Jean Bérard

Brieuc Frénais

THE COMONOTONE FLOW OF A STOCHASTICALLY MONOTONE FELLER PROCESS ON THE REAL LINE

We show that any stochastically monotone Feller semigroup on ℝ can be extended by a consistent family of order-preserving Feller semigroups on the successive powers of ℝ. We exhibit a specific such family, which is uniquely characterized by a maximality property with respect to the super-modular order on ℝ 𝑛 .

INTRODUCTION

We start by recalling various necessary definitions and properties in Subsections 1.1 to 1.4, then state our main results in Subsection 1.5 and discuss them in Subsection 1.6.

1.1. Markov kernels. A Markov kernel 𝐾 on a measurable space (𝑆, 𝒮 ) is a map from 𝑆 × 𝒮 to ℝ such that (i) for all 𝑥 ∈ 𝑆, 𝐾(𝑥, ⋅) is a probability measure on (𝑆, 𝒮 );

(ii) for all 𝐵 ∈ 𝒮 , 𝐾(⋅, 𝐵) is a measurable real-valued function on 𝑆. When there is no ambiguity regarding the choice of the 𝜎-algebra 𝒮 , we simply say that 𝐾 is a Markov kernel on 𝑆. Given a bounded measurable real-valued function 𝑓 on 𝑆, we define a bounded measurable real-valued function 𝐾𝑓 on 𝑆 by 𝐾𝑓 (𝑥) = ∫ 𝑆 𝑓 (𝑦)𝑑𝐾(𝑥, 𝑦). Denoting by  𝑏 (𝑆) the vector space of bounded real-valued measurable functions on 𝑆, equipped with the sup-norm ‖𝑓 ‖ ∞ = sup 𝑥∈𝑆 |𝑓 (𝑥)|, the map 𝑓 ↦ 𝐾𝑓 defines a linear operator from  𝑏 (𝑆) into itself, and satisfies (𝛼) ‖𝐾𝑓 ‖ ∞ ≤ ‖𝑓 ‖ ∞ ;

(𝛽) 𝐾𝑓 ≥ 0 when 𝑓 ≥ 0;

(𝛾) 𝐾𝟏 = 𝟏 (where 𝟏 denotes the constant fonction equal to 1).

1.2. Markov and Feller semigroups. Given two Markov kernels 𝐾, 𝐿 on 𝑆, the composition of the two kernels is yet another Markov kernel 𝐾𝐿 defined by (𝐾𝐿)(𝑥, 𝐵) = ∫ 𝑆 𝐿(𝑦, 𝐵)𝑑𝐾(𝑥, 𝑦). The composition is an associative (but in general non-commutative) operation on Markov kernels. Moreover, for 𝑓 ∈  𝑏 (𝑆), we have that (𝐾𝐿)𝑓 = 𝐾(𝐿𝑓 ). We say that a family of Markov kernels (𝐾 𝑡 ) 𝑡∈ℝ + on 𝑆 is a Markov semigroup if (I) for all 𝑥 ∈ 𝑆 and 𝐵 ∈ 𝒮 , 𝐾 0 (𝑥, 𝐵) = 𝛿 𝑥 (𝐵);

(II) the Chapman-Kolmogorov equation holds: for all 𝑠, 𝑡 ≥ 0, 𝐾 𝑠+𝑡 = 𝐾 𝑠 𝐾 𝑡 . Moreover, we say that a family (𝑋 𝑡 ) 𝑡∈ℝ + of 𝑆-valued random variables defined on the same probability space (Ω,  , ℙ) is a Markov process governed by the Markov semigroup (𝐾 𝑡 ) 𝑡∈ℝ + when, for all 𝑠, 𝑡 ∈ ℝ + , and all 𝐵 ∈ 𝒮 , one has 𝔼 ( 𝟏 𝐵 (𝑋 𝑠+𝑡 )|𝜎(𝑋 𝑢 ; 𝑢 ∈ [0, 𝑠]) ) a.s. = 𝐾 𝑡 (𝑋 𝑠 , 𝐵).

Now assume that 𝑆 is a locally compact separable metric space (abbreviated lcsm in the sequel), equipped with the corresponding Borel 𝜎-algebra. We denote by  0 (𝑆) the vector space of continuous real-valued functions on 𝑆 vanishing at infinity. Note that  0 (𝑆) equipped with the sup-norm is a Banach space.

We say that a Markov semigroup (𝐾 𝑡 ) 𝑡∈ℝ + on 𝑆 enjoys the Feller property (or, more succintly, that it is a Feller semigroup) when:

(Fa) ∀𝑡 ≥ 0, ∀𝑓 ∈  0 (𝑆), 𝐾 𝑡 𝑓 ∈  0 (𝑆);

(Fb) ∀𝑓 ∈  0 (𝑆), lim 𝑡→0+ ‖ ‖ 𝐾 𝑡 𝑓 -𝑓 ‖ ‖∞ = 0.

Let us now denote by  𝑏 (𝑆) the vector space of bounded continuous real-valued functions on 𝑆, and observe that property (Fa) implies 1 , but is not in general equivalent2 to (Fa') ∀𝑡 ≥ 0, ∀𝑓 ∈  𝑏 (𝑆), 𝐾 𝑡 𝑓 ∈  𝑏 (𝑆). Moreover, (Fb) may be replaced by the apparently weaker assumption of pointwise (instead of uniform) convergence (Fb') ∀𝑓 ∈  0 (𝑆), ∀𝑥 ∈ 𝑆, lim 𝑡→0+ 𝐾 𝑡 𝑓 (𝑥) = 𝑓 (𝑥), but it turns out 3 that a Markov semigroup satisfying (Fa) and (Fb') also satisfies (Fb).

1.3. Consistent families. For every integer 𝑛 ≥ 2, equip the product space 𝑆 𝑛 with the product 𝜎-algebra 𝒮 ⊗𝑛 . Given integers 1 ≤ 𝑘 ≤ 𝑛, and 𝑖 1 , … , 𝑖 𝑘 ∈ 1, 𝑛 , we denote by 𝜋 𝑛 𝑖 1 ,…,𝑖 𝑘 the projection from 𝑆 𝑛 to 𝑆 𝑘 defined by 𝜋 𝑛 𝑖 1 ,…,𝑖 𝑘 (𝑥 1 , … , 𝑥 𝑛 ) = (𝑥 𝑖 1 , … , 𝑥 𝑖 𝑘 ). We say that a family of Markov kernels (𝐾 (𝑛) ) 𝑛≥1 , where, for each integer 𝑛, 𝐾 (𝑛) is a Markov kernel on 𝑆 𝑛 , is a consistent family if, for all 1 ≤ 𝑘 ≤ 𝑛, all 𝑖 1 , … , 𝑖 𝑘 ∈ 1, 𝑛 , all 𝐱 ∈ 𝑆 𝑛 and all measurable subset 𝐵 of 𝑆 𝑘 , ( 1)

𝐾 (𝑛) ( 𝐱, (𝜋 𝑛 𝑖 1 ,…,𝑖 𝑘 ) -1 (𝐵) ) = 𝐾 (𝑘) ( 𝜋 𝑛 𝑖 1 ,…,𝑖 𝑘 (𝐱), 𝐵 ) .
We say that a family of Markov kernels (𝐾 (𝑛) ) 𝑛≥2 is a consistent extension of a Markov kernel 𝐾 on 𝑆, if the family (𝐾 (𝑛) ) 𝑛≥1 , with 𝐾 (1) = 𝐾, is a consistent family. A family of Markov semigroups

( (𝐾 (𝑛) 𝑡 ) 𝑡∈ℝ + ) 𝑛≥1
is said to be consistent if, for every 𝑡 ∈ ℝ + , (𝐾 (𝑛) 𝑡 ) 𝑛≥1 is a consistent family of Markov kernels; a consistent extension of a Markov semigroup on 𝑆 is defined accordingly.

Note that equal indices are allowed in [START_REF] Anderson | Continuous-time Markov chains[END_REF], which implies in particular that, for all 𝑥 ∈ 𝑆, we have

(2) 𝐾 (2) ((𝑥, 𝑥), {(𝑦, 𝑦); 𝑦 ∈ 𝑆}) = 1.

1.4. Stochastic orders. Now assume that 𝑆 = ℝ, and denote by  ↗ 𝑏 (ℝ) the set of bounded nondecreasing real-valued functions on ℝ. The usual stochastic order 𝜇 ≤ 𝐬𝐭 𝜈 between (Borel) probability measures on ℝ (see [START_REF] Müller | Comparison methods for stochastic models and risks[END_REF]) is defined by the fact that, for all 𝑓 ∈  ↗ 𝑏 (ℝ), one has ∫ ℝ 𝑓 (𝑥)𝑑𝜇(𝑥) ≤ ∫ ℝ 𝑓 (𝑥)𝑑𝜈(𝑥). We say that a Markov semigroup (𝐾 𝑡 ) 𝑡∈ℝ + on ℝ is stochastically monotone when:

(M) ∀𝑡 ≥ 0, ∀𝑓 ∈  ↗ 𝑏 (ℝ), 𝐾 𝑡 𝑓 ∈  ↗ 𝑏 (ℝ). An immediately equivalent formulation of (M) in terms of ≤ 𝐬𝐭 is that (3) ∀𝑥, 𝑦 ∈ ℝ, 𝑥 ≤ 𝑦 ⇒ 𝐾 𝑡 (𝑥, ⋅) ≤ 𝐬𝐭 𝐾 𝑡 (𝑦, ⋅).
Given 𝐱 = (𝑥 1 , … , 𝑥 𝑛 ) ∈ ℝ 𝑛 and 𝐲 = (𝑦 1 , … , 𝑦 𝑛 ) ∈ ℝ 𝑛 , we say that 𝐲 is order-compatible with 𝐱 if, for all 1 ≤ 𝑖, 𝑗 ≤ 𝑛, 𝑥 𝑖 ≤ 𝑥 𝑗 ⇒ 𝑦 𝑖 ≤ 𝑦 𝑗 . We then denote ℝ 𝑛 𝐱 = {𝐲 ∈ ℝ 𝑛 such that 𝐲 is order-compatible with 𝐱}, and we say that a Markov kernel 𝐾 (𝑛) on ℝ 𝑛 is order-preserving when, for all 𝐱 ∈ ℝ 𝑛 , we have that

𝐾 (𝑛) ( 𝐱, ℝ 𝑛 𝐱 ) = 1. Denote by  𝑏 (ℝ 𝑛 ) the set of real-valued bounded Borel super-modular functions on ℝ 𝑛 , i.e. bounded Borel functions 𝑓 ∶ ℝ 𝑛 → ℝ such that, for all 𝐱, 𝐲 ∈ ℝ 𝑛 , 𝑓 (𝐱) + 𝑓 (𝐲) ≤ 𝑓 (𝐱 ∨ 𝐲) + 𝑓 (𝐱 ∧ 𝐲), where, for 𝐱 = (𝑥 1 , … , 𝑥 𝑛 ) and 𝐲 = (𝑦 1 , … , 𝑦 𝑛 ), we set 𝐱 ∨ 𝐲 = (max(𝑥 1 , 𝑦 1 ), … , max(𝑥 𝑛 , 𝑦 𝑛 )) and 𝐱 ∧ 𝐲 = (min(𝑥 1 , 𝑦 1 ), … , min(𝑥 𝑛 , 𝑦 𝑛 )).
Given 𝑛 ≥ 2 and two (Borel) probability measures 𝜇, 𝜈 on ℝ 𝑛 , we say that 𝜇 ≤ 𝐬𝐦 𝜈 (see e.g. [START_REF] Müller | Comparison methods for stochastic models and risks[END_REF]) when

(4) ∀𝑓 ∈  𝑏 (ℝ 𝑛 ), ∫ ℝ 𝑛 𝑓 (𝐱)𝑑𝜇(𝐱) ≤ ∫ ℝ 𝑛 𝑓 (𝐱)𝑑𝜈(𝐱).
Note that ≤ 𝐬𝐦 defines a partial order on the set of probability measures on ℝ 𝑛 , and that two probability measures that are comparable with respect to ≤ 𝐬𝐦 must have the same marginal distributions. ) 𝑡∈ℝ + is itself a Markov process on ℝ 𝑛 governed by a Feller Markov semigroup comes as an important additional property (see [START_REF] Bérard | Hydrodynamic limit of N-branching Markov processes[END_REF], which was indeed our original motivation for investigating the present question).

Results of this kind have been established in a variety of contexts, including 7 discrete-time Markov chains on partially ordered Polish spaces ( [START_REF] Kamae | Stochastic inequalities on partially ordered spaces[END_REF]), continuous-time Markov chains on countable partially ordered spaces ( [START_REF] López | Stochastic domination and Markovian couplings[END_REF]), jump processes on partially ordered Polish spaces ( [START_REF] Zhang | Existence and application of optimal Markovian coupling with respect to non-negative lower semicontinuous functions[END_REF]), interacting particle systems on {0, 1} ℤ 𝑑 ( [START_REF] Thomas | Interacting particle systems[END_REF]).

In the setting of continuous-time Markov processes on the real line, let us mention two important special cases for which the conclusion of Theorem 1 can be established relatively easily:

• Lévy processes (see e.g. [START_REF] Bertoin | Lévy processes[END_REF]): using the parallel coupling, obtained by setting, for all 𝑥 ∈ ℝ and 𝑡 ≥ 0, 𝑋 𝑥 𝑡 = 𝑥 + 𝑋 0 𝑡 , where (𝑋 0 𝑡 ) 𝑡≥0 is a version of the Lévy process starting at 0, see Figure 1a; • Feller processes with continuous paths: one defines 𝑃 (𝑛) 𝑡 as the distribution at time 𝑡 of a family 8 of 𝑛 trajectories that evolve independently until they meet, and stick together thereafter (this is 4 A priori, this may not be unique, but it turns out that it is in view of the present statement. 5 Strassen's theorem is valid in the general context of probability measures on Polish spaces. See [START_REF] Strassen | The existence of probability measures with given marginals[END_REF] for Strassen's original paper, and [START_REF] Lindvall | On Strassen's theorem on stochastic domination[END_REF] for useful additional elements. 6 From classical results (see e.g. [START_REF] Revuz | Continuous martingales and Brownian motion, volume 293 of Grundlehren der mathematischen Wissenschaften[END_REF] or [START_REF] Kallenberg | Foundations of modern probability[END_REF]), we have the existence of an ℝ 𝑛 -valued Markov process (𝑋 7 For the sake of brevity, we limit ourselves to a small sample of references, and point the interested reader to the works quoted by, or quoting, these references for a more exhaustive view of this subject. 8 A construction for general Feller processes (not necessarily with continous paths) leading to a strong Markov process is given in [START_REF] Steven | Coalescing systems of non-Brownian particles[END_REF] Section 2.1. Continuity of paths easily shows that 𝑃 (𝑛) 𝑡 is order-preserving. The Feller property of (𝑃 (𝑛) 𝑡 ) 𝑡≥0 can then be established using the fact that we have a consistent extension of a Feller semigroup on the real line by order-preserving semigroups, exactly as in the Proof of Theorem 1.

(A) Parallel coupling (B) Doeblin coupling FIGURE 1. Special cases of Theorem 1 with easier constructions often called the Doeblin coupling), see Figure 1b; the construction with 𝑛 = 2 is a nice way of proving the not-so-obvious fact that any Feller process on the real line with continuous paths is stochastically monotone. In the case of jump processes ( [START_REF] Chen | From Markov chains to non-equilibrium particle systems[END_REF]), and more generally, of Lévy-type processes ( [START_REF] Wang | Stochastic comparison for Lévy-type processes[END_REF][START_REF] Kolokoltsov | Stochastic monotonicity and duality of one-dimensional Markov processes[END_REF]), necessary and sufficient explicit conditions for stochastic monotonicity, expressed in terms of the infinitesimal generator of the process, have been obtained. One nice feature of Theorem 1 is that it holds in a completely general setting, regardless of the specific structure of the underlying process, as soon as it is stochastically monotone and enjoys the Feller property, which are also necessary conditions for the conclusion of the theorem to hold.

Theorem 2 shows that, for all 𝑛 ≥ 2, 𝑠 ∈ ℝ + and 𝐱 = (𝑥 1 , … , 𝑥 𝑛 ), P (𝑛) 𝑠 (𝐱, ⋅) produces an 𝑛-dimensional coupling of the distributions 𝑃 𝑠 (𝑥 1 , ⋅), … , 𝑃 𝑠 (𝑥 𝑛 , ⋅), enjoying a maximality property with respect to the super-modular order. Without the constraint that this coupling has to be the distribution at time 𝑠 of a Feller process on ℝ 𝑛 whose marginals evolve according to (𝑃 𝑡 ) 𝑡∈ℝ + , the maximum would be achieved by the classical comonotone coupling of 𝑃 𝑠 (𝑥 1 , ⋅), … , 𝑃 𝑠 (𝑥 𝑛 , ⋅) (see [START_REF] Müller | Comparison methods for stochastic models and risks[END_REF]). Loosely speaking, P (𝑛) is obtained (as a well-defined limit) by composing infinitely many such comonotone couplings associated with an infinitesimal time interval 𝑠, so we choose to call the resulting object the comonotone flow associated with (𝑃 𝑡 ) 𝑡≥0 . Strictly speaking, we have not defined a flow but only a consistent family

( (𝑃 (𝑛) 𝑡 ) 𝑡∈ℝ + ) 𝑛≥1
of Feller Markov semigroups. However, it is shown9 in [START_REF] Lejan | Flows, coalescence and noise[END_REF] that such a family of semigroups corresponds to a stochastic flow, in a precise sense. We do not push this question further here, and postpone the study of the various properties enjoyed by this stochastic flow to future work.

1.7. Organization of the paper. In Section 2, we collect various results related to the Feller property of Markov semigroups, on general (locally compact separable metric, or compact metric) spaces, then on ℝ. Section 3 is devoted to the proof of Theorem 1, and Section 4 to the proof of Theorem 2. Section 5 is an appendix where we explain why we think an element is missing in the proof of some results in [START_REF] Kamae | Stochastic inequalities on partially ordered spaces[END_REF] which are related to the conclusion of Theorem 1 in the present paper. Section 7 shows some numerical illustrations.

FELLER-RELATED RESULTS

This section is devoted to results related to the Feller property of Markov semigroups. In Subsection 2.1, we provide alternative formulations of the Feller property that turn out to be more suitable for the arguments developed in Section 3, namely, transferring the assumed regularity properties of the family of kernels (𝑃 𝑡 ) 𝑡∈ℝ + on ℝ, to families of kernels on ℝ 𝑛 that extend 𝑃 𝑡 . In Subsection 2.2, we show that the combination of stochastic monotonicity with the Feller property allows one to extend (𝑃 𝑡 ) 𝑡∈ℝ + to a Feller semigroup ( P𝑡 ) 𝑡∈ℝ + on the extended real line ℝ, whose advantage (in our context) over the usual one-point compactification ℝ ∪ {∞} is its compatibility with the underlying order on ℝ.

General properties.

Given a lcsm space, we denote by (𝑆) the set of (Borel) probability measures on 𝑆, equipped with the topology of weak convergence, which is metrizable (see below). To a Markov kernel 𝐾 on 𝑆, we associate the map Ǩ from 𝑆 to (𝑆), defined as 𝑥 ↦ 𝐾(𝑥, ⋅). Conversely, to a map 𝐊 from 𝑆 to (𝑆), we associate the map K from 𝑆 × 𝒮 to ℝ defined as (𝑥, 𝐵) ↦ [𝐊(𝑥)] (𝐵), which may or may not be a Markov kernel, depending on whether the map K(⋅, 𝐵) is measurable for every 𝐵.

Lemma 1. Given a lcsm space 𝑆, and a continuous map

𝐊 ∶ 𝑆 → (𝑆), K is a Markov kernel.
Proof. Arguing as in the proof of Proposition 1 below, we have that, for all 𝑓 ∈  𝑏 (𝑆), the map 𝑥 ↦ K𝑓 (𝑥) = ∫ 𝑆 𝑓 (𝑦)𝑑 [𝐊(𝑥)] (𝑦) is itself an element of  𝑏 (𝑆), and in particular is a Borel map.

We now apply Theorem 0.2.2 in [START_REF] Revuz | Continuous martingales and Brownian motion, volume 293 of Grundlehren der mathematischen Wissenschaften[END_REF], which is a functional version of the monotone class theorem: the set ℋ of functions 𝑓 ∈  𝑏 (𝑆) such that 𝑥 ↦ K𝑓 (𝑥) is Borel contains  𝑏 (𝑆) -which plays the role of 𝒞 in [START_REF] Revuz | Continuous martingales and Brownian motion, volume 293 of Grundlehren der mathematischen Wissenschaften[END_REF] -and the assumptions of the theorem are met 10 , so that ℋ contains all 𝜎( 𝑏 (𝑆))-measurable functions, and thus all bounded real-valued Borel functions on 𝑆. In particular K𝟏 𝐵 is a Borel function for any Borel set 𝐵 in 𝑆, so K is indeed a Markov kernel on 𝑆. □ Proposition 1. Given a Markov kernel 𝐾 on a lcsm space 𝑆, the following three properties are equivalent:

(𝐢) Ǩ is a continuous map; (𝐢𝐢) ∀𝑓 ∈  𝑏 (𝑆), 𝐾𝑓 ∈  𝑏 (𝑆); (𝐢𝐢𝐢) ∀𝑓 ∈  0 (𝑆), 𝐾𝑓 ∈  𝑏 (𝑆).
Proof. Remember that, for a Markov kernel 𝐾 and 𝑓 ∈  𝑏 (𝑆), one always has 𝐾𝑓 ∈  𝑏 (𝑆), so that the only stake in (𝐢𝐢) and (𝐢𝐢𝐢) is the continuity of 𝐾𝑓 . Now, by definition, continuity of the map 𝑥 ↦ 𝐾(𝑥, ⋅) means that, for any sequence (𝑥 𝑘 ) 𝑘≥1 such that lim 𝑘→+∞ 𝑥 𝑘 = 𝑥, we have the weak convergence

𝐾(𝑥 𝑘 , ⋅) 𝐰 ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← → 𝑘→+∞ 𝐾(𝑥, ⋅).
In turn, weak convergence means that, for every 𝑓 ∈  𝑏 (𝑆), lim 𝑘→+∞ 𝐾𝑓 (𝑥 𝑘 ) = 𝐾𝑓 (𝑥), which reads as the continuity of 𝑃 𝑓 at 𝑥. As a consequence, (𝐢) and (𝐢𝐢) are equivalent. The equivalence of (𝐢) and (𝐢𝐢𝐢) stems from the fact that, since 𝑆 is a lcsm space, the weak convergence

𝐾(𝑥 𝑘 , ⋅) 𝐰 ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← → 𝑘→+∞ 𝐾(𝑥, ⋅) is equivalent to the fact that lim 𝑘→+∞ 𝐾𝑓 (𝑥 𝑘 ) = 𝐾𝑓 (𝑥) for every 𝑓 ∈  0 (𝑆). □ Proposition 2. A Markov kernel 𝐾 on a lcsm space 𝑆 satisfies (5) ∀𝑓 ∈  0 (𝑆), 𝐾𝑓 ∈  0 (𝑆),
if and only if the following two conditions are satisfied:

(𝐢) Ǩ is a continuous map; (𝐢𝐢) for every compact subset 𝐶 of 𝑆, lim 𝑥→∞ 𝐾(𝑥, 𝐶) = 0.
Proof. We start with the "if" part, assuming that (𝐢) and (𝐢𝐢) are satisfied. Consider 𝑓 ∈  0 (𝑆). Since  0 (𝑆) ⊂  𝑏 (𝑆), Proposition 1 and (𝐢) show that 𝐾𝑓 ∈  𝑏 (𝑆), and it remains to prove that 𝐾𝑓 goes to zero at infinity. Given 𝜀 > 0 and a compact 𝐶 such that |𝑓 | ≤ 𝜀 outside 𝐶, we have |𝐾𝑓 (𝑥)| ≤ ‖𝑓 ‖ ∞ 𝐾(𝑥, 𝐶) + 𝜀, so that, thanks to (𝐢𝐢), |𝐾𝑓 (𝑥)| ≤ 2𝜀 as soon as 𝑥 is outside a sufficiently large compact set 𝐶 ′ . Now for the "only if" part, assuming that (5) is satisfied. From Proposition (1), we have (𝐢), since  0 (𝑆) ⊂  𝑏 (𝑆). As for (𝐢𝐢), consider a compact set 𝐶. By compactness of 𝐶 and local compactness of 𝑆, there exists11 a continuous function 𝑓 with compact support such that 𝑓 ≥ 𝟏 𝐶 . Then 𝐾(𝑥, 𝐶) = 𝐾𝟏 𝐶 (𝑥) ≤ 𝐾𝑓 (𝑥), and, since 𝑓 ∈  0 (𝑆), [START_REF] Bérard | Hydrodynamic limit of N-branching Markov processes[END_REF] implies that 𝐾𝑓 ∈  0 (𝑆) so that lim 𝑥→∞ 𝐾𝑓 (𝑥) = 0. □ We now focus on the case where 𝑆 is a compact (hence separable) metric space, denoting by (𝑆) the space of continuous real-valued functions on 𝑆 (due to the fact that 𝑆 is compact, we have (𝑆) =  𝑏 (𝑆) =  0 (𝑆)). Among the many possible metrics compatible with the weak convergence topology on (𝑆), we choose to use the Wasserstein (or Kantorovich-Rubinstein) 𝑊 1 distance (see [START_REF] Villani | Optimal transport, volume 338 of Grundlehren der mathematischen Wissenschaften[END_REF], Chapter 6). Specifically, consider the space 𝐿𝑖𝑝(𝑆) of real-valued Lipschitz functions on 𝑆. Denoting by 𝑑 𝑆 the metric on 𝑆, for 𝑓 ∈ 𝐿𝑖𝑝(𝑆), we let ‖𝑓 ‖ 𝐿𝑖𝑝 = sup{|𝑓 (𝑦) -𝑓 (𝑥)|∕𝑑 𝑆 (𝑥, 𝑦); 𝑥, 𝑦 ∈ 𝑆, 𝑥 ≠ 𝑦}. Note that, since 𝑆 is compact, every 𝑓 ∈ 𝐿𝑖𝑝(𝑆) is bounded. The 𝑊 1 distance on (𝑆) is then given by ( 6)

𝑊 1 (𝜇, 𝜈) = sup { | | | | ∫ 𝑆 𝑓 (𝑥)𝑑𝜇(𝑥) -∫ 𝑆 𝑓 (𝑥)𝑑𝜈(𝑥) | | | | ; ‖𝑓 ‖ 𝐿𝑖𝑝 ≤ 1 } .
Lemma 2. Let 𝐾, 𝐿, 𝑀 be Markov kernels on 𝑆, then

sup 𝑥∈𝑆 𝑊 1 (𝐾𝐿(𝑥, ⋅), 𝐾𝑀(𝑥, ⋅)) ≤ sup 𝑥∈𝑆 𝑊 1 (𝐿(𝑥, ⋅), 𝑀(𝑥, ⋅)).
Proof. Consider 𝑓 ∈ 𝐿𝑖𝑝(𝑆) such that ‖𝑓 ‖ 𝐿𝑖𝑝 ≤ 1. By definition, for all 𝑥 ∈ 𝑆, we have that

|𝐿𝑓 (𝑥) -𝑀𝑓 (𝑥)| ≤ 𝑊 1 (𝐿(𝑥, ⋅), 𝑀(𝑥, ⋅)) , so that ‖𝐿𝑓 -𝑀𝑓 ‖ ∞ ≤ sup 𝑥∈𝑆 𝑊 1 (𝐿(𝑥, ⋅), 𝑀(𝑥, ⋅)). Us- ing the contraction property (𝛼) of Markov kernels, ‖𝐾𝐿𝑓 -𝐾𝑀𝑓 ‖ ∞ ≤ ‖𝐿𝑓 -𝑀𝑓 ‖ ∞ , so that sup 𝑥∈𝑆 sup 𝑓 ∈𝐿𝑖𝑝(𝑆), ‖𝑓 ‖ 𝐿𝑖𝑝 ≤1 |𝐾𝐿𝑓 (𝑥) -𝐾𝑀𝑓 (𝑥)| ≤ sup 𝑥∈𝑆 𝑊 1 (𝐿(𝑥, ⋅), 𝑀(𝑥, ⋅)). □ Proposition 3.
Given a family of Markov kernels (𝐾 𝑡 ) 𝑡∈𝐽 on a compact metric space 𝑆, where 𝐽 is a subset of ]0, +∞[ such that inf 𝐽 = 0, the following properties are equivalent:

(𝐢) ∀𝑓 ∈ (𝑆), lim 𝑡→0+ ‖ ‖ 𝐾 𝑡 𝑓 -𝑓 ‖ ‖∞ = 0 (𝐢𝐢)
for all 𝜀 > 0, and all 𝑥 ∈ 𝑆, 𝐾 𝑡 (𝑥, 𝐵 𝑆 (𝑥, 𝜀) 𝑐 ) goes 12 to 0 as 𝑡 goes to 0, uniformly over 𝑥 ∈ 𝑆; (𝐢𝐢𝐢) for all 𝑥 ∈ 𝑆, 𝑊 1 (𝐾 𝑡 (𝑥, ⋅), 𝛿 𝑥 ) goes to 0 as 𝑡 goes to 0, uniformly over 𝑥 ∈ 𝑆.

Proof. Before we start the proof, let us introduce a few notations and definitions. For a non-empty set 𝐴 ⊂ 𝑆 and 𝑥 ∈ 𝑆, we let 𝑑 𝑆 (𝑥, 𝐴) = inf {𝑑 𝑆 (𝑥, 𝑦); 𝑦 ∈ 𝐴}, and, for 𝑟 > 0, we denote 𝐴 𝑟 = {𝑥 ∈ 𝑆; 𝑑 𝑆 (𝑥, 𝐴) < 𝑟}. We then define the function 𝐼 𝑟 𝐴 on 𝑆 by 𝐼 𝑟 𝐴 (𝑥) = 1 -min(𝑑 𝑆 (𝑥,𝐴),𝑟) 𝑟

. We have that 0 ≤ 𝐼 𝑟 𝐴 ≤ 1, 𝐼 𝑟 𝐴 = 1 on 𝐴, 𝐼 𝑟 𝐴 = 0 outside 𝐴 𝑟 , and

‖ ‖ ‖ 𝐼 𝑟 𝐴 ‖ ‖ ‖𝐿𝑖𝑝 ≤ 1∕𝑟.
We start with (𝐢) ⇒ (𝐢𝐢). Assume that (𝐢) holds, and consider 𝜀 > 0, 𝑥 ∈ 𝑆, and the function 𝑓 = 𝐼 𝜀∕3 𝐵 𝑆 (𝑥,𝜀∕3) . For all 𝑦 ∈ 𝐵 𝑆 (𝑥, 𝜀∕3) one has that 𝑓 (𝑦) = 1, and, for all 𝑧 ∉ 𝐵 𝑆 (𝑦, 𝜀), one has that 𝑧 ∉ 𝐵 𝑆 (𝑥, 𝜀∕3) 𝜀∕3 so that 𝑓 (𝑧) = 0. Since 𝑓 ≤ 1, we deduce that 𝐾 𝑡 (𝑦, 𝐵 𝑆 (𝑦, 𝜀) 𝑐 ) ≤ 1 -𝐾 𝑡 𝑓 (𝑦). From (𝐢), 𝐾 𝑡 𝑓 (𝑦) goes to 𝑓 (𝑦) = 1 as 𝑡 goes to 0, uniformly with respect to 𝑦 ∈ 𝐵 𝑆 (𝑥, 𝜀∕3), so that 𝐾 𝑡 (𝑦, 𝐵 𝑆 (𝑦, 𝜀) 𝑐 ) goes to 0 as 𝑡 goes to 0, uniformly with respect to 𝑦 ∈ 𝐵 𝑆 (𝑥, 𝜀∕3). Covering the compact set 𝑆 by a finite number of balls of the form 𝐵 𝑆 (𝑥, 𝜀∕3), we have proved that (𝐢𝐢) holds.

We now prove that (𝐢𝐢) ⇒ (𝐢). Assume that (𝐢𝐢) holds. Consider 𝑓 ∈ (𝑆), and 𝜀 > 0. Since 𝑆 is compact, 𝑓 is uniformly continuous and there is a 𝛿 > 0 such that, whenever 𝑑 𝑆 (𝑥, 𝑦) ≤ 𝛿, one has 𝑐 ), so (𝐢𝐢) leads to the desired conclusion.

|𝑓 (𝑥) -𝑓 (𝑦)| ≤ 𝜀. As a consequence, for all 𝑥 ∈ 𝑆, |𝐾 𝑡 𝑓 (𝑥) -𝑓 (𝑥)| ≤ 𝜀 + 2 ‖𝑓 ‖ ∞ 𝐾 𝑡 (𝑥, 𝐵 𝑆 (𝑥, 𝜀)
The proof that (𝐢𝐢) ⇒ (𝐢𝐢𝐢) is similar, where Lipschitz continuity replaces uniform continuity. Indeed, for 𝑓 ∈ 𝐿𝑖𝑝(𝑆), we have, for all 𝜀 > 0, and for all 𝑥 ∈ 𝑆, that |𝐾 𝑡 𝑓 (𝑥) -𝑓 (𝑥)| ≤ ‖𝑓 ‖ 𝐿𝑖𝑝 𝜀 + is locally compact, for every 𝑥 ∈ 𝐶, there exists a number 𝑟 > 0 such that 𝐵(𝑥, 𝑟) is compact. By compactness of 𝐶, we deduce that there exist 𝑚 ≥ 1, 𝑥 1 , … , 𝑥 𝑚 ∈ 𝐾 and 𝑟 1 , … , 𝑟 𝑚 > 0 such that 𝐶 ⊂ 𝑈 , where 𝑈 = ⋃ 𝑚 𝑘=1 𝐵(𝑥 𝑘 , 𝑟 𝑘 ). By compactness of 𝐶 again, the continuous function 𝑥 ↦ 𝑑(𝑥, 𝑈 𝑐 ) has a minimum value 𝜌 on 𝐶, and, since 𝑈 𝑐 is closed and 𝐶 ∩ 𝑈 𝑐 = ∅, we must have 𝜌 > 0. Choosing 𝜀 ∈]0, 𝜌[ ensures that 𝐶 𝜀 ⊂ 𝑈 . As a consequence, 𝐶 𝜀 is included in the compact set ⋃ 𝑚 𝑘=1 𝐵(𝑥 𝑘 , 𝑟 𝑘 ), so that 𝐶 𝜀 is compact. One can then take 𝑓 = 𝐼 𝜀 𝐶 . 12 We denote by 𝐵 𝑆 (𝑥, 𝑟) the open ball of radius 𝑟 centered at 𝑥. Δ 𝑆 ‖𝑓 ‖ 𝐿𝑖𝑝 𝐾 𝑡 (𝑥, 𝐵 𝑆 (𝑥, 𝜀) 𝑐 ), using the fact that, for all 𝑥, 𝑦 ∈ 𝑆, |𝑓 (𝑦) -𝑓 (𝑥)| ≤ ‖𝑓 ‖ 𝐿𝑖𝑝 Δ 𝑆 , where Δ 𝑆 = sup 𝑥,𝑦∈𝑆 𝑑 𝑆 (𝑥, 𝑦) is finite thanks to the fact that 𝑆 is compact.

We now prove that (𝐢𝐢𝐢) ⇒ (𝐢𝐢). Given 𝑥 ∈ 𝑆 and 𝜀 > 0, consider the function 𝑓 = 𝐼 𝜀∕3 𝐵 𝑆 (𝑥,𝜀∕3) used in the proof that (𝐢) ⇒ (𝐢𝐢). We have that ‖𝑓 ‖ 𝐿𝑖𝑝 ≤ 3∕𝜀, so that 𝑔 = (3∕𝜀) -1 𝑓 , sastifies ‖𝑔‖ 𝐿𝑖𝑝 ≤ 1, and as a consequence, for all 𝑦 ∈ 𝑆, |𝐾 𝑡 𝑓 (𝑦) -𝑓 (𝑦)| ≤ (3∕𝜀)𝑊 1 (𝐾 𝑡 (𝑦, ⋅), 𝛿 𝑦 ), and we can argue as in the proof that (𝐢) ⇒ (𝐢𝐢), thanks to (𝐢𝐢𝐢). 5) and

(7) ∀𝑓 ∈  ↗ 𝑏 (ℝ), 𝐾𝑓 ∈  ↗ 𝑏 (ℝ), also satisfies (8) ∀𝑓 ∈  𝑢,𝑣 (ℝ), 𝐾𝑓 ∈  𝑢,𝑣 (ℝ).
Proof.

We first prove that, for all 𝑎 > 0,

Lemma 3. Given Borel probability measures

𝜇, 𝜇 1 , 𝜇 2 , … on ℝ, 𝜇 𝑘 𝐰 ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← → 𝑘→+∞ 𝜇 if and only if μ𝑘 𝐰 ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← → 𝑘→+∞ μ. Proof. Assume that μ𝑘 𝐰 ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← → 𝑘→+∞ μ.
For 𝑓 ∈  0 (ℝ), 𝑓 can be extended to a function f ∈ (ℝ) by letting f (±∞) = 0. For any probability measure 𝜈 on ℝ, we have that ∫ ℝ 𝑓 (𝑥)𝑑𝜈(𝑥) = ∫ ℝ f (𝑥)𝑑𝜈(𝑥). As a consequence, the fact that

lim 𝑘→+∞ ∫ ℝ f (𝑥)𝑑μ 𝑘 (𝑥) = ∫ ℝ f (𝑥)𝑑 μ(𝑥) implies that lim 𝑘→+∞ ∫ ℝ 𝑓 (𝑥)𝑑𝜇 𝑘 (𝑥) = ∫ ℝ 𝑓 (𝑥)𝑑𝜇(𝑥). So we have proved that 𝜇 𝑘 𝐰 ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← → 𝑘→+∞ 𝜇. Now assume that 𝜇 𝑘 𝐰 ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← → 𝑘→+∞ 𝜇.
For 𝑓 ∈ (ℝ), the restriction of 𝑓 to ℝ is a bounded continuous function, so we have that lim 𝑘→+∞ ∫ ℝ 𝑓 (𝑥)𝑑𝜇 𝑘 (𝑥) = ∫ ℝ 𝑓 (𝑥)𝑑𝜇(𝑥), whence lim 𝑘→+∞ ∫ ℝ 𝑓 (𝑥)𝑑μ 𝑘 (𝑥) = ∫ ℝ 𝑓 (𝑥)𝑑 μ(𝑥). So we have proved that μ𝑘

𝐰 ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← → 𝑘→+∞ μ. □
We extend (𝑃 𝑡 ) 𝑡∈ℝ + to a family of Markov kernels ( P𝑡 ) 𝑡∈ℝ + on ℝ by setting, for all 𝑥 ∈ ℝ: P𝑡 (𝑥, ⋅) = P𝑡 (𝑥, ⋅), P𝑡 (+∞, ⋅) = 𝛿 +∞ (⋅) and P𝑡 (-∞, ⋅) = 𝛿 -∞ (⋅). We now prove (Fb'), i.e. ∀𝑓 ∈ (ℝ) and 𝑥 ∈ ℝ, lim 𝑡→0 P𝑡 𝑓 (𝑥) = 𝑓 (𝑥). This is immediate for 𝑥 = ±∞ since then P𝑡 𝑓 (𝑥) = 𝑓 (𝑥) for all 𝑡. For all 𝑥 ∈ ℝ, we have that 𝑃 𝑡 (𝑥, ⋅) converges weakly to 𝛿 𝑥 as 𝑡 → 0, so that, since

𝑓 |ℝ ∈  𝑏 (ℝ), lim 𝑡→0 𝑃 𝑡 𝑓 ℝ (𝑥) = 𝑓 ℝ (𝑥),

PROOF OF THEOREM 1

This section is devoted to the proof of Theorem 1. First, in Subsection 3.1, we define the kernels 𝑄 (𝑛) 𝑡 by using, for 𝐱 = (𝑥 1 , … , 𝑥 𝑛 ) and 𝑡 ≥ 0, the classical comonotone coupling of 𝑃 𝑡 (𝑥 1 , ⋅), … , 𝑃 𝑡 (𝑥 𝑛 , ⋅), which turns out to be order-preserving thanks to the stochastic monotonicity of (𝑃 𝑡 ) 𝑡≥0 (more precisely, we work with the extended semigroup ( P𝑡 ) 𝑡≥0 on ℝ). Then, in Subsection 3.2, we construct the kernels 𝑃 (𝑛) 𝑡 by iterating the composition of 𝑄 (𝑛) 𝑠 a large number of times for smaller and smaller 𝑠, then taking the limit. One key tool to prove that this procedure leads to a well-behaved limit is Proposition 7, which enables the transfer of regularity properties of the kernels 𝑃 𝑡 to the kernels 𝑄 (𝑛) 𝑡 , thanks to the interplay between order-preserving properties of kernels and estimates in the 𝑊 (𝑛) 1 distance.

3.1. Discrete-time flow. For 𝑥 ∈ ℝ and 𝑡 ≥ 0, let 𝐹 [-1] 𝑥,𝑡 ∶ [0, 1] → ℝ denote the quantile function of the probability distribution 𝑃 𝑡 (𝑥, ⋅), i.e. 𝐹 [-1] 𝑥,𝑡 (𝑢) = inf {𝑦 ∈ ℝ; 𝑃 𝑡 (𝑥, ] -∞, 𝑦]) ≥ 𝑢}. We extend the definition to 𝑥 ∈ ℝ by letting 𝐹 [-1] -∞,𝑡 ≡ -∞ and 𝐹 [-1] +∞,𝑡 ≡ +∞.

Lemma 4. The family of functions 𝐹 [-1]

𝑥,𝑡 enjoys the following properties: (a) for all 𝑥 ∈ ℝ and 𝑡 ≥ 0, the map 𝐹 [-1] 𝑥,𝑡 is non-decreasing; (b) for all 𝑥 ∈ ℝ, 𝑡 ≥ 0 and 𝑢 ∈]0, 1[, 𝐹 [-1] 𝑥,𝑡 (𝑢) ∈ ℝ; (c) for all 𝑡 ≥ 0 and 𝑢 ∈]0, 1[, the map 𝑥 ↦ 𝐹 [-1] 𝑥,𝑡 (𝑢) is non-decreasing; (d) If 𝑈 is a random variable following the uniform distribution on the interval [0, 1], then, for all 𝑥, 𝑡, the distribution of the random variable 𝐹 [-1] 𝑥,𝑡 (𝑈 ) is P𝑡 (𝑥, ⋅); (e) for all 𝑡 ∈ ℝ, and 𝑥 ∈ ℝ, except for an at most countable set of values of 𝑢 ∈ [0, 1], the map

𝑦 ↦ 𝐹 [-1] 𝑦,𝑡 (𝑢) is continuous at 𝑥. Proof.
Properties (a) and (b) are immediate consequences of the definition and of the fact that the cdf of a real-valued random variable is non-decreasing, with limits 0 and 1 at -∞ and +∞, respectively. Property (c) is a consequence of stochastic monotonicity: if 𝑥 1 , 𝑥 2 ∈ ℝ are such that 𝑥 1 ≤ 𝑥 2 , then, for all 𝑦 ∈ ℝ, 𝑃 𝑡 (𝑥 1 , ] -∞, 𝑦]) ≥ 𝑃 𝑡 (𝑥 2 , ] -∞, 𝑦]). Extension to ℝ is immediate in view of the definition of 𝐹 [-1] 𝑥,𝑡 when 𝑥 = ±∞. Property (d) is obvious when 𝑥 = ±∞, and classical when 𝑥 ∈ ℝ, we e.g. refer to the proof of Theorem 25.6 in [START_REF] Billingsley | Probability and measure[END_REF]. We now deal with property (e). Given 𝑡 ≥ 0, 𝑥 ∈ ℝ and a sequence (𝑥 𝑘 ) 𝑘≥1

of real numbers such that lim 𝑘→+∞ 𝑥 𝑘 = 𝑥, we have the weak convergence 𝑃 𝑡 (𝑥 𝑘 , ⋅)

𝐰 ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← → 𝑘→+∞ 𝑃 𝑡 (𝑥, ⋅),
so that, refering again to the proof of Theorem 25.6 in [START_REF] Billingsley | Probability and measure[END_REF], whenever 𝐹 [-1] 𝑥,𝑡 is continuous at a certain 𝑢 ∈]0, 1[, we have that lim 𝑘→+∞ 𝐹 [-1] 𝑥 𝑘 ,𝑡 (𝑢) = 𝐹 [-1] 𝑥,𝑡 (𝑢). Since 𝐹 [-1] 𝑥,𝑡 is non-decreasing, the number of points of discontinuity is at most countable. Assume now that 𝑥 = +∞. From the proof of Proposition 4, we have that, for any 𝑎 > 0, lim 𝑥→+∞ 𝑃 𝑡 (𝑥, ] -∞, 𝑎]) = 0, so that, for any 𝑢 ∈]0, 1], lim 𝑥→+∞ 𝐹 [-1] 𝑥,𝑡 (𝑢) = +∞ = 𝐹 [-1] +∞,𝑡 (𝑢). The case 𝑥 = -∞ is treated symmetrically. □

Given 𝑡 ≥ 0 and 𝐱 = (𝑥 1 , … , 𝑥 𝑛 ) ∈ ℝ 𝑛 , define 𝑄 (𝑛) 𝑡 (𝐱, ⋅) as the distribution on ℝ 𝑛 of the random vector

𝑍 𝐱 = ( 𝐹 [-1] 𝑥 𝑖 ,𝑡 (𝑈 ) ) 1≤𝑖≤𝑛
, where 𝑈 is a random variable following the uniform distribution on the interval

[0, 1].
Proposition 6. For all 𝑡 ≥ 0, and 𝑛 ≥ 2, 𝑄 (𝑛) 𝑡 is an order-preserving 13 ), and define 𝑄 (𝑛) 𝑡 𝑓 (𝐱) = ∫ ℝ 𝑛 𝑓 (𝐲)𝑑𝑄 (𝑛) 𝑡 (𝐱, 𝑦). Given 𝐱 ∈ ℝ and a sequence (𝐱 𝑘 ) 𝑘≥1 such that lim 𝑘→+∞ 𝐱 𝑘 = 𝐱, we deduce from Lemma 4 (e) that, almost surely, lim 𝑘→+∞ 𝑍 𝐱 𝑘 = 𝑍 𝐱 , so that, by continuity of 𝑓 , almost surely, lim 𝑘→+∞ 𝑓 (𝑍 𝐱 𝑘 ) = 𝑓 (𝑍 𝐱 ), whence, by dominated convergence, lim 𝑘→+∞ 𝑄 (𝑛) 𝑡 𝑓 (𝐱 𝑘 ) = 𝑄 (𝑛) 𝑡 𝑓 (𝐱). We deduce that 𝑄 (𝑛) 𝑡 𝑓 ∈ (ℝ

𝑛

), then invoke Lemma 1 to deduce that 𝑄 (𝑛) 𝑡 is a Markov kernel. That 𝑄 (𝑛) 𝑡 is an extension of P𝑡 is a direct consequence of Lemma 4 (d). Consistency is a consequence of the fact that, by definition, 𝜋 𝑛 𝑖 1 ,…,𝑖 𝑘

(𝑍 𝐱 ) = ( 𝐹 [-1]
𝑥 𝑖 𝑗 ,𝑡 (𝑈 )

) 1≤𝑗≤𝑘 = 𝑍 𝜋 𝑛 𝑖 1 ,…,𝑖 𝑘
(𝐱) . □

Construction of the limit 𝑅 (𝑛)

𝑡 . Now we call 𝐷 + the set of positive dyadic rational numbers. Given 𝑡 ∈ 𝐷 + , we write 𝑡 = 𝑘2 -𝑚 0 ∈ 𝐷 + where 𝑘 ≥ 1 and 𝑚 0 ≥ 0 are integers, and 𝑚 0 has the minimum possible value in such an expression. Then, for every integer 𝑚 ≥ 𝑚 0 , we let

𝑄 (𝑛),𝑚 𝑡 = [ 𝑄 (𝑛) 2 -𝑚 ] 𝑘2 𝑚-𝑚 0 , i.e. 𝑄 (𝑛),𝑚

𝑡

is the repeated composition of kernels 𝑄 (𝑛) 2 -𝑚 ⋯ 𝑄 (𝑛) 2 -𝑚 with a total of 𝑘2 𝑚-𝑚 0 kernels in the composition.

As a result, (𝑄 (𝑛),𝑚

𝑡

) 𝑚≥𝑚 0 is a sequence of Markov kernels on ℝ 𝑛 . Moreover, from Proposition 6, for all 𝑡 ∈ 𝐷 + , 𝑛 ≥ 2, and 𝑚 ≥ 𝑚 0 , 𝑄 (𝑛),𝑚 𝑡 is order-preserving 14 , and, for all 𝑡 ∈ 𝐷 + and 𝑚 ≥ 𝑚 0 , the family

(𝑄 (𝑛),𝑚

𝑡

) 𝑛≥2 is a consistent extension 15 of P𝑡 . Now let 𝐐 (𝑛),𝑚 𝑡 = Q(𝑛),𝑚 𝑡 . Denote by 𝑊 (𝑛) 1 the Wasserstein (or Kantorovich-Rubinstein) distance on (ℝ

𝑛

), where ℝ 𝑛 is equipped with the distance 𝑑 𝑛 defined in Subsection 2.2. 13 We extend the definition of an order-preserving Markov kernel on ℝ 𝑛 to that of an order-preserving Markov kernel on ℝ 𝑛 in the obvious way. 14 Using the fact that the composition of order-preserving Markov kernels is still an order-preserving Markov kernel, see Lemma 5 in Section 6. 15 Using the fact that, if (𝐾 (𝑛) ) 𝑛≥2 and (𝐿 (𝑛) ) 𝑛≥2 are consistent extensions, respectively of 𝐾 and 𝐿, then (𝐾 (𝐳, ⋅). By the consistency property, the distribution of (𝑋 1 , … , 𝑋 𝑛 ) is 𝑄 (𝑛),𝑚 𝑡 (𝐱, ⋅) and the distribution of (𝑌 1 , … , 𝑌 𝑛 ) is 𝑄 (𝑛),𝑚 𝑡 (𝐲, ⋅). Moreover, the order-preserving property implies that, with probability 1, for all 1 ≤ 𝑖 ≤ 𝑛, the relative order of 𝑋 𝑖 and 𝑌 𝑖 is the same 16 as the relative order between 𝑥 𝑖 and 𝑦 𝑖 . Now consider a function 𝑓 ∈ 𝐿𝑖𝑝(ℝ 𝑛 ) such that ‖𝑓 ‖ 𝐿𝑖𝑝 ≤ 1. Let 𝐗 = (𝑋 1 , … , 𝑋 𝑛 ) and 𝐘 = (𝑌 1 , … , 𝑌 𝑛 ). We have that

| | | 𝑄 (𝑛),𝑚 𝑡 (𝐲, ⋅) -𝑄 (𝑛),𝑚 𝑡 (𝐱, ⋅) | | | = |𝔼𝑓 (𝐘) -𝔼𝑓 (𝐗)| ≤ 𝔼 |𝑓 (𝐘) -𝑓 (𝐗)| . Since ‖𝑓 ‖ 𝐿𝑖𝑝 ≤ 1, 𝔼 |𝑓 (𝐘) -𝑓 (𝐗)| ≤ 𝔼 𝑛 ∑ 𝑖=1 |𝜙(𝑌 𝑖 ) -𝜙(𝑋 𝑖 )| = 𝑛 ∑ 𝑖=1 𝔼|𝜙(𝑌 𝑖 ) -𝜙(𝑋 𝑖 )|.
Since 𝜙 is non-decreasing and the relative order of 𝑋 𝑖 and 𝑌 𝑖 is non-random, we have that Remember that ℝ 𝑛 is a compact metric space, so that (ℝ

𝔼|𝜙(𝑌 𝑖 ) -𝜙(𝑋 𝑖 )| = | | | 𝔼 ( 𝜙(𝑌 𝑖 ) -𝜙(𝑋 𝑖 ) ) | | | = | | 𝔼𝜙(𝑌 𝑖 ) -𝔼𝜙(𝑋 𝑖 ) | | . Since 𝑄 (𝑛),𝑚

𝑛

) is also a compact metric space, hence a complete metric space. As a consequence, for each 𝑡 ∈ 𝐷 + and 𝑛 ≥ 2, in view of Corollary 1, we can invoke the Arzelà-Ascoli theorem (see e.g. [START_REF] Dixmier | General topology[END_REF]) to show the convergence along a subsequence: the sequence of maps ( 𝐐

(𝑛),𝑚 𝑘 𝑡 ) 𝑘≥1
converges, as 𝑘 → +∞, to a limiting map 𝐑 (𝑛) 𝑡 , in the sense of uniform convergence of continuous maps from ℝ 𝑛 to (ℝ 𝑛 ). Moreover, by diagonal extraction, we can assume that convergence occurs simultaneously for every 𝑡 in (the countable set) 𝐷 + and every 𝑛 ≥ 2. To sum up, we have a strictly increasing sequence of integers (𝑚 𝑘 ) 𝑘≥1 and, for every 𝑡 ∈ 𝐷 + and 𝑛 ≥ 2, an element 𝐑 (𝑛) 𝑡 of ℭ 𝑛 , such that 17 [START_REF] Feller | Two singular diffusion problems[END_REF] lim , for each 𝑡 ∈ 𝐷 + , the family of Markov kernels (𝑅 (𝑛) 𝑡 ) 𝑛≥2 inherits the property of being a family of order-preserving kernels forming a consistent extension of P𝑡 .

𝑘→+∞ 𝑑 ℭ 𝑛 (𝐐 (𝑛),𝑚 𝑘 𝑡 , 𝐑 (𝑛) 𝑡 ) = 0. Now let 𝑅 (𝑛) 𝑡 = R(𝑛) 𝑡 ,
Let us now check that the semigroup property (restricted to 𝐷 + ) holds for our family of Markov kernels:

(11) ∀𝑠, 𝑡 ∈ 𝐷 + , 𝑅 (𝑛) 𝑠+𝑡 = 𝑅 (𝑛) 𝑠 𝑅 (𝑛)
𝑡 . 16 To be explicit: if

(𝐱, ⋅) 𝐰 ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← → 𝑘→+∞ 𝑅 (𝑛)
𝑠 (𝐱, ⋅), we deduce that [START_REF] Kamae | Stochastic inequalities on partially ordered spaces[END_REF] lim

𝑘→+∞ 𝑄 (𝑛),𝑚 𝑘 𝑠 𝑅 (𝑛) 𝑡 𝑓 (𝐱) = 𝑅 (𝑛) 𝑠 𝑅 (𝑛) 𝑡 𝑓 (𝐱).
Combining ( 10), ( 12) and ( 13), we have that, for all 𝐱 ∈ ℝ 𝑛 , lim 𝑘→+∞ 𝑄

(𝑛),𝑚 𝑘 𝑠 𝑄 (𝑛),𝑚 𝑘 𝑡 𝑓 (𝐱) = 𝑅 (𝑛) 𝑠 𝑅 (𝑛) 𝑡 𝑓 (𝐱). Now the weak convergence 𝑄 (𝑛),𝑚 𝑘 𝑠+𝑡 (𝐱, ⋅) 𝐰 ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← → 𝑘→+∞ 𝑅 (𝑛) 𝑠+𝑡 (𝐱, ⋅) implies that lim 𝑘→+∞ 𝑄 (𝑛),𝑚 𝑘 𝑠+𝑡 𝑓 (𝐱) = 𝑅 (𝑛) 𝑠+𝑡 𝑓 (𝐱). Using the identity 𝑄 (𝑛),𝑚 𝑘 𝑠 𝑄 (𝑛),𝑚 𝑘 𝑡 = 𝑄 (𝑛),𝑚 𝑘 𝑠+𝑡 , we deduce that lim 𝑘→+∞ 𝑄 (𝑛),𝑚 𝑘 𝑠 𝑄 (𝑛),𝑚 𝑘 𝑡 𝑓 (𝐱) = 𝑅 (𝑛)
𝑠+𝑡 𝑓 (𝐱), so that 𝑅 (𝑛) 𝑠 𝑅 (𝑛) 𝑡 𝑓 (𝐱) = 𝑅 (𝑛) 𝑠+𝑡 𝑓 (𝐱), and ( 11) is established 18 . Now for 𝐱 = (𝑥 1 , … , 𝑥 𝑛 ) and 𝜀 > 0, observe that 𝐵 ℝ 𝑛 (𝐱, 𝜀) 𝑐 ⊂ ⋃ 𝑛 𝑖=1 𝐵 ℝ (𝑥 𝑖 , 𝜀∕𝑛) 𝑐 , so that, using the fact that 𝑅 (𝑛) 𝑡 is an extension of P𝑡 and the union bound,

𝑅 (𝑛) 𝑡 (𝐱, 𝐵 ℝ 𝑛 (𝐱, 𝜀) 𝑐 ) ≤ 𝑛 ∑ 𝑖=1 P𝑡 (𝑥 𝑖 , 𝐵 ℝ (𝑥 𝑖 , 𝜀∕𝑛) 𝑐 ).
Using the Feller property of ( P𝑡 ) 𝑡≥0 and Proposition 3 (ii), we deduce that 𝑅 (𝑛) 𝑡 (𝐱, 𝐵 ℝ 𝑛 (𝐱, 𝜀) 𝑐 ) goes to 0 as 𝑡 goes to 0, uniformly over 𝐱 ∈ ℝ 𝑛 , which, thanks to Proposition 3 again, shows that [START_REF] Kirstein | Monotonicity and comparability of time-homogeneous Markov processes with discrete state space[END_REF] lim

𝑡→0 𝑡∈𝐷 + sup 𝑥∈ℝ 𝑛 𝑊 (𝑛) 1 (𝑅 (𝑛) 𝑡 (𝑥, ⋅), 𝛿 𝑥 ) = 0.
Using Lemma 2, we have that [START_REF] Kolokoltsov | Stochastic monotonicity and duality of one-dimensional Markov processes[END_REF] sup

𝑥∈ℝ 𝑛 𝑊 (𝑛) 1 (𝑅 (𝑛) 𝑠+𝑡 (𝑥, ⋅), 𝑅 (𝑛) 𝑠 (𝑥, ⋅)) ≤ sup 𝑥∈ℝ 𝑛 𝑊 (𝑛) 1 (𝑅 (𝑛) 𝑡 (𝑥, ⋅), 𝛿 𝑥 ).
Combining ( 14) and [START_REF] Kolokoltsov | Stochastic monotonicity and duality of one-dimensional Markov processes[END_REF] shows that the map 𝑠 ↦ 𝐑 (𝑛) 𝑠 from 𝐷 + to ℭ 𝑛 is uniformly continuous. Since ℭ 𝑛 is a complete metric space, and 𝐷 + is a dense subset of ℝ + , there is a unique extension (see [START_REF] Dixmier | General topology[END_REF]) to a uniformly continuous map 𝑠 ↦ 𝐑 (𝑛) 𝑠 from ℝ + to ℭ 𝑛 . This allows us to extend the definition 𝑅 (𝑛) 𝑡 = R(𝑛) 𝑡 to every 𝑡 ∈ ℝ + . We now check that the semigroup property holds for

(𝑅 𝑡 ) 𝑡∈ℝ + , i.e. ( 16
) ∀𝑠, 𝑡 ∈ ℝ + , 𝑅 (𝑛) 𝑠+𝑡 = 𝑅 (𝑛) 𝑠 𝑅 (𝑛)
𝑡 . Let (𝑠 𝑘 ) 𝑘≥1 and (𝑡 𝑘 ) 𝑘≥1 be sequences of elements in 𝐷 + which converge to 𝑠 and 𝑡 respectively. By [START_REF] Allen | Stochastic monotonicity and realizable monotonicity[END_REF], we have that 𝑅 (𝑛) 𝑠 𝑘 𝑅 (𝑛)

𝑡 𝑘 = 𝑅 (𝑛) 𝑠 𝑘 +𝑡 𝑘 . Now, for 𝑓 ∈ 𝐿𝑖𝑝(ℝ 𝑛
) such that ‖𝑓 ‖ 𝐿𝑖𝑝 ≤ 1, we write

𝑅 (𝑛) 𝑠 𝑘 𝑅 (𝑛) 𝑡 𝑘 𝑓 -𝑅 (𝑛) 𝑠 𝑅 (𝑛) 𝑡 𝑓 = 𝑅 (𝑛) 𝑠 𝑘 𝑅 (𝑛) 𝑡 𝑘 𝑓 -𝑅 (𝑛) 𝑠 𝑘 𝑅 (𝑛) 𝑡 𝑓 + 𝑅 (𝑛) 𝑠 𝑘 𝑅 (𝑛) 𝑡 𝑓 -𝑅 (𝑛) 𝑠 𝑅 (𝑛) 𝑡 𝑓 .
Using Lemma 2, we have that, for all

𝐱 ∈ ℝ 𝑛 , | | | 𝑅 (𝑛) 𝑠 𝑘 𝑅 (𝑛) 𝑡 𝑘 𝑓 (𝐱) -𝑅 (𝑛) 𝑠 𝑘 𝑅 (𝑛) 𝑡 𝑓 (𝐱) | | | ≤ 𝑑 ℭ 𝑛 (𝐑 (𝑛)
𝑡 𝑘 , 𝐑 (𝑛) 𝑡 ).

18 Indeed, we have established that, for all 𝐱 ∈ ℝ 𝑛 , 𝑊 (𝑛) 1 (𝑅 (𝑛) 𝑠 𝑅 (𝑛) 𝑡 (𝐱, ⋅), 𝑅 (𝑛) 𝑠+𝑡 (𝐱, ⋅)) = 0.

Arguing exactly as in the proof of [START_REF] Allen | Stochastic monotonicity and realizable monotonicity[END_REF], we have that 𝑅 (𝑛) 𝑡 𝑓 ∈ (ℝ

𝑛

). Moreover, by continuity of the extension to ℝ + , we have the weak convergence 𝑅 (𝑛) 𝑠 𝑘 (𝐱, ⋅)

𝐰 ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← → 𝑘→+∞ 𝑅 (𝑛) 𝑠 (𝐱, ⋅), so that lim 𝑘→+∞ 𝑅 (𝑛)
𝑠 𝑘 𝑅 (𝑛) 𝑡 𝑓 (𝐱) = 𝑅 (𝑛) 𝑠 𝑅 (𝑛) 𝑡 𝑓 (𝐱), and similarly lim 𝑘→+∞ 𝑅 (𝑛) 𝑠 𝑘 +𝑡 𝑘 𝑓 (𝐱) = 𝑅 (𝑛) 𝑠+𝑡 𝑓 (𝐱), so that 𝑅 (𝑛) 𝑠+𝑡 𝑓 (𝐱) = 𝑅 (𝑛) 𝑠 𝑅 (𝑛) 𝑡 𝑓 (𝐱), and ( 16) is established. We now observe that, by continuity of the projection maps 𝜋 𝑛 𝑖 1 ,…,𝑖 𝑘 , and the Feller property of ( P𝑡 ) 𝑡∈ℝ + , the property that (𝑅 (𝑛) 𝑡 ) 𝑛≥2 is a family of order-preserving kernels forming a consistent extension of P𝑡 , already established for every 𝑡 ∈ 𝐷 + , holds for every 𝑡 ∈ ℝ + . Also, thanks to the continuity of 𝑡 ↦ 𝐑 (𝑛) 𝑡 at 𝑡 = 0, and by Proposition 3, the family of Markov kernels (𝑅 (𝑛) 𝑡 ) 𝑛≥2 satisfies property (Fb). For 𝑛 ≥ 2, 𝑡 ∈ ℝ + , 𝐱 ∈ ℝ 𝑛 and Borel set 𝐵 of ℝ 𝑛 , we define 𝑃 (𝑛) 𝑡 (𝐱, 𝐵) = 𝑅 (𝑛) 𝑡 (𝐱, 𝐵). Since 𝑅 (𝑛) 𝑡 is an extension of P𝑡 and P𝑡 (𝑥 𝑖 , ℝ) = 1, we have that 𝑅 (𝑛) 𝑡 (𝐱, ℝ 𝑛 ) = 1, so 𝑃 (𝑛) 𝑡 is indeed a Markov kernel on ℝ, and the fact that, for all 𝑡, (𝑃 (𝑛) 𝑡 ) 𝑛≥2 is a consistent extension of 𝑃 𝑡 by order-preserving Markov kernels is an immediate consequence of the fact that (𝑅 (𝑛) 𝑡 ) 𝑛≥2 is a consistent extension of P𝑡 by order-preserving Markov kernels. Since a function 𝑓 ∈  0 (ℝ 𝑛 ) immediately extends to a function f ∈ (ℝ 𝑛 ), we deduce the fact that 𝑃 (𝑛) 𝑡 𝑓 ∈  𝑏 (ℝ 𝑛 ) and the convergence lim 𝑡→0 ‖ ‖ ‖

𝑃 (𝑛) 𝑡 𝑓 -𝑓 ‖ ‖ ‖∞
= 0 from the corresponding properties of 𝑅 (𝑛) 𝑡 . To complete the proof of the Feller property of (𝑃 (𝑛) 𝑡 ) 𝑡∈ℝ + , it remains to prove that 𝑃 (𝑛) 𝑡 𝑓 ∈  0 (ℝ 𝑛 ), i.e. lim 𝐱→∞ 𝑃 (𝑛) 𝑡 𝑓 (𝐱) = 0, since we already know that 𝑃 (𝑛) 𝑡 𝑓 ∈  𝑏 (ℝ 𝑛 ). To this end, using the fact that 𝑃 (𝑛) 𝑡 is an extension of 𝑃 𝑡 , we have that, for all 𝑎 > 0:

𝑃 (𝑛) 𝑡 (𝐱, [-𝑎, +𝑎] 𝑛 ) ≤ min 1≤𝑖≤𝑛 𝑃 𝑡 (𝑥 𝑖 , [-𝑎, +𝑎]).
Thanks to Proposition 2 and the Feller property of 𝑃 𝑡 , we deduce that lim 𝐱→∞ 𝑃 (𝑛) 𝑡 (𝐱, [-𝑎, +𝑎] 𝑛 ) = 0, and, thanks to Proposition 2 again, that lim 𝐱→∞ 𝑃 (𝑛) 𝑡 𝑓 (𝐱) = 0.

PROOF OF THEOREM 2

This (short) section is devoted to the proof of Theorem 2. We start with a general result showing that the ≤ 𝐬𝐦 order is preserved by consistent families of order-preserving Markov kernels. Combined with the maximal property of the co-monotone coupling with respect to ≤ 𝐬𝐦 and the fact that ( P (𝑛) 𝑡 ) 𝑡∈ℝ + is obtained as a limit of the composition of order-preserving kernels constructed using the co-monotone coupling, we deduce Theorem 2.

Proposition 9. Let (𝑀 (𝑛)

𝑡 ) 𝑡≥0 ∈ 𝔐 𝑛 , and let (𝑄 (𝑛) 𝑡 ) 𝑡≥0 be the family of Markov kernels introduced in Subsection 3.1. Then, for all 𝑡 ≥ 0 and 𝐱 ∈ ℝ 𝑛 , 𝑀 (𝑛) 𝑡 (𝐱, ⋅) ≤ 𝐬𝐦 𝑄 (𝑛) 𝑡 (𝐱, ⋅).

Proof. By construction (see Theorem 3.1.1 in [START_REF] Müller | Comparison methods for stochastic models and risks[END_REF]), 𝑄 (𝑛) 𝑡 (𝐱, ⋅) is a comonotone probability distribution on ℝ 𝑛 . Moreover, 𝑄 (𝑛) 𝑡 (𝐱, ⋅) and 𝑀 (𝑛) 𝑡 (𝐱, ⋅) have the same marginals: 𝑃 𝑡 (𝑥 1 , ⋅), … , 𝑃 𝑡 (𝑥 𝑛 , ⋅). By Theorem 3.9.8 in [START_REF] Müller | Comparison methods for stochastic models and risks[END_REF] (using property (P5)), we deduce that 𝑀 (𝑛) 𝑡 (𝐱, ⋅) ≤ 𝐬𝐦 𝑄 (𝑛) 𝑡 (𝐱, ⋅). □

We now prove by induction that, for all 𝑓 ∈  𝑏 (ℝ 𝑛 ), 𝑠 ≥ 0, and integer 𝑘 ≥ 0, 𝑓 converges pointwise to P (𝑛) 𝑡 𝑓 , so that, taking the limit in our inequality, we have that 𝑀 (𝑛) 𝑡 𝑓 ≤ P (𝑛) 𝑡 𝑓 . Given 𝑡 ∈ ℝ + , consider a sequence (𝑠 𝑚 ) 𝑚≥1 of elements of 𝐷 + such that lim 𝑚→+∞ 𝑠 𝑚 = 𝑡. By the Feller property of (𝑀 (𝑛) 𝑠 ) 𝑠∈ℝ + and ( P (𝑛) 𝑠 ) 𝑠∈ℝ + , we have that, as 𝑚 → +∞, 𝑀 (𝑛) 𝑠 𝑚 𝑓 converges pointwise to 𝑀 (𝑛) 𝑡 𝑓 , and P (𝑛) 𝑠 𝑚 𝑓 converges pointwise to P (𝑛) 𝑡 𝑓 , so that, taking the limit in the inequality 𝑀 (𝑛) 𝑠 𝑚 𝑓 ≤ P (𝑛) 𝑠 𝑚 𝑓 , we have that 𝑀 (𝑛) 𝑡 𝑓 ≤ P (𝑛) 𝑡 𝑓 . Using [START_REF] Müller | Comparison methods for stochastic models and risks[END_REF] (Theorems 3.9.10 and 3.9.11), Theorem 2 is proved. 5.2. About Theorem 5. Now Theorem 5 in [START_REF] Kamae | Stochastic inequalities on partially ordered spaces[END_REF] assumes that (𝑋 𝑡 ) 𝑡∈ℝ + and (𝑌 𝑡 ) 𝑡∈ℝ + are (possibly timeinhomogeneous) Markov processes, and does not directly assume [START_REF] Thomas | Interacting particle systems[END_REF], but instead the following stochastic monotonicity condition: for all 𝑠, 𝑡 ≥ 0, and all 𝑥, 𝑦 ∈ 𝐸 such that 𝑥 ≼ 𝑦, [START_REF] López | Stochastic domination and Markovian couplings[END_REF] ℙ(𝑋 𝑡+𝑠 ∈ ⋅|𝑋 𝑡 = 𝑥) ≼ 𝐬𝐭 ℙ(𝑌 𝑡+𝑠 ∈ ⋅|𝑌 𝑡 = 𝑦), with the same conclusion as Theorem 4. The proof of Theorem 5 consists in observing that, using the Markov property, (20) implies [START_REF] Thomas | Interacting particle systems[END_REF], then invoking Theorem 4. However, in view of our previous remark, this approach does not seem to lead to a complete proof of Theorem 5 unless one can prove that (20) implies [START_REF] Lindvall | On Strassen's theorem on stochastic domination[END_REF]. Unfortunately, such an implication is not true in general, as we show in the following counterexample.

Take 𝐸 = {𝑎, 𝑏, 𝑐}, where 𝑎 ≼ 𝑏 ≼ 𝑐, and where 𝑋 and 𝑌 are two versions of the same continuous-time Markov chain on 𝐸 with distinct starting points. The infinitesimal generator of the chain is prescribed by the 𝑄-matrix (see e.g. [START_REF] Anderson | Continuous-time Markov chains[END_REF])

𝐿 = (𝐿 𝑥𝑦 ) 𝑥∈𝐸,𝑦∈𝐸 = 𝑎 𝑏 𝑐 ( ) 𝑎 -2.5 1.75 0.75 𝑏 1.5 -2.5 1 𝑐 0.5 0 -0.5
and defines a Markov semigroup on 𝐸, given, for all 𝑡 ≥ 0, by the transition matrix 𝑃 𝑡 = exp(𝑡𝐿). To check that 𝐿 indeed defines a stochastically monotone Markov semigroup on 𝐸, we check the condition stated in [START_REF] Anderson | Continuous-time Markov chains[END_REF] (Theorem 3.4 page 249, attributed to [START_REF] Kirstein | Monotonicity and comparability of time-homogeneous Markov processes with discrete state space[END_REF]), which in our setting reduces to the two conditions 𝐿 𝑎𝑐 ≤ 𝐿 𝑏𝑐 and 𝐿 𝑏𝑎 ≥ 𝐿 𝑐𝑎 , both visibly satisfied. We now check numerically that

ℙ(𝑋 1 ∈ ⋅|𝑋 0 = 𝑎, 𝑋 2 = 𝑎) ⋠ ℙ(𝑌 1 ∈ ⋅|𝑌 0 = 𝑏, 𝑌 2 = 𝑏).
Denoting 𝑃 = exp(𝐿), we have that

ℙ(𝑋 1 ≼ 𝑎|𝑋 0 = 𝑎, 𝑋 2 = 𝑎) = ℙ(𝑋 1 = 𝑎|𝑋 0 = 𝑎, 𝑋 2 = 𝑎) = 𝑃 𝑎𝑎 𝑃 𝑎𝑎 𝑃 𝑎𝑎 𝑃 𝑎𝑎 + 𝑃 𝑎𝑏 𝑃 𝑏𝑎 + 𝑃 𝑎𝑐 𝑃 𝑐𝑎 ,
and

ℙ(𝑌 1 ≼ 𝑎|𝑌 0 = 𝑏, 𝑌 2 = 𝑏) = ℙ(𝑌 1 = 𝑎|𝑌 0 = 𝑏, 𝑌 2 = 𝑏) = 𝑃 𝑏𝑎 𝑃 𝑎𝑏 𝑃 𝑏𝑎 𝑃 𝑎𝑏 + 𝑃 𝑏𝑏 𝑃 𝑏𝑏 + 𝑃 𝑏𝑐 𝑃 𝑐𝑏 .
Numerically computing 𝑃 with the SciPy open-source software 19 , we get, rounding to 3 decimal places, that ℙ(𝑋 1 ≼ 𝑎|𝑋 0 = 𝑎, 𝑋 2 = 𝑎) ≈ 0.362 and ℙ(𝑌 1 ≼ 𝑎|𝑌 0 = 𝑏, 𝑌 2 = 𝑏) ≈ 0.374, while the stochastic domination ℙ(𝑋

1 ∈ ⋅|𝑋 0 = 𝑎, 𝑋 2 = 𝑎) ≼ ℙ(𝑌 1 ∈ ⋅|𝑌 0 = 𝑏, 𝑌 2 = 𝑏) would imply the inequality ℙ(𝑌 1 ≼ 𝑎|𝑌 0 = 𝑏, 𝑌 2 = 𝑏) ≤ ℙ(𝑋 1 ≼ 𝑎|𝑋 0 = 𝑎, 𝑋 2 = 𝑎).
To sum up, an element seems to be missing in the proof of Theorem 5. We observe that, in the specific case where 𝐸 = ℝ and where (𝑋 𝑡 ) 𝑡≥0 and (𝑌 𝑡 ) 𝑡≥0 are two versions of the same Feller process with distinct starting points, Theorem 1 in the present paper can be used to deduce the conclusion of Theorem 5.

MISCELLANEOUS LEMMAS

Lemma 5. If 𝐾 (𝑛) and 𝐿 (𝑛) are order-preserving kernels on ℝ 𝑛 , so is their composition 𝐾 (𝑛) 𝐿 (𝑛) . Proof. By definition, for all 𝐱 ∈ ℝ 𝑛 , we have that 𝐾 (𝑛) 𝐿 (𝑛) (𝐱, ℝ 𝑛 𝐱 ) = ∫ ℝ 𝑛 𝐿 (𝑛) (𝐲, ℝ 𝑛 𝐱 )𝑑𝐾 (𝑛) (𝐱, 𝐲). Since 𝐾 (𝑛) is order-preserving, we have that 𝐾 (𝑛) (𝐱, ℝ 𝑛 𝐱 ) = 1, so the previous integral can be rewritten as ∫ ℝ 𝑛 𝐱 𝐿 (𝑛) (𝐲, ℝ 𝑛 𝐱 )𝑑𝐾 (𝑛) (𝐱, 𝐲). For all 𝐲 ∈ ℝ 𝑛 𝐱 , the definition shows that 20 ℝ 𝑛 𝐲 ⊂ ℝ 𝑛 𝐱 , so that, since 𝐿 (𝑛) is order-preserving, 𝐿 (𝑛) (𝐲, ℝ 𝑛 𝐱 ) = 1, and we have proved that 𝐾 (𝑛) 𝐿 (𝑛) (𝐱, ℝ 𝑛 𝐱 ) = 1. □ Lemma 6. If (𝐾 (𝑛) ) 𝑛≥2 and (𝐿 (𝑛) ) 𝑛≥2 are consistent extensions, respectively of 𝐾 and 𝐿, then (𝐾 (𝑛) 𝐿 (𝑛) ) 𝑛≥2 is a consistent extension of 𝐾𝐿. 19 Specifically, we used scipy.linalg.expm to compute the matrix exponential. 20 If 𝐱 = (𝑥 Proof. Let 𝐾 (1) = 𝐾 and 𝐿 (1) = 𝐿. Given integers 1 ≤ 𝑘 ≤ 𝑛, 𝑖 ) -1 (𝐵)) = (𝐾 (𝑘) 𝐿 (𝑘) )(𝜋 𝑛 𝑖 1 ,…,𝑖 𝑘 (𝐱), 𝐵), which proves the conclusion of the Lemma. □

NUMERICAL ILLUSTRATIONS

In this section, we show some numerical simulations to illustrate our construction, when the underlying process is a square-root diffusion as defined by the following s.d.e. [START_REF] Revuz | Continuous martingales and Brownian motion, volume 293 of Grundlehren der mathematischen Wissenschaften[END_REF] 𝑑𝑋 𝑡 = 𝑎(𝑏 -𝑋 𝑡 )𝑑𝑡 + 𝜎 √ 𝑋 𝑡 𝑑𝑊 𝑡 , where (𝑊 𝑡 ) 𝑡≥0 is a standard one-dimensional brownian motion. Provided that the condition 2𝑎𝑏 ≥ 𝜎 2 is satisfied, [START_REF] Revuz | Continuous martingales and Brownian motion, volume 293 of Grundlehren der mathematischen Wissenschaften[END_REF] defines an explicit Feller semigroup on ]0, +∞[: for 𝑡 > 0, 𝑃 𝑡 (𝑥, ⋅) is the distribution of 𝑍∕(2 * 𝑐), where 𝑐 = 2𝑎

(1-𝑒 -𝑎𝑡 )𝜎 2 , and 𝑍 follows a non-central chi-square distribution with 4𝑎𝑏 𝜎 2 degrees of freedom and non-centrality parameter 2𝑐𝑥𝑒 -𝑎𝑡 . (This class of processes has been studied by Feller [START_REF] Feller | Two singular diffusion problems[END_REF], and gained popularity in financial mathematics as a model for the dynamics of interest rates [START_REF] Cox | A theory of the term structure of interest rates[END_REF]). To stick to the framework of the present paper, that is, Feller processes on ℝ, and not on ]0, +∞[, one may consider the semigroup associated with the process (log(𝑋 𝑡 )) 𝑡≥0 instead, which leads to identical results up to applying a log change of scale. For the sake of simplicity, we keep working with the semigroup associated to (𝑋 𝑡 ) 𝑡≥0 in the following illustrations.

We work with the set of parameters 𝑎 = 3, 𝑏 = 2 and 𝜎 2 = 8. The following figures depict the values of 𝑁 = 5000 simulated i.i.d. pairs distributed according to 𝑄 (2),𝑚 𝑡 ((𝑥 1 , 𝑥 2 ), ⋅), for 𝑥 1 = 0.5, 𝑥 2 = 2, 𝑡 = 0.5, and 𝑚 = 1, … , 6. Fig. 2 displays each pair (𝑥, 𝑦) as a point with coordinates (𝑥, 𝑦), while Fig. 3 uses the point with coordinates (𝑥, 𝑦 -𝑥) instead. Simulations were done using the R open-source software.

For 𝑚 = 1, 𝑄 (2),𝑚 𝑡 ((𝑥 1 , 𝑥 2 ), ⋅) is just the classical monotonic coupling, which is the reason why the points are lying on a curve. As 𝑚 gets larger, the array of points gets more spread out, with barely distinguishable differences between the few last consecutive graphs. Email address: jberard@unistra.fr / brieuc.frenais@math.unistra.fr

□ 2 . 2 .Proposition 4 .

 224 Extension of (𝑃 𝑡 ) to ℝ. For 𝑢, 𝑣 ∈ ℝ, denote by  𝑢,𝑣 (ℝ) the set of continuous functions 𝑓 ∶ ℝ → ℝ such that lim 𝑥→-∞ 𝑓 (𝑥) = 𝑢 and lim 𝑥→+∞ 𝑓 (𝑥) = 𝑣. A Markov kernel 𝐾 on ℝ satisfying (

  lim 𝑥→-∞ 𝐾(𝑥, [-𝑎, +∞[) = 0 and lim 𝑥→+∞ 𝐾(𝑥, ] -∞, 𝑎]) = 0.Given 𝑎 > 0, consider 𝜀 > 0. Then choose 𝑏 ≥ 𝑎 such that 𝐾(0, [-𝑏, +𝑏] 𝑐 ) ≤ 𝜀. Thanks to[START_REF] Bérard | Hydrodynamic limit of N-branching Markov processes[END_REF] and Proposition (2), there exists 𝑥 0 ≤ 0 such that, for all 𝑥 ≤ 𝑥 0 , 𝐾(𝑥, [-𝑏, +𝑏]) ≤ 𝜀. By the monotonicity assumption[START_REF] Cox | A theory of the term structure of interest rates[END_REF], 𝐾(𝑥, ]𝑏, +∞[) ≤ 𝐾(0, ]𝑏, +∞[)) ≤ 𝜀. Combining the two inequalities for 𝐾(𝑥, ⋅), we deduce that 𝐾(𝑥, [-𝑏, +∞[) ≤ 2𝜀, and, since 𝑏 ≥ 𝑎, we have that, for all 𝑥 ≤ 𝑥 0 , 𝐾(𝑥, [-𝑎, +∞[) ≤ 𝐾(𝑥, [-𝑏, +∞[) ≤ 2𝜀. We have proved the first part of (9). The second part is proved symmetrically. Now consider 𝑓 ∈  𝑢,𝑣 (ℝ). Given 𝜖 > 0, there exists 𝑎 > 0 such that, for all 𝑧 < -𝑎, |𝑓 (𝑧) -𝑢| ≤ 𝜀. As a consequence, |𝐾𝑓 (𝑥) -𝑢| ≤ 𝜀 + (||𝑓 || ∞ + 𝑢)𝐾(𝑥, [-𝑎, +∞[). Using the first part of (9), we deduce that lim sup 𝑥→-∞ |𝐾𝑓 (𝑥) -𝑢| ≤ 𝜀. Since 𝜀 is arbitrary, we have thus proved that lim 𝑥→-∞ 𝐾𝑓 (𝑥) = 𝑢. A similar argument proves that lim 𝑥→+∞ 𝐾𝑓 (𝑥) = 𝑣. Finally, since 𝑓 ∈  𝑏 (ℝ) and 𝐾 satisfies (5), Proposition 2 shows that 𝐾𝑓 is a continuous function. □ We equip the extended real line ℝ = ℝ ∪ {-∞, +∞} = [-∞, +∞] with the metric 𝑑 1 (𝑥, 𝑦) = |𝜙(𝑦) -𝜙(𝑥)|, where 𝜙(𝑥) = tanh(𝑥) (with tanh(+∞) = 1 and tanh(-∞) = -1), which makes it a separable compact metric space. For 𝑛 ≥ 2, we equip ℝ 𝑛 with the metric 𝑑 𝑛 (𝐱, 𝐲) = ∑ 𝑛 𝑖=1 |𝜙(𝑦 𝑖 ) -𝜙(𝑥 𝑖 )|. Given a Borel probability measure 𝜇 on ℝ, we extend it to a probability measure μ on ℝ by letting, for every Borel set 𝐵 of ℝ, μ(𝐵) = 𝜇(𝐵 ∩ ℝ).

Proposition 5 .

 5 If (𝑃 𝑡 ) ℝ + is stochastically monotone and enjoys the Feller property, then ( P𝑡 ) 𝑡≥0 has the Feller property.Proof. We first prove (Fa), i.e. ∀𝑓 ∈ (ℝ) and 𝑡 ≥ 0, P𝑡 𝑓 ∈ (ℝ). Given 𝑓 ∈ (ℝ), we denote by 𝑓 |ℝ the restriction of 𝑓 to ℝ, and observe that 𝑓 |ℝ ∈  𝑢,𝑣 (ℝ), where 𝑢 = 𝑓 (-∞) and 𝑣 = 𝑓 (+∞), so, by Proposition 4, 𝑃 𝑡 𝑓 |ℝ ∈  𝑢,𝑣 (ℝ). For 𝑥 ∈ ℝ, P𝑡 𝑓 (𝑥) = 𝑃 𝑡 𝑓 |ℝ (𝑥), while P𝑡 𝑓 (-∞) = 𝑢 and P𝑡 𝑓 (+∞) = 𝑣. We deduce that P𝑡 𝑓 is continuous on ℝ.

Corollary 1 .

 1 𝑡is an extension of P𝑡 , we have that 𝔼𝜙(𝑋 𝑖 ) = P𝑡 𝜙(𝑥 𝑖 ) and 𝔼𝜙(𝑌 𝑖 ) = P𝑡 𝜙(𝑦 𝑖 ).□ For all 𝑛 ≥ 2 and 𝑡 ∈ 𝐷 + , the family ( Note that 𝜙 ∈ (ℝ), so the Feller property of P𝑡 implies that P𝑡 𝜙 is continuous. □ Denote by ℭ 𝑛 the set of continuous functions from ℝ 𝑛 to (ℝ 𝑛 ), equipped with the distance 𝑑 ℭ 𝑛 (𝐊, 𝐋) = sup 𝑥∈ℝ 𝑛 𝑊 (𝑛) 1 (𝐊(𝑥), 𝐋(𝑥)).

FIGURE 2 .

 2 FIGURE 2. 5000 simulated pairs following the distribution 𝑄(2),𝑚 𝑡 ((𝑥 1 , 𝑥 2 ), ⋅).

FIGURE 3 .

 3 FIGURE 3. Transformation (𝑥, 𝑦) ↦ (𝑥, 𝑦 -𝑥) applied to the graphs in Fig. 2.

  For 𝑛 ≥ 2, denote by 𝔐 𝑛 the set of Feller semigroups (𝑀(𝑛) 𝑡 ) 𝑡∈ℝ + on ℝ 𝑛 such that, for every 𝑡 ≥ 0, every 𝐱 = (𝑥 1 , … , 𝑥 𝑛 ) ∈ ℝ 𝑛 , and every 1 ≤ 𝑖 ≤ 𝑛, 𝑀 𝑛 𝑡 (𝐱, (𝜋 𝑛 𝑖 ) -1 (𝐵)) = 𝑃 𝑡 (𝑥 𝑖 , 𝐵). Theorem 2. Let (𝑃 𝑡 ) 𝑡∈ℝ + be a stochastically monotone Feller semigroup on ℝ, and, for 𝑛 ≥ 2, denote by ( P (𝑛) 𝑡 ) 𝑡∈ℝ + the4 Markov semigroup constructed in the proof of Theorem 1. Then, for every (𝑀 (𝑛) 𝑡 ) 𝑡∈ℝ + ∈ 𝔐 𝑛 , 𝑠 ∈ ℝ + and 𝐱 ∈ ℝ 𝑛 , one has that 𝑀 (𝑛) 𝑡∈ℝ + ∈ 𝔐 𝑛 , and since ≤ 𝐬𝐦 is a partial order, this shows that ( P (𝑛) 𝑡 ) 𝑡∈ℝ + is uniquely characterized as the solution of the following maximization problem with respect to ≤ 𝐬𝐦 :

		P (𝑛) 𝑠 (x, ⋅) =	max (𝑀 (𝑛) 𝑡 ) 𝑡∈ℝ + ∈𝔐 𝑛	𝑀 (𝑛) 𝑠 (x, ⋅).
		𝑥 1 𝑡 , … , 𝑋	𝑥 𝑛
	𝑥 1 𝑡 , … , 𝑋	𝑥 𝑛
	The fact that (𝑋 𝑡 , … , 𝑋 𝑥 1 𝑡 𝑥 𝑛

1.5. Statement of the main results. Theorem 1. If (𝑃 𝑡 ) 𝑡∈ℝ + is a stochastically monotone Feller semigroup on ℝ, there exists a consistent extension of (𝑃 𝑡 ) 𝑡≥0 by a family ( (𝑃 (𝑛) 𝑡 ) 𝑡∈ℝ + ) 𝑛≥1 of order-preserving Feller semigroups. 𝑠 (x, ⋅) ≤ 𝐬𝐦 P (𝑛) 𝑠 (x, ⋅). Since by construction ( P (𝑛) 𝑡 ) 1.6. Discussion. Broadly speaking, Theorem 1 is a monotonicity equivalence result in the sense of [11]: starting from a monotonicity property within a family of probability measures, one deduces the existence of a monotone coupling, i.e. a family of random variables providing an effective realization of the monotonicity property. The archetype for such results is Strassen's theorem 5 : if 𝜇 ≤ 𝐬𝐭 𝜈, there exists a pair of random variables (𝑋, 𝑌 ) such that Law(𝑋) = 𝜇, Law(𝑌 ) = 𝜈, and almost surely 𝑋 ≤ 𝑌 . In our context, the existence of the Feller semigroup (𝑃 (𝑛) 𝑡 ) 𝑡∈ℝ + shows that 6 , given 𝐱 = (𝑥 1 , … , 𝑥 𝑛 ) ∈ ℝ 𝑛 , one can define a family of real-valued random variables (𝑋 𝑡 ) 𝑡∈ℝ + on the same probability space, such that: • for 1 ≤ 𝑖 ≤ 𝑛, 𝑋 𝑥 𝑖 𝑡 is a real-valued Markov process with càdlàg paths, starting at 𝑥 𝑖 , and governed by (𝑃 𝑡 ) 𝑡∈ℝ + , • for all 𝑡 ∈ ℝ + , (𝑋 𝑡 ) is order-compatible with (𝑥 1 , … , 𝑥 𝑛 ).

  whence lim 𝑡→0 P𝑡 𝑓 (𝑥) = 𝑓 (𝑥). □ Remark 1. Observe that it is not true that, without additional assumptions, the extension to ℝ of a Feller Markov semigroup on ℝ always inherits the Feller property (this is true nonetheless for the extension to the one-point compactification ℝ ∪ {∞}). Consider for instance the semigroup (𝑃 𝑡 ) associated with a standard brownian motion (𝐵 𝑡 ) 𝑡∈ℝ + on ℝ whose sign is reversed at each occurrence of an independent rate 1 Poisson process (𝑁 𝑡 ) 𝑡∈ℝ

+ , i.e. 𝑋 𝑡 = 𝐵 𝑡 ⋅ (-1) 𝑁 𝑡 . Indeed, for 𝑓 ∈  𝑎,𝑏 (ℝ) with 𝑎 ≠ 𝑏, lim 𝑥→-∞ 𝑃 𝑡 𝑓 (𝑥) = 𝑎ℙ(𝑁 𝑡 is even) + 𝑏ℙ(𝑁 𝑡 is odd), so 𝑃 𝑡 𝑓 ∉  𝑎,𝑏 (ℝ) for 𝑡 > 0.

  Markov kernel on ℝ

				𝑛	. Moreover,
	for all 𝑡 ≥ 0, the family (𝑄 (𝑛) 𝑡 ) 𝑛≥2 is a consistent extension of P𝑡 .
	Proof. By Lemma 4 (c), we have that, for any 𝑢 ∈ [0, 1[,	(	𝐹 [-1] 𝑥 𝑖 ,𝑡 (𝑢)	)
	𝑛

1≤𝑖≤𝑛

is order-compatible with 𝑥 1 , … , 𝑥 𝑛 . To check that 𝑄

(𝑛)

𝑡 is a Markov kernel, we note that, by definition, for all 𝐱, 𝑄 (𝑛) 𝑡 (𝐱, ⋅) is a probability measure on ℝ 𝑛 . Now consider a function 𝑓 ∈ (ℝ

  For all 𝑚 ≥ 𝑚 0 , the map 𝐐 (𝑛),𝑚 Proof. Let 𝐳 = (𝐱, 𝐲) = (𝑥 1 , … , 𝑥 𝑛 , 𝑦 1 , … , 𝑦 𝑛 ). Let (𝑋 1 , … , 𝑋 𝑛 , 𝑌 1 , … , 𝑌 𝑛 ) be a random vector on ℝ

				𝑡	from ℝ	𝑛	to (ℝ	𝑛	) satisfies:
	𝑊 (𝑛) 1	(	𝐐 (𝑛),𝑚 𝑡	(𝐱), 𝐐 (𝑛),𝑚 𝑡	(𝐲)	)	≤

(𝑛)

𝐿

(𝑛)

) 𝑛≥2 is a consistent extension of 𝐾𝐿, see Lemma 6 in Section 6. Proposition 7. 𝑛 ∑ 𝑖=1 | P𝑡 𝜙(𝑦 𝑖 ) -P𝑡 𝜙(𝑥 𝑖 )|. 𝑛 whose distribution is 𝑄 (2𝑛),𝑚 𝑡

  and note that 𝑅(𝑛) 𝑡 is a Markov kernel (e.g. since 𝐑(𝑛) 𝑡 is a continuous map from ℝ

		𝑛
	to (ℝ 𝑛	), see Lemma 1), and that, by continuity of the projection maps 𝜋 𝑛 𝑖 1 ,…,𝑖 𝑘

  𝑥 𝑖 ≤ 𝑦 𝑖 , then almost surely 𝑋 𝑖 ≤ 𝑌 𝑖 , while, if 𝑦 𝑖 ≤ 𝑥 𝑖 , then almost surely 𝑌 𝑖 ≤ 𝑋 𝑖 .By construction, given 𝑠, 𝑡 ∈ 𝐷 + , we have that 𝑄

	consider 𝑓 ∈ 𝐿𝑖𝑝(ℝ 𝑛	(𝑛),𝑚 𝑘 𝑠 ) such that ‖𝑓 ‖ 𝐿𝑖𝑝 ≤ 1, and write	𝑄 (𝑛),𝑚 𝑘 𝑡	= 𝑄	(𝑛),𝑚 𝑘 𝑠+𝑡	for all large enough 𝑘. Now
	𝑄 (𝑛),𝑚 𝑘 𝑠	𝑄 (𝑛),𝑚 𝑘 𝑡	𝑓 -𝑅 (𝑛) 𝑠 𝑅 (𝑛) 𝑡 𝑓 = 𝑄 (𝑛),𝑚 𝑘 𝑠	𝑄 (𝑛),𝑚 𝑘 𝑡	𝑓 -𝑄 (𝑛),𝑚 𝑘 𝑠	𝑅 (𝑛) 𝑡 𝑓 + 𝑄 (𝑛),𝑚 𝑘 𝑠	𝑅 (𝑛) 𝑡 𝑓 -𝑅 (𝑛) 𝑠 𝑅 (𝑛) 𝑡 𝑓 .
	Using Lemma 2, we have that, for all 𝐱 ∈ ℝ 𝑛	,
	(12) On the other hand, 𝑅 (𝑛) | | | 𝑄 (𝑛),𝑚 𝑘 𝑠 𝑡 𝑓 ∈ (ℝ 𝑄 (𝑛),𝑚 𝑘 𝑡 𝑛 ) since 𝑓 ∈ (ℝ 𝑓 (𝐱) -𝑄 (𝑛),𝑚 𝑘 𝑠 𝑛 ) and 𝐑 (𝑛) 𝑅 (𝑛) 𝑡 𝑓 (𝐱) | | | 𝑡 is a continuous map from ℝ (𝑛),𝑚 𝑘 𝑡 , 𝐑 (𝑛) 𝑡 ). ≤ 𝑑 ℭ 𝑛 (𝐐 𝑛 Proposition 1). Since, for every 𝐱 ∈ ℝ 𝑛 (𝑛),𝑚 𝑘 , we have the weak convergence 𝑄 𝑠	to (ℝ 𝑛	) (see

17

Note that 𝐐 (𝑛),𝑚 𝑘 𝑡 is defined as soon as 𝑚 𝑘 ≥ 𝑚 0 , where 𝑚 0 depends on 𝑡.

  Assume that 𝑓 is continuous in addition to being in  𝑏 (ℝ 𝑛 ). Given 𝑡 ∈ 𝐷 + , remember from Subsection 3.2 that, for 𝑚 ≥ 𝑚 0 , 𝑄 (𝑛),𝑚

	)															𝑀 (𝑛) 𝑠𝑘 𝑓 ≤	[ 𝑄 (𝑛) 𝑠	] 𝑘 𝑓 .
	For 𝑘 = 0, this is obvious since the property to be proved is that 𝑓 ≤ 𝑓 . Now write	[	𝑄 (𝑛) 𝑠	] 𝑘+1	𝑓 =
	𝑄 (𝑛) 𝑠	[	𝑄 (𝑛) 𝑠	] 𝑘	𝑓 . Thanks to Proposition 8,	[ 𝑄 (𝑛) 𝑠	] 𝑘	𝑓 is super-modular. We deduce from Proposition 9 that
	𝑀 (𝑛) 𝑠	[ 𝑄 (𝑛) 𝑠	] 𝑘	𝑓 ≤ 𝑄 (𝑛) 𝑠	[	𝑄 (𝑛) 𝑠	] 𝑘	𝑓 =	[ 𝑄 (𝑛) 𝑠	] 𝑘+1	𝑓 . Assuming that 𝑀 (𝑛) 𝑠𝑘 𝑓 ≤	[	𝑄 (𝑛) 𝑠	] 𝑘	𝑓 , positivity implies
	that 𝑀 (𝑛) 𝑠(𝑘+1) 𝑓 = 𝑀 (𝑛) 𝑠 𝑀 (𝑛) 𝑠𝑘 𝑓 ≤ 𝑀 (𝑛) 𝑠	[ 𝑄 (𝑛) 𝑠	] 𝑘	𝑓 , so that, combining the previous inequalities, we deduce
	that 𝑀 (𝑛) 𝑠(𝑘+1) 𝑓 ≤	[	𝑄 (𝑛) 𝑠	] 𝑘+1	𝑓 , and (17) is proved by induction.
																𝑡	=	[	2 -𝑚 𝑄 (𝑛)	] 𝑘2 𝑚-𝑚 0

, where 𝑡 = 𝑘2 -𝑚 0 ∈ 𝐷 + , and 𝑘 ≥ 1 and 𝑚 0 ≥ 0 are integers. As a consequence,

[START_REF] Lejan | Flows, coalescence and noise[END_REF]

implies that 𝑀

(𝑛)

𝑡 𝑓 ≤ 𝑄 (𝑛),𝑚 𝑡 𝑓 . Remember that P (𝑛) 𝑡 is defined as the limit for 𝑘 → +∞ of a subsequence ( 𝑄

(𝑛),𝑚 𝑘 𝑡 ) 𝑘≥1

. Since 𝑓 is continuous, we have, by weak convergence, that, as 𝑘 → +∞, 𝑄 (𝑛),𝑚 𝑘 𝑡

  𝑛, 𝑥 𝑖 ≤ 𝑥 𝑗 ⇒ 𝑦 𝑖 ≤ 𝑦 𝑗 (since 𝐲 ∈ ℝ 𝑛 𝐱 ), hence 𝑥 𝑖 ≤ 𝑥 𝑗 ⇒ 𝑧 𝑖 ≤ 𝑧 𝑗 (since 𝐳 ∈ ℝ 𝑛 𝐲 ).

1 , … , 𝑥 𝑛 ), 𝐲 = (𝑦 1 , … , 𝑦 𝑛 ), 𝐳 = (𝑧 1 , … , 𝑧 𝑛 ) are such that 𝐲 ∈ ℝ 𝑛 𝐱 and 𝐳 ∈ ℝ 𝑛 𝐲 , we have that, for all 1 ≤ 𝑖, 𝑗 ≤

  1 , … , 𝑖 𝑘 ∈ 1, 𝑛 , 𝐱 ∈ 𝑆 𝑛 and measurable subset 𝐵 of 𝑆 𝑘 , we have by definition 𝐾 (𝑛) 𝐿 (𝑛) (𝐱, (𝜋 𝑛 𝑖 1 ,…,𝑖 𝑘 ) -1 (𝐵)) = ∫ 𝑆 𝑛 𝐿 (𝑛) (𝐲, (𝜋 𝑛 𝑖 1 ,…,𝑖 𝑘 ) -1 (𝐵))𝑑𝐾 (𝑛) (𝐱, 𝐲). (𝐿 (𝑛) ) 𝑛≥2 is a consistent extension of 𝐿, we have that 𝐿 (𝑛) (𝐲, (𝜋 𝑛 𝑖 1 ,…,𝑖 𝑘 ) -1 (𝐵)) = 𝐿 (𝑘) (𝜋 𝑛 𝑖 1 ,…,𝑖 𝑘 𝐾 (𝑛) (𝐱, ⋅) the image probability measure on 𝑆 𝑘 defined by On the other hand, since (𝐾 (𝑛) ) 𝑛≥2 is a consistent extension of 𝐾, we have that

	Since (𝐲), 𝐵),
	so that
	(21)	𝐾 (𝑛) 𝐿 (𝑛) (𝐱, (𝜋 𝑛 𝑖 1 ,…,𝑖 𝑘	) -1 (𝐵)) = ∫ 𝑆 𝑛	𝐿 (𝑘) (𝜋 𝑛 𝑖 1 ,…,𝑖 𝑘	(𝐲), 𝐵)𝑑𝐾 (𝑛) (𝐱, 𝐲).
	Denoting by (𝜋 𝑛 𝑖 1 ,…,𝑖 𝑘	) [ (𝜋 𝑛 𝑖 1 ,…,𝑖 𝑘	) * 𝐾 (𝑛) (𝐱, ⋅) ]	(𝐴) = 𝐾 (𝑛) (𝐱, (𝜋 𝑛 𝑖 1 ,…,𝑖 𝑘	) -1 (𝐴),
	(21) rewrites as
		𝐾 (𝑛) 𝐿 (𝑛) (𝐱, (𝜋 𝑛 𝑖 1 ,…,𝑖 𝑘	) -1 (𝐵)) = ∫ 𝑆 𝑘	𝐿 (𝑘) (𝐳, 𝐵)𝑑	[ (𝜋 𝑛 𝑖 1 ,…,𝑖 𝑘	) * 𝐾 (𝑛) (𝐱, ⋅)	]	(𝐳).
			[	(𝜋 𝑛 𝑖 1 ,…,𝑖 𝑘	)

* * 𝐾

(𝑛)

(𝐱, ⋅) ] (𝐴) = 𝐾 (𝑘) (𝜋 𝑛 𝑖 1 ,…,𝑖 𝑘 (𝐱), 𝐴), so that 𝐾 (𝑛) 𝐿 (𝑛) (𝐱, (𝜋 𝑛 𝑖 1 ,…,𝑖 𝑘 ) -1 (𝐵)) = ∫ 𝑆 𝑘 𝐿 (𝑘) (𝐳, 𝐵)𝑑𝐾 (𝑘) (𝜋 𝑛 𝑖 1 ,…,𝑖 𝑘 (𝐱), 𝐳),

and thus 𝐾 (𝑛) 𝐿 (𝑛) (𝐱, (𝜋 𝑛 𝑖 1 ,…,𝑖 𝑘

  INSTITUT DE RECHERCHE MATHÉMATIQUE AVANCÉE, UMR 7501, UNIVERSITÉ DE STRASBOURG ET CNRS, 7 RUE RENÉ DESCARTES, 67 000 STRASBOURG, FRANCE

See Proposition 1 below.

See e.g.[START_REF] Beznea | Strong Feller semigroups and Markov processes: A counter example[END_REF].

See e.g.[START_REF] Kallenberg | Foundations of modern probability[END_REF] Theorem 19.6, or[START_REF] Revuz | Continuous martingales and Brownian motion, volume 293 of Grundlehren der mathematischen Wissenschaften[END_REF] Chapter III, Proposition (2.4).

See also the correction[START_REF] Le | Flows, coalescence and noise. A correction[END_REF] and the fully corrected version arXiv:math/0203221v6. Condition (1) and its consequence[START_REF] Bertoin | Lévy processes[END_REF] precisely match the definition in[START_REF] Lejan | Flows, coalescence and noise[END_REF] of a compatible family of Feller semigroups satisfying 𝑃[START_REF] Bertoin | Lévy processes[END_REF] 𝑡 𝑓 ⊗2 (𝑥, 𝑥) = 𝑃 𝑡 𝑓 2 (𝑥) for all 𝑥 ∈ 𝑆 and 𝑓 ∈  0 (𝑆).

The set ℋ is a vector space that contains the constant functions and the supremum of any bounded non-decreasing sequence of its non-negative elements (thanks to the monotone convergence theorem); the set 𝒞 is stable under pointwise multiplication.

This is a rather standard result. Here is an elementary argument, using the notations of the proof of Proposition 3. Since 𝑆

Proposition 8. If (𝐾 (𝑛) ) 𝑛≥1 is a consistent family of order-preserving Markov kernels on the successive powers of ℝ, then, for all 𝑛 ≥ 2 and 𝑓 ∈  𝑏 (ℝ 𝑛 ), 𝐾 (𝑛) 𝑓 ∈  𝑏 (ℝ 𝑛 ).

Proof. Consider 𝐱 = (𝑥 1 , … , 𝑥 𝑛 ) and 𝐲 = (𝑦 1 , … , 𝑦 𝑛 ). Let 𝐳 = (𝐱, 𝐲) = (𝑥 1 , … , 𝑥 𝑛 , 𝑦 1 , … , 𝑦 𝑛 ). Let (𝑋 1 , … , 𝑋 𝑛 , 𝑌 1 , … , 𝑌 𝑛 ) be a random vector on ℝ 𝑛 whose distribution is 𝐾 (2𝑛) (𝐳, ⋅), and let X = (𝑋 1 , … , 𝑋 𝑛 ) and 𝐘 = (𝑌 1 , … , 𝑌 𝑛 ). By the consistency property, the distribution of 𝐗 is 𝐾 (𝑛) (𝐱, ⋅) and the distribution of 𝐘 is 𝐾 (𝑛) (𝐲, ⋅), so that 𝐾 (𝑛) 𝑓 (𝐱) = 𝔼𝑓 (𝐗) and 𝐾 (𝑛) 𝑓 (𝐲) = 𝔼𝑓 (𝐘). Now, for 1 ≤ 𝑖 ≤ 𝑛, let 𝑍 + 𝑖 = 𝑋 𝑖 and 𝑍 - 𝑖 = 𝑌 𝑖 when 𝑥 𝑖 ≤ 𝑦 𝑖 , 𝑍 + 𝑖 = 𝑌 𝑖 and 𝑍 - 𝑖 = 𝑋 𝑖 when 𝑥 𝑖 > 𝑦 𝑖 , so that, thanks to the consistency property again, the distribution of (𝑍 + 1 , … , 𝑍 + 𝑛 ) is 𝐾 (𝑛) (𝐱∨𝐲, ⋅) and the distribution of

Moreover, by the order-preserving property of 𝐾 (2𝑛) , we have that 𝐗∨𝐘 = (𝑍 + 1 , … , 𝑍 + 𝑛 ) and 𝐗∧𝐘 = (𝑍 - 1 , … , 𝑍 - 𝑛 ). By the super-modularity property of 𝑓 , 𝑓 (𝐗 ∨ 𝐘) + 𝑓 (𝐗 ∧ 𝐘) ≥ 𝑓 (𝐗) + 𝑓 (𝐘), and we thus deduce that

Taking expectations, we deduce that 𝐾 (𝑛) 𝑓 (𝐱 ∨ 𝐲) + 𝐾 (𝑛) 𝑓 (𝐱 ∧ 𝐲) ≥ 𝐾 (𝑛) 𝑓 (𝐱) + 𝐾 (𝑛) 𝑓 (𝐲), so we have proved that 𝐾 (𝑛) 𝑓 is super-modular. □ [START_REF] Kamae | Stochastic inequalities on partially ordered spaces[END_REF] In this section, we explain why we think an element is missing in the proof of Theorem 4 in [START_REF] Kamae | Stochastic inequalities on partially ordered spaces[END_REF], and hence of Theorem 5 in [START_REF] Kamae | Stochastic inequalities on partially ordered spaces[END_REF], which is a corollary. 5.1. About Theorem 4. Theorem 4 in [START_REF] Kamae | Stochastic inequalities on partially ordered spaces[END_REF] considers a partially ordered Polish space (𝐸, ≼), two families of 𝐸-valued random variables (𝑋 𝑡 ) 𝑡∈ℝ + and (𝑌 𝑡 ) 𝑡∈ℝ + such that 𝑡 ↦ 𝑋 𝑡 and 𝑡 ↦ 𝑌 𝑡 are (random) càdlàg paths from ℝ + to 𝐸, and, assumes that, for all 𝑛 ≥ 2, 𝑥 1 , … , 𝑥 𝑛-1 ∈ ℝ, 𝑦 1 , … , 𝑦 𝑛-1 ∈ ℝ such that 𝑥 𝑖 ≼ 𝑦 𝑖 for 𝑖 = 1, … , 𝑛 -1, and ordered time indices 0 ≤ 𝑡 1 < ⋯ < 𝑡 𝑛 ∈ ℝ, one has [START_REF] Thomas | Interacting particle systems[END_REF] ℙ(𝑋

ABOUT THEOREMS 4 AND 5 IN

where ≼ 𝐬𝐭 denotes stochastic domination between probability measures on (𝐸, ≼) (see [START_REF] Kamae | Stochastic inequalities on partially ordered spaces[END_REF]). Assuming moreover that ℙ(𝑋 0 ∈ ⋅) ≼ 𝐬𝐭 ℙ(𝑌 0 ∈ ⋅), the conclusion of Theorem 4 is that one can define a family of pairs of 𝐸-valued random variables ( X𝑡 , Ỹ𝑡 ) 𝑡≥0 on the same probability space, in such a way that 𝑡 ↦ X𝑡 and 𝑡 ↦ Ỹ𝑡 are (random) càdlàg paths from ℝ + to 𝐸, that almost surely for all 𝑡, X𝑡 ≼ Ỹ𝑡 , that ( X𝑡 ) 𝑡≥0 has the same (joint) distribution as (𝑋 𝑡 ) 𝑡≥0 , and that ( Ỹ𝑡 ) 𝑡≥0 has the same (joint) distribution as (𝑌 𝑡 ) 𝑡≥0 .

We now consider the proof of Theorem 4. Given an increasing sequence 0 = 𝑡 1 < 𝑡 2 < ⋯, one can use ( 18) and Strassen's theorem to define inductively a sequence of pairs of 𝐸-valued random variables ( X𝑡 𝑛 , Ỹ𝑡 𝑛 ) 𝑛≥1 in such a way that, almost surely, for all 𝑛 ≥ 1, X𝑡 𝑛 ≼ Ỹ𝑡 𝑛 , and such that ( X𝑡 1 , … , X𝑡 𝑛 ) has the same distribution as (𝑋 𝑡 1 , … , 𝑋 𝑡 𝑛 ), and ( Ỹ𝑡 1 , … , Ỹ𝑡 𝑛 ) has the same distribution as (𝑌 𝑡 1 , … , 𝑌 𝑡 𝑛 ). (This mirrors the argument used to prove Theorem 2 in [START_REF] Kamae | Stochastic inequalities on partially ordered spaces[END_REF].)

The problem is the claim (without an explanation), in the proof of Theorem 4, that this approach can be used to produce a sequence of random variables ( X𝑡 𝑛 , Ỹ𝑡 𝑛 ) 𝑛≥1 with the above properties, where (𝑡 𝑛 ) 𝑛≥1 is an enumeration of a countable dense subset of ℝ + , since clearly such an enumeration cannot be done using an increasing sequence. Thus, to make the proof of Theorem 4 work, it seems that the following strengthening of Assumption (18) would be needed, allowing for unordered sequences of time indices: for every 𝑛 ≥ 2, 𝑥 1 , … , 𝑥 𝑛-1 ∈ ℝ, 𝑦 1 , … , 𝑦 𝑛-1 ∈ ℝ such that 𝑥 𝑖 ≼ 𝑦 𝑖 for 𝑖 = 1, … , 𝑛 -1, and pairwise distinct time indices 𝑡 1 , … , 𝑡 𝑛 ∈ ℝ + , one has [START_REF] Lindvall | On Strassen's theorem on stochastic domination[END_REF] ℙ(𝑋 𝑡 𝑛 ∈ ⋅|𝑋 𝑡 1 = 𝑥 1 , ⋯ , 𝑋 𝑡 𝑛-1 = 𝑥 𝑛-1 ) ≼ 𝐬𝐭 ℙ(𝑌 𝑡 𝑛 ∈ ⋅|𝑌 𝑡 1 = 𝑦 1 , ⋯ , 𝑌 𝑡 𝑛-1 = 𝑦 𝑛-1 ).


Keywords: axial flux, in-wheel machine, seven-phase machine, multi-phase machine, bi-harmonic machine

For in-wheel machine, outer rotor machines appear as a natural solution. Practically these machines are either radial-flux with one rotor or axial-flux with two rotors. The paper is proposing a machine with three outer rotors with two different polarities in order to reduce useless end-windings while keeping an acceptable thickness for the radial-flux rotor and high torque quality. This Hybrid Flux Permanent Magnet original structure (named HFPM) is possible thanks to the use of seven phases. The third rotor can be considered as an option of an initial double-rotor axial-flux machine in order to increase the torque density. First, the machine structure and the winding design are presented; then, based on 3D finite element method, comparison between the two machines, with two or three rotors, are provided in terms of torque densities and qualities.

I. INTRODUCTION

OR in-wheel machine, outer rotor machines appear as a natural solution [START_REF] Wang | Modular threephase permanent-magnet brushless machines for in-wheel applications[END_REF]. Practically these machines are either radial-flux with one rotor or axial-flux with two-rotors [START_REF] Ifedi | Fault-tolerant in-wheel motor topologies for high-performance electric vehciles[END_REF]. Radial-flux are classical and easy to make with stacked iron sheets but when the base speed is weak such as for bicycle or scooter applications [START_REF] Yang | Development of an External Rotor V-Shape Permanent Magnet Machine for E-Bike Application[END_REF], it is remaining unused space inside the stator [START_REF] Zhang | Comparative Study on Two Modular Spoke-Type PM Machines for In-wheel Traction Applications[END_REF] (Fig. 1 (a)) except with integrated drive such as a commercial e-bike machine [START_REF] Dhawan | Multi-Phase Multi-Pole Electric Machine[END_REF], as shown in Fig. 1 (b). Axial-flux machine with two rotors (Fig. 2 (a)) are attractive to improve the compactness [START_REF] Gieras | Axial Flux Permanent Magnet Brushless Machines[END_REF] [START_REF] Hemeida | Applicability of Fractional Slot Axial Flux Permanent Magnet Synchronous Machines in Field Weakening Region[END_REF]. Nevertheless, even if two rotors allow compensating globally the axial forces between stator and rotor, it is necessary to link together mechanically the two axial rotors in order to give sufficient mechanical strength.

The first idea of the paper is to make active this mechanical connection between the two axial rotors by transforming it into a third rotor as in [START_REF] Gieras | Axial Flux Permanent Magnet Brushless Machines[END_REF] (Fig. 2 (b)), thus also making active part of the end windings. Unfortunately, when the number of poles is the same for this third rotor, the necessary thickness of the rotor yoke, in order to verify magnetic constraint of saturation, can be important [START_REF] Zhu | Investigation Study of the Influence of Pole Numbers on Torque Density and Flux-weakening Ability of Fractional Slot Concentrated Winding Wheel-Hub Machines[END_REF] in comparison with a simple linking mechanical piece. As consequence, the second idea is to use a higher polarity for the radial-flux rotor while maintaining a high quality of torque as it is possible with multiphase machines [START_REF] Gong | Design Considerations of Five-Phase Machine with Double p/3p Polarity[END_REF] thanks to decoupling between harmonics.

Considering now the choice of the axial-flux configuration it is existing different solutions such as NN type slotted Torus machine (Torus-S) or NS type Torus-S [START_REF] Adyn | Axial Flux Permanent Magnet Disc Machines: A Review[END_REF], as shown in Fig. 3. The last one seems to be more attractive since the thickness of the stator is only determined by the depth of the slot and mechanical constraints for the rigidity of the stator [START_REF] Zhang | Development of a Yokeless and Segmented Armature Axial Flux Machine[END_REF]. In this case, 3-dimensionnal Soft Magnetic Composites (SMC) powder are difficult to use because of their weak mechanical properties. Costly enrolled iron sheets are more adapted. On contrary NN type Torus-S stator thickness is constrained by the magnetic polarity of the axial rotor: large unnecessary end-windings can be then induced [START_REF] Locment | Vectorial approach-based control of a seven-phase axial flux machine designed for fault operation[END_REF]. In this paper, the third rotor makes active the end-windings: the stator thickness is then less a drawback. Moreover, low-cost material with modest mechanical properties such as SMC powder can be advantageously used.

In the paper, the machine structure and the winding design are first presented; then, based on 3D finite element method, the initial machine with two axial rotors is compared to the proposed three-rotor machine.

A. Machine topology

The proposed machine with NN type Torus-S structure is shown in Fig. 4, including the 1/4 of the structure in Fig. 4 (a) and the exploded structure of the whole machine in Fig. 4 (b). It is a sandwich structure i.e. two outer rotors with PMs surface-mounted and one stator with toroidal tooth concentrated winding. By adding a third radial rotor, the upper end-windings of this NN-type Torus-S structure become active with an even greater contribution as is the axial length of the stator core made of SMC material. In order to enhance the torque density and power density, the radial rotor with V-shape magnets is designed. Moreover, the number of poles of the radial rotor is 36, i.e. 3 times more than the other two axial-flux rotors with 12 poles. This choice is possible keeping a high torque quality by using a machine with seven phases thanks to the decoupling between the first and third harmonics. The characteristics of this particular machine will be investigated the following section.

B. Non-overlapping winding design

The first constraint imposed to the toroidal windings of the machine is the making simplicity: a tooth concentrated winding without overlapping is chosen. As consequence, the winding is then often a fractional slot concentrated winding (FSCW) [START_REF] El-Refaie | Fractional slot concentrated windings synchronous permanent magnet machines: opportunities and challenges[END_REF] whose main challenges is the rich content of space harmonics of magneto-motive force (mmf), including both sub-and high-orders. These parasite spatial harmonics can lead to eddy current losses, vibration and noise, local core saturation etc., which deteriorate the performance of the machine. It is necessary to choose carefully the slot/pole combination in order to avoid these drawbacks.

Two criteria are used to select the slot/pole combination of the proposed machine:

 The first one is the high value of winding factor, which is related to the torque density. Due to the specificity of the machine, the winding factors of both the fundamental and the third harmonic are expected to be important.  The second one is the content of the parasitic harmonics of mmf under the current injection of both the fundamental and the third harmonic. Considering the first criterion, 14-slot/12-pole combination is selected [START_REF] Aslan | Slot/Pole Combinations Choice for Concentrated Multiphase Machines Dedicated to Mild-Hybrid Applications[END_REF]. The winding factors of the fundamental and the third harmonic are 0.97 and 0.78 respectively. The mmf spectrum is then analyzed and presented in Fig. 5. As it can be seen, the working harmonic is the 6th with the fundamental current supply, and the 18th with the 3rd harmonic current. Unfortunately, the parasitic harmonics of mmf are rich, especially the ones with big value of magnitude, such as the 8th and the 20th. In order to reduce the negative effects induced by the parasitic harmonics, stator shift technique is applied [START_REF] Reddy | Generalized Approach of Stator Shifting in Interior Permanent-Magnet Machines Equipped With Fractional-Slot Concentrated Windings[END_REF]. A novel winding topology with 28-slot/12-pole combination is achieved. It can be noted that in Fig. 5, the mmf spectrums under the injection of both the fundamental and the 3rd harmonic are improved. Meanwhile, the winding factors are also with the same high value, i.e. K1=0.97 and and K3=0.78. Therefore, a higher torque density is expected.

C. Prototype specifications

Fig. 7 shows the SMC stator core and the third laminated radial rotor of the prototype. The back iron of the two axial rotors are made of non-laminated iron 1010 and the third radial rotor is laminated with sheets.

Fig. 8 shows the B-H curves of three magnetic materials used for the prototype: SMC, laminated sheets, and 1010 iron. It can be seen that the difference of magnetic property of the three materials is important. The low cost SMC material is characterized with low values both in permeability and saturation flux density. Therefore, the axial length of the stator with SMC is greater than that with enrolled iron sheets.

III. MATHEMATICAL MODEL

Neglecting saturation and reluctance effects, the sevenphase machine can be considered as three two-phase fictitious machines and one single-phase machine magnetically decoupled [START_REF] Semail | Right harmonic spectrum for the back-electromotive force of an n-phase synchronous motor[END_REF]. Each fictitious machine is characterized by a set of harmonics as shown in Table II. Under assumption of no reluctance effect, the fictitious machines are magnetically independent, however they are mechanically linked together, i.e. the total output torque of the machine is the sum of the fictitious machines. It should be noted that the zero-sequence machine is not taken into account when the machine is wye connected.

A. Electromagnetic torque expression

The electromagnetic torque of the seven-phase machine is derived in this part, using fictitious machines approach [START_REF] Locment | Vectorial approach-based control of a seven-phase axial flux machine designed for fault operation[END_REF] with the formula (1). In each fictitious machine, it can be considered that only the main harmonic is contributing significantly to the torque. As consequence the expression of the electromotive force in each fictitious machine can be approximated by:

      k k k Mk x kp x kp E e     cos sin 2 2 7    (2)
with Ek the RMS value of the harmonic k of back-emf.

As the adopted strategy chosen for the supply in the design step is classically MTPA (Maximum Torque Per Ampere), the vector current is co-linear with the back-emf vector. The vector current is then:

      k k k Mk x kp x kp E K i     cos sin 2 2 7 0    (3)
By ( 1), the obtained torque with three injected harmonics is then:

      / ) / ( ) / ( 1 7 2 1 5 2 1 3 2 1 0 E E E E E K T (4)
And the value of K0 is linked to the AL linear current density chosen for the design of the machine.

B. V-shape PM machine modeling

In order to reinforce the flux density in the air gap and improve the flux-weakening capability, the radial rotor is designed using V-shaped poles. An equivalent magnetic circuit using Thevenin's theorem [START_REF] Thelin | Analytical calculation of the airgap flux density of PM synchronous motors with buried magnets including axial leakage, tooth and yoke saturations[END_REF] is therefore built for the modeling of the radial rotor, as shown in Fig. 9.

In Fig. 9, Rtr, Rts, Rys, Ryr are related to the reluctance of stator and rotor teeth and yokes; Rg is the reluctance of the air gap; Rb is the reluctance of flux barriers; RFe is the reluctance of the saturated iron bridges under one pole. Using Thevenin-equivalent theory, the following relationship can be obtained:

th m th sat F F R    ( 5
)
Where Fth is the equivalent Thevenin mmf; Fm is the mmf generated from PMs; Rth is equivalent Thevenin reluctance which can be obtained by the sum of PMs reluctance and internal airgap around the PMs; Фsat is the flux required to saturate the iron bridges which can be calculated as follows,

= sat sat Fe f B w k L  ( 6
)
Where Bsat is the flux density in saturated iron and Kf is the stacking factor for the iron lamination; L is the axial length of the magnet; wFe is the sum of the iron bridge widths under one pole.

IV. 3D FINITE ELEMENT MODEL ANALYSIS

A. Flux density distribution

The 3D-FEM is used to validate the pre-design of the HFPM machine. In order to reduce computation time, 1/4 of machine model is built and analyzed. The generated mesh and the flux density distribution under no-load condition is shown in Fig. 10. As can be seen, in the stator core and the two axial rotors, the maximum of magnetic flux density is about 1.6T. In the radial rotor, the iron bridge is saturated and the maximum value of the magnetic flux density is 2T.

B. Flux Linkage

Fig. 11 shows the flux linkage and its harmonic spectrum of one phase in no-load condition. It can be noted that the 1 st and the 3 rd harmonic component are dominant. The ratio between them Φ3/Φ1 is equal to 0.41. The other harmonics, such as the 5 th and the 9 th harmonics, are both of very small magnitude and can be neglected in the following analytical analysis.

C. Cogging Torque

Fig. 12 shows the 3D-FEM predicted cogging torque, in which the red curve is related to the proposed 3-rotor HFPM machine. The blue curve is related to the initial machine called AFPM obtained by suppressing the magnets of the third radial rotor of HFPM. It can be seen that the cogging torque is increased by adding the magnet in the 3 rd radial rotor, however, they are both small values (less than 6% for HFPM and 3% for AFPM) in comparison with the rated torque value.

D. Electromagnetic Torque

As it can be considered that the flux linkage contains only the first and third harmonic, only the first and third harmonics of current are injected. For a given RMS value of current, a comparison of the torques obtained with MTPA strategy as presented in III-A is made considering three kinds of supply: with only the first, only the third, and both the first and the third sinusoidal currents as shown in Fig. 13. It can be remarked that the peak value of the currents is of same order. The waveforms of the torque at 600 rpm speed are then shown in Fig. 14. The torque ripples can be considered as low even when two harmonics of close amplitudes are injected (about 6 %), and the main frequency of the pulsation is 840Hz as expected (14*p*600/60). It confirms that the first and third harmonic interactions are weak because of the orthogonality of the two "" planes associated to these two harmonics. Thanks to (4) and for the considered seven-phase machine whose Φ3/Φ1=0.4, the torque under bi-harmonic current supply is 53% (resp. 26%) higher than this one obtained with only the first (resp. the third) harmonic current: the global torque is the sum of the torques produced in each "" plane. With a same level of copper losses and about same peak current, the bi-harmonic supply is obviously interesting.

V. PERFORMANCE COMPARISON

The performance of the HFPM machine is compared to the AFPM machine presented in IV-C.. Fig. 15 shows the comparison of back-EMF at no-load. It can be seen that the amplitudes of the 3 rd harmonic back-EMF for the HFPM machine are significantly increased, 5 times higher than the one of AFPM. However, the fundamental component is slightly reduced 5%, due to the saturation of the SMC material with the addition of the 3 rd radial rotor.

Table III shows the volume, mass, volume of magnet, volume torque density in volume, and the mass torque density.

Under the same 3.15A RMS current injection, the output torque of the HFPM is increased by 51% under bi-harmonic current supply, as shown in Fig. 16 (a). As consequence, the torque densities in volume and in mass are both significantly improved by 50% and 54%, as shown in Fig. 16 (b). The main drawback is a higher cost because of the addition of magnets. The third rotor can be considered as an option to increase easily the torque keeping the same volume of the machine.

VI. CONCLUSION

In this paper, by simply adding magnets inside a mechanical linking between two axial rotors of a 7-phase axial flux permanent magnet machine, a hybrid flux permanent magnet machine is obtained. Using specific property of multiphase machine, the chosen number of poles of the third radial rotor is three times this one of the two other axial rotors: the corresponding small thickness of this third radial allows to increase by more than 50% mass and volume torque densities. It is then possible to propose two machines with many identical components and whose geometry and mass are similar but whose performances are quite different. As example of application, two versions of a scooter could be then easily obtained with only a different quantity of Permanent Magnets inside. Results have been only presented for MTPA strategy. The degrees of freedom of the machine for torque production allow other strategies. Flux weakening capabilities will be studied in next paper.

Fig. 1 .Fig. 2 .

 12 Fig. 2. Outer rotor PM machine from [6] (3/iron-4/magnets)), (a) twoaxial rotors, (b) with a supplementary radial-flux rotor

  Fig.6shows the novel winding topology. The coils of the same color belong to the winding of the same phase. It is toroidal type and the non-overlapping one, even with stator shift technique application.

Fig. 6 .

 6 Fig. 6. Novel 28-slot/12-pole winding topology

7 .

 7 Fig. 8. B-H curves of three materials

  the rotating speed of the rotor in rad/s; Mk e  and Mk i  are back-emf and current vectors of fictitious machine Mk with k=1, 3, 5. They are obtained by vector projection of back-emf and current vectors in the associated planes of each fictitious machine, defined by the classical Concordia transformation.

Fig. 9 .

 9 Fig. 9. Equivalent magnetic circuit based on Thevenin theorem

Fig. 10 .

 10 Fig. 10. 3D FEM analysis results with (a) mesh generation and (b) flux density distribution of HFPM machine

Fig. 11 .

 11 Fig. 12. FE predicted cogging torque at 600 rpm

Fig. 13 .

 13 Fig.[START_REF] El-Refaie | Fractional slot concentrated windings synchronous permanent magnet machines: opportunities and challenges[END_REF]. For a given RMS current, phase currents for the MTPA supplies at 600 rpm (62.8rd/s) with a period of 15.9ms for the first harmonic

Fig. 14 .

 14 Fig. 14. For given RMS current, comparison of the torques under three kinds of supply: only the first, only the third, and both the first and the third harmonic current at 600 rpm (62.8 rd/s)

Fig. 15 .

 15 Comparison of back-EMF with (a) curves and (b) spectrum

Fig. 16 .

 16 Fig. 16. for a given RMS current, performance comparison with (a) torque and (b) torque densities of both SMC-stator and PM rotors machines with MTPA strategy

TABLE I

 I

		Axial length of axial rotor core (mm)	5.3
		thickness of axial magnet (mm)	1.9
		thickness of air gaps (mm)	1
		Outer diameter of radial rotor (mm)	183.54
		Inner diameter of radial rotor (mm)	161
		Axial length of the radial rotor (mm)	47.6
		Thickness of V-shape magnet (mm)	2
		Nominal speed (rpm)	600
		Nominal torque (Nm)	8
	PARAMETER AND MACHINE DIMENSIONS
	Number of phases	7
	Number of slots	28
	Number of pairs of poles (axial rotors) p	6
	Number of pairs of poles (radial rotors) 3p	18
	Stator core outer diameter (mm)	145.9
	Stator core inner diameter (mm)	50.6
	Slot dimension (mm)	4.3*13.1
	Number of wires per slot	22
	Axial length of stator core (mm)	47.6

TABLE II THE

 II HARMONICS CORRESPONDING TO FICTITIOUS MACHINES

TABLE III IMPACT

 III OF ADDITION OF MAGNETS IN THE THRID RADIAL ROTOR

	Machine	Volume (dm 3 )	Mass (kg)	Volume magnet (dm 3 )	Torque N.m)	Torque density (Nm/m 3 )	Torque density (Nm/kg)
	AFPM	1.693	9.21	0.03	6.3	3721	0.68
	HFPM	1.693	9.01	0.10	9.5	5595	1.05
	Variation (%)	0	-2	+233	+51	+50	+54