely many times (many non-Markovian limits exist in general). The striking fact is that the construction requires no regularity for the family µ. We rely on order arguments, which seems to be completely new for the purpose. 
 
We also prove new results the Markov-quantile process yields in two contemporary frameworks: 
– In case µ is increasing for the stochastic order, X has increasing trajectories. This is an analogue of a result of Kellerer dealing with the convex order, peacocks and martingales. Modiﬁying Kellerer’s proof, we also prove simultaneously his result and ours in this case. 
– If µ is absolutely continuous in Wasserstein space P2(R) then X is solution of a Benamou–Brenier transport problem with marginals µt. It 
provides a Markov probabilistic representation of the continuity equation, unique in a certain sense.