Let µ = (µt) t∈R be any 1-parameter family of probability measures on R. Its quantile process (Gt) t∈R : ]0, 1[ → R R , 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.

On a (pseudo-)Riemannian manifold (M, g), some fields of endomorphisms i.e. sections of End(T M) may be parallel for g. They form an associative algebra e, which is also the commutant of the holonomy group of g. As any associative algebra, e is the sum of its radical and of a semi-simple algebra s. This s may be of eight different types, see [8]. Then, for any self adjoint nilpotent element N of the commutant of such an s in End(T M), the set of germs of metrics such that e ⊃ s ∪ {N } is non-empty. We parametrize it. Generically, the holonomy algebra of those metrics is the full commutant o(g) s∪{N } and then, apart from some "degenerate" cases, e = s ⊕ (N ), where (N ) is the ideal spanned by N . To prove it, we introduce an analogy with complex Differential Calculus, the ring R[X]/(X n ) replacing the field C. This treats the case where the radical of e is principal and consists of self adjoint elements. We add a glimpse on the case where this radical is not principal.

Abstract. On a (pseudo-)Riemannian manifold (M, g), some fields of endomorphisms i.e. sections of End(T M) may be parallel for g. They form an associative algebra e, which is also the commutant of the holonomy group of g. As any associative algebra, e is the sum of its radical and of a semi-simple algebra s. Here we study s: it may be of eight different types, including the generic type s = R Id, and the Kähler and hyperkähler types s ≃ C and s ≃ H. This is a result on real, semi-simple algebras with involution. For each type, the corresponding set of germs of metrics is non-empty; we parametrise it. We give the constraints imposed to the Ricci curvature by parallel endomorphism fields.Keywords: Pseudo-Riemannian, Kähler, hyperkähler, parakähler metrics, holonomy group, parallel endomorphism, nilpotent endomorphism, commutant, Ricci curvature, real algebra with involution, semi-simple associative algebra.M.S.C. 2010: 53B30, 53C29, 16K20, 16W10 secondary 53B35, 53C10, 53C12, 15A21.We classify here the germs of (pseudo-)Riemannian metric, after the semi-simple part of their algebra of parallel endomorphism fields. Our motivation is the following.Motivation. A Kähler metric g on some manifold M may be defined as a Riemannian metric admitting an almost complex structure J which is parallel: DJ = 0 with D the LeviCivita connection of g. A natural question is to ask whether other fields of endomorphisms, i.e. sections of End(T M), may be parallel for a Riemannian metric. The answer is nearly immediate. First, one restricts the study to metrics that do not split into a non trivial Riemannian product, called here "indecomposable". Otherwise, any parallel endomorphism field is the direct sum of parallel such fields on each factor (considering as a unique factor the possible flat factor). Then a brief reasoning ensures that only three cases occur: g may be generic i.e. admit only the homotheties as parallel endomorphisms, be Kähler, or be hyperkähler i.e. admit two (hence three) anticommuting parallel complex structures. The brevity of this list is due to a simple fact: the action of the holonomy group H of an indecomposable Riemannian metric is irreducible i.e. does not stabilise any proper subspace. In particular, this compels any parallel endomorphism field to be of the form λ Id +µJ with J some parallel, skew adjoint almost complex structure. Now, such irreducibility fails in general for an indecomposable pseudo-Riemannian metric, so that a miscellany of other parallel endomorphism fields may appear. This gives rise to the question tackled here:

We prove here, by geometric, or rather dynamical, methods, the following theorem. Let G be a non-compact connected Lie subgroup of the isometry group Isom(H n ) of the real hyperbolic space H n , which does not fix any point at infinity, i.e. on ∂H n