COMPLEX INTERTWININGS AND QUANTIFICATION OF DISCRETE FREE MOTIONS

. The traditional quantiﬁcation of free motions on Euclidean spaces into the Laplacian is revisited as a complex intertwining obtained through Doob transforms with respect to complex eigenvectors. This approach can be applied to free motions on ﬁnitely generated discrete Abelian groups: Z m , with m P N , ﬁnite tori and their products. It leads to a proposition of Markov quantiﬁcation. It is a ﬁrst attempt to give a probability-oriented interpretation of exp p ξL q , when L is a (ﬁnite) Markov generator and ξ is a complex number of modulus 1


Introduction
Broadly speaking, quantification is the mathematical link between classical and quantum mechanics and has led to the tremendous development of semi-classical limits, see e.g. the book of Zworski [6] and the references therein.Usually, the underlying state spaces and dynamics are continuous, since it is important to be able to consider high frequencies or large values in the spectrum of the associated Hamiltonian operators, quantifying the classical energies.Nevertheless, quantification has also been considered for discrete spaces, specially in the context of wave equations, see for instance Macià [3,4], Mielke et al. [2,5].In general the authors also "quantify" space, by requiring that the distance separating nearest neighbors is of the same order as the semi-classical parameter h ą 0, in some sense their results concern approximation of continuous spaces.Here, we would like to consider fixed discrete state spaces, even finite sets, e.g.just on Z 3 := Z{p3Zq.It leads us to propose a general definition of quantification in the context of Markov process theory and to see that it is meaningful, at least for the simplest examples of free motions.Of course this first attempt will have to be tested with more interesting examples, but it gives a feeling of what we are looking for.

Definition of a Markovian quantification
Let V be a metric space, whose distance is denoted ρ, endowed with a (non-negative) measure µ.Consider a Markov generator L which is self-adjoint in L 2 pµ, Cq, in probabilistic terminology, we say that µ is reversible for L. Functional calculus enables one to define the operator P ξ := exppξLq for any ξ P C. At least for ξ P C `, the set of complex numbers whose real part is non-negative, the domain of P ξ is the whole space L 2 pµ, Cq, since L. MICLO the spectrum of L is included into p´8, 0s.When L is a jump generator with bounded jump rates (in particular when V is finite), the domain of P ξ is L 2 pµ, Cq for all ξ P C.
Let V be another topological space endowed with a continuous and onto mapping π : V Ñ V .It means that V can be seen as a weak kind of bundle over V : for any y P V , denote V y := π ´1pyq, it will be convenient to write the elements of V under the form py, zq, where y P V and z P V y .The base component y should be thought of as a position and the fiber component z as a generalized speed or impulsion.A priori no assumption is made on the fibers V y , for y P V , they could be not all the same.Let L be a Markov generator on V such that for any py 0 , z 0 q P V, the martingale problem associated to L and to the initial point py 0 , z 0 q is well-posed.We will denote by X := pX psqq sě0 := pY psq, Zpsqq sě0 a corresponding càdlàg Markov process and P py0,z0q will stand for the underlying probability measure.It is convenient to add the time as a coordinate to V: on V := R `ˆV , whose generic elements are denoted by pt, y, zq, consider the generator L := B t `L. (1.1) A corresponding Markov process starting from pt 0 , y 0 , z 0 q is X := pXpsqq sě0 := pt 0 `s, Y psq, Zpsqq sě0 under P py0,z0q .The latter underlying probability will also be denoted P pt0,y0,z0q to indicate that X is starting from pt 0 , y 0 , z 0 q (and P py0,z0q will stand for P p0,y0,z0q ).
We say that L is a quantification of L if there exists a family pF h q hą0 of continuous and bounded mappings from V ˆV to C satisfying the three following conditions: (H1) Probability density: for any h ą 0 and any pt, y, zq P V, we have (H3) Intertwining: for any py 0 , z 0 q P V, there exists ξ 0 P T, the circle of complex numbers of modulus 1, such that, @ h ą 0, @ s ě 0, @ x P V, E p0,y0,z0q rF h pXpsq; xqs " P ξ0s rF h p0, y 0 , z 0 ; ¨qspxq.
Note that the rhs is a priori only defined µ-a.s. in x P V , since P ξ0hs is an operator on L 2 pµ, Cq.Nevertheless, as an immediate consequence of dominated convergence, the lhs is continuous with respect to x P V .The meaning of the above equality is thus that there exists a continuous version of P ξ0s rF h p0, y 0 , z 0 ; ¨qs given by the lhs.In all our examples of discrete free motions, ξ 0 will in fact only be dependent on z 0 .
A drawback of Assumption (H3) is its lack of stability by tensorization (except in the traditional situation where ξ 0 does not depend on py 0 , z 0 q P V) and the following extension shall more generally be considered.We say that a commuting family pL pjq q jPJ of reversible Markov generators in L 2 pµ, Cq, where J is a finite index set, is a multi-dimensional quantification of L, when there exists a family pF h q hą0 of mappings as above satisfying (H1), (H2) and (H4) Multi-dimensional intertwining: for any py 0 , z 0 q P V, there exists a family pξ pjq 0 q jPJ P T J , such that, @ h ą 0, @ s ě 0, @ x P V, E p0,y0,z0q rF h pXpsq; xqs " exp ˜s ÿ jPJ ξ pjq 0 L pjq ¸rF h p0, y 0 , z 0 ; ¨qspxq.
Let us make a few comments about the definition of quantification by (H1), (H2) and (H3).The multidimensional extension will be justified by Theorem 1.8 below and discussed at the end of Section 5.
Remark 1.1.In the usual real free motion Schrödinger case recalled in Theorem 1.5 below, ξ 0 " i (or ξ 0 " ´i) does not depend on py 0 , z 0 q P V.Under natural assumptions, it is equivalent to the fact that for any h ą 0 and x P V , the process pF h pX psq; xqq sě0 is deterministic.Indeed, for the direct implication, note that P ξ0 is a unitary operator for any ξ 0 P iR, so we get for all h ą 0, py 0 , z 0 q P V and s ě 0, where we used (H1).
It follows from the above computations that we must have equality in the Cauchy-Schwarz' inequality (1.2).This is only possible if F h pXphsq; xq is a deterministic quantity under P py0,z0q , for any fixed h ą 0, py 0 , z 0 q P V, s ě 0 and x P V .We can go further and conclude that the process X itself is deterministic, under the assumption that the family pF h q hą0 is discriminating, in the sense that for any pt, y, zq " pt, y 1 , z 1 q P p0, `8q ˆV, there exist h ą 0 and x P V such that F h pt, y, z; xq " F h pt, y 1 , z 1 ; xq (this condition will be satisfied in all our examples).Conversely, when pF h pX psq; xqq sě0 is deterministic, we get in (H3) @ h ą 0, @ py 0 , z 0 q P V, @ s ě 0, @ x P V, F h pXphsq; xq " P ξ0hs rF h p0, y 0 , z 0 ; ¨qspxq.
Taking the square of the modulus and integrating, we deduce that Now assume that L is irreducible, namely its eigenvalue 0 is of multiplicity 1.Its eigenspace is then reduced to the constant functions.Spectral calculus implies that for ξ 0 R iR and s, h ą 0, all the eigenvalues, except 1, of P ξ0hs have a modulus smaller (respectively larger) than 1 if pξ 0 q ą 0 (resp.pξ 0 q ă 0).Thus for the above equality to hold, the mapping V Q x Þ Ñ F h p0, y 0 , z 0 ; xq must be constant, for any h ą 0 and py 0 , z 0 q P V. Assumption (H2) then leads to a contradiction, except in the trivial case where V is a singleton.Thus, we must have ξ 0 P piRq X T " t˘iu.These considerations show why when V is discrete (then X cannot be deterministic if we want it to be Markovian), one has to consider for ξ 0 other elements of T than ˘i.The fact that we imposed ξ 0 P T is just a normalization taking into account that it is always possible to multiply a Markov generator by a positive constant: it amounts to multiply the time by the same constant.
Remark 1.2.Let us discuss the meaning of the above quantification, in a very heuristic way.According to (H1) and (H2), for any s ě 0 and x P V , we have for h ą 0 small and where for any y P V , δ y stands for the Dirac mass at y. Taking into account the previous remark, we "infer", in the case where ξ 0 " i, that (H3) writes namely, |P ihs r¨s| has approximatively transported a δ y0 into a δ Y phsq , if a sense could be given to these expressions.More rigorously, an equality cannot hold in (1.3), since we would deduce that the evolution of Y phsq, for s ě 0, does not depend on z 0 , which is typically not true.In fact, an important motivation for the quantification procedure is to remove the fiber components of the evolution of pY phsq, Zphsqq sě0 and to include them (as "phases") into the evolving complex-valued distributions on V given by the family pP ihs rF h p0, y 0 , z 0 ; ¨qsq sě0 .According to the interpretation of y and z as position and speed components, the stochastic process pY phsqq sě0 could be seen as a second order Markov process (while pY phsq, Zphsqq sě0 is a usual, i.e. first order, Markov process).Quantification then looks like a crazy attempt to approximate the second order Markov process pY phsqq sě0 by a "first order Markov process" on V whose semi-group would be pP ihs q sě0 , puzzling as the imaginary-valued times remain.A similar heuristic could be proposed in the case of discrete state spaces V : then for all y P V , a δ y " δ y , interpreting the Dirac mass as a density with respect to the counting measure (i.e. the Kronecker delta), and (1.3) would have to be modified into the statement ErexppiΘphsqqδ Y phsq s ≈ P ξ0hs rδ y0 s, where Θphsq is a random real phase, with Θp0q " 0. A difference with the usual Schrödinger case is that a part of the information about the fiber component will be transferred to the complex time, since ξ 0 will depend on z 0 .
Remark 1.3.The concentration property of (H2) is of order 1, in the sense that a concentration of order k P Z `can be defined by requiring that for any r ą 0 and any compact set K Ă R `ˆV , lim In the usual Schrödinger case, the concentration will even be strong, in the sense that it is of any order k P Z `.
The order 1 is the smallest meaningful order for quantification: for fixed s ą 0 and small h, the movement of X phsq from X p0q will typically be of order h (either in distance or in probability to have jumped to a distance of order 1), because we are looking for generators L leading to "ballistic" behaviors, even if the associated motions are not deterministic.As a consequence, to get a pertinent result, the concentration should be stronger than h and this is exactly what is requiring (H2).In our simple discrete examples, we will have for h ą 0 small, with the notations of (H2), sup pt,y,zqPV : pt,yqPK ż ρpx,yqěr |F h pht, y, z; xq| 2 µpdxq " Oph 2 q.
Remark 1.4.The above definition of quantification is not completely satisfactory, because a Markov process on V can be seen as its own quantification.Indeed, let be given a jump generator L on a discrete space V .It is the quantification of L := L on V := V ˆt0u " V , by choosing ξ 0 " 1 and @ h ą 0, @ pt, y, 0q P V, @ x P V, F h pt, y, 0; where the rhs is the Kronecker delta.A similar result holds on general state spaces V and Markov generators L, by considering a family pF h q hą0 such that |F h pt, y, 0; xq| 2 µpdxq is an approximation of the Dirac mass at y and F h pt, y, 0; xq is symmetrical in y, x and does not depend on t.
This kind of degeneracy could be avoided by requiring in (H3) that ξ 0 P Tzt´1, 1u, the case ξ " ´1 appearing for instance when L generates deterministic motions that can be reversed in time.This prohibition of real values for ξ 0 is in the spirit of Remark 1.2, where we are trying to get a probabilistic interpretation of P ξ0t , for t ą 0, task which is particularly puzzling when ξ 0 R R. Furthermore, we are looking for generators L whose associated motions contain the smallest possible quantity of randomness (the most ballistic, with the wording of Rem.1.3) and according to Remark 1.1, this is reflected by the least real possible ξ 0 .Nevertheless, it will become apparent in Sections 4 and 5 that real values should be allowed for ξ 0 for some py 0 , z 0 q P V.
Another drawback of the present definition of quantification, is that given the Markov generator L, there is not a unique L to which L is the quantification, up to natural identifications.Indeed, given a quantification, another one can be obtained by enlarging the fibers (for instance by duplicating them).
A notion of minimal quantification up to appropriate isomorphisms is thus missing and this minimization should concern first the real part of ξ 0 and next the size of the fibers of V.
In the present paper, we will investigate some simple situations of free motions: the generator L will only act on the base component y, but in a way parametrized by the fiber component z.Thus the fiber component will not move: for any s ě 0, Zpsq " z 0 .This feature imposes that all the fibers are the same, property which was not required in the general definition given above.We will have that V " V ˆW , where W is another topological space, even if its topology will not play a role when the fiber component does not move.
As already mentioned, our main goal is to define quantification on finite state spaces V .This demand is the reason why we allow the process pX psqq sě0 to be stochastic and for the complex number ξ 0 P T not to be necessary equal to ˘i, in accordance with Remark 1.1.In fact our initial motivation is an inverse problem: given a finite irreducible Markov generator L, we are wondering if we can find a "natural" dynamics to which it is a quantification.It leads to other questions that are out of the scope of this paper, in particular because the limitations pointed out in Remark 1.4 have to be overcome first: is there always a "semi-classical limit" (i.e. a corresponding minimal generator L), is it unique?These interrogations will be interesting even for non-reversible generators, in fact, the reversibility assumption was mainly adopted to simplify the definition of the operators exppξLq, for ξ P C, but there is no such difficulty when V is finite.Here we will answer the first question only for the usual Laplacian on a discrete multidimensional torus V " Z n1 ˆZn2 ˆ¨¨¨ˆZ nm , where m P N and n 1 , n 2 , ..., n m P Nzt1u.In future works, we hope to deal with the challenges of adding potential energy terms or of considering general weighted graphs.This is not just for the sake of generality: to define (stochastic)

L. MICLO
Hamiltonian dynamics on graphs whose quantification corresponds to Metropolis algorithms is an interesting perspective in the field of optimizing and sampling stochastic algorithms.

Results on free motions
After recalling the classical example of the free motion on R, we will present the quantification of discrete free motions, on Z and on finite tori.
For the free motion on R, we take V := R endowed with the Laplacian L " B 2 .The fiber space W is also equal to R and we consider the operator L " 2zB y acting on V ˆW " R 2 .We have Theorem 1.5.The generator L is a quantification of L and in (H3), ξ 0 " i is independent of py 0 , z 0 q.Here is our first example of quantification of a discrete free motion.Take V := Z endowed with the generator L acting on bounded functions f on Z via The fiber space is W " r´π, πq and we consider the operator L acting on bounded functions f defined on Z ˆr´π, πq by @ py, zq P Z ˆr´π, πq, Lrf spy, zq := 2 |sinpzq| pf py `signpzq, zq ´f py, zq, where signpzq := 1 when z ě 0 and signpzq " ´1 when z ă 0.
We have the following theorem.
Theorem 1.6.The generator L is a quantification of L and in (H3), ξ 0 " i expp´i |z|q.
Our second example of quantification of a discrete free motion is on the finite circle Z n , where n P Nzt1u.Consider on Z n the discrete Laplacian L, equally given by (1.4) for x P Z n .The fiber space is now and we consider the operator L acting on any function f defined on the finite set Z n ˆWn via (1.5),where py, zq P Z n ˆWn .In this context, Theorem 1.6 is valid, apparently without amendment, but the domains of the operators are not the same.
Theorem 1.7.The generator L is a quantification of L and in (H3), ξ 0 " i expp´i |z|q.
To see the interest of (H4), we now come to a "multi-dimensional" example.We take V " Z n1 ˆZn2 ˆ¨¨¨Ẑ nm , where m P N and n 1 , n 2 , ..., n m P Nzt1u.Let J := rrmss := t1, 2, ..., mu and for j P J, consider the generator L pjq acting on functions f defined on V via @ x P V, L pjq rf spxq := f px `ej q `f px ´ej q ´2f pxq, where e j is the element of V whose coordinates are all 0, except the j-th one equal to 1 P Z nj .Let W " W n1 ˆWn2 ˆ¨¨¨ˆW nm , where the factor spaces are defined as in (1.6).Consider the operator L acting on functions f defined on V ˆW via @ py, zq P V ˆW, Lrf spy, zq := 2 ÿ jPJ |sinpz j q| pf py `signpz j q, zq ´f py, zq, where pz 1 , z 2 , ..., z m q are the coordinates of a generic element z P W .
The following result can be deduced from Theorem 1.7 by tensorization.
Theorem 1.8.The family of generators pL pjq q jPJ is a multi-dimensional quantification of L and in (H4), ξ pjq 0 " i expp´i |z j |q, for all j P J.
All these results will be in proven in the same manner, via an intertwining using complex kernels.It will also be applied to the free motion on the circle V , seen as R{p2πZq, endowed with the Laplacian L " B 2 .The fiber space is now Z and we consider the operator L " 2zB y acting on V ˆW " R{p2πZq ˆZ.Theorem 1.5 is valid in this context: Theorem 1.9.The generator L is a quantification of L and in (H3), ξ 0 " i is independent of py 0 , z 0 q.
The discreteness of the fiber space Z is now quite suspect and we are wondering if Theorem 1.9 is true with the same operator L but with W " R (V ˆW should then be seen as the cotangent space of the circle).If it is not, it would mean that our definition of quantification is too strict: in (H3) and (H4), the equality should be required only up to additive terms negligible with respect to h, as in (H2).In the present paper, we fostered the investigation of the perfect intertwinings (H3) and (H4), valid for all times.
The plan of the paper is as follows.The next section presents our intertwining method: it is based on a Doob transform with respect to complex valued eigenvectors which do not vanish.Sections 3 and 4 treat of the free motions on R and Z, respectively.The last section deal with the cases of tori.We will also see that there is no difficulty in tensorizing Theorems 1.5 and 1.9 and that the tensorization of Theorem 1.6 can be done as in Theorem 1.8.

An intertwining relation
Here, the intertwining relation at the base of the construction of the family pF h q hą0 is deduced via a Doob transformation applied with complex-valued eigenfunctions, contrary to the usual ground state transforms, which are usually considered relatively to positive eigenvectors.The framework is a little more general than in Section 1.1, since reversibility is not required, nor even invariance with respect to a measure µ.
On a state space V , consider a Markov generator L " 0 defined as an endomorphism on a unitary algebra A of C-valued functions.Let ϕ P A be an eigenvector of L associated to an eigenvalue λ P C (in the applications of the following sections, we will only consider reversible L with real eigenvalues, nevertheless, it will be convenient to deal with C-valued functions).Assume that ϕ does not vanish on V and that ϕ ´1 P A, to be able to consider the Doob transform of L by ϕ, acting on A: Assume that we can find χ, ζ P T (the circle of complex numbers of modulus 1) and a Markov generator Remark 2.1.In general such a decomposition is not unique, as we will see in Remarks 3.1 and 4.1 of the next sections.
Suppose furthermore that L and p L commute.This assumption is very strong, but will be satisfied by our free motion examples.In more general situations, as those mentioned at the end of Section 1.1, some commutation relations will enter into play, we hope to investigate them in future works.
In the introduction, P η stood for exppηLq for η P C and we would like to define similarly r P η and p P η .In the present framework, the meaning of the exponential is not so clear.Not wanting to obscure the simplicity of the following arguments, let us first assume that A is a Banach algebra and that L is a bounded operator on A. With these hypotheses, P η , r P η and p P η are naturally defined as exponentials for any η P C.These assumptions hold It follows that for any η, η 1 P C p P η 1 r P ´ζpη`η 1 q " p P η 1 p P ´pη`η 1 q P ξpη`η 1 q " p P ´η P ξη P ξη 1 " r The above constructions depend on the choice of ϕ, but only up to a factor and it is tempting to parametrize them by the eigenvalue λ.Its multiplicity, as well as the possible choices of χ, ζ and p L mentioned in Remark 2.1, has to be taken into account.It leads us to consider W a parametrization of a multiset of eigenvalues of L to which we can associate non-vanishing eigenvectors ϕ as above (it seems preferable for the corresponding eigenvectors to be independent in case of multiplicity).As in the introduction, define V := R `ˆV ˆW , whose generic elements are denoted pt, y, zq.By definition, to each z P W , we associate an eigenvector ϕ z , an eigenvalue λ z , but also the complex numbers χ z , ζ z , ξ z , the generator p L z , etc. Consider the Markovian generator L given on A, the unitary algebra of mappings on V which are C 1 in t and belong to A as function of y, by In this formula, the variable z enters in the definition of p L z , which is acting on y.Note that the variable z is not modified by the dynamic generated by L. With the notations of the introduction, the generator L is just p L z , so that (1.1) is valid.
The algebra A is not preserved by L, anyway, for any s ě 0, there is no difficulty to define P s := exppsLq on A directly via @ F P A, @ pt, y, zq P V, P s rF spt, y, zq " p P z,s rF pt `s, ¨, zqspyq, where p P z,s := expps p L z q, similar notations will be used below.Define the operator R from V to V through @ pt, y, zq P V, @ f P A, Rrf spt, y, zq := r P z,´ζzt rf spyq " exppλ z ζz tq ϕ z pyq where we used (2.1), which implies that @ η P C, r P z,η r¨s " expp´ηλ z q ϕ z P η rϕ z ¨s.
Lemma 2.2 can be partially rewritten under the form.
Lemma 2.3.We have This relation is equivalent to the generator intertwining Proof On the other hand, we compute that RP ξs rf spt, y, zq " r P z,´ζzt rP ξs rf sspyq " p r P z,´ζzt P ξs qrf spyq, so the first announced equality is a direct consequence of Lemma 2.2.
The second equality is obtained by differentiation with respect to s at 0 `.Conversely, the first equality is recovered from the second one by integration.Remark 2.4.The assumptions made before Lemma 2.2 are too strong, e.g. to deal with the classical free motion Schrödinger equation, but they can be relaxed at the expense of further notations.Let B be a Banach space of functions defined on V and containing A X B as a dense subset.Let C Ă C be a cone containing R ànd pP η q ηPC be a family of continuous operators on B such that for any η P C, pP sη q sě0 is the semi-group associated to the pregenerator ηL (acting on A X B), in the sense of Hille-Yoshida.Make  The operator R will be important to construct the family of functions pF h q hą0 presented in the introduction.More precisely, for any h ą 0 and py, zq P V ˆW , we will find an appropriate function f h,y,z P A, concentrating around y 0 for h ą 0 small and F h will be defined by @ pt, y, zq P V, @ x P V, F h pt, y, z; xq := Rrf h,y,z spt, x, zq. (2.6) Before investigating more precisely these functions in the examples of the following sections, let us come back to the decomposition (2.2).The carré du champs Γ associated to L is the bilinear functional on A ˆA defined by @ f, g P A, Γrf, gs := Lrf gs ´f Lrgs ´gLrf s We compute that for any f P A, where This alternative writing is particularly important when L is a diffusion generator.A Markov generator L is of diffusion type when A is stable by composition with smooth mappings F : R Ñ R and that In this context, recall that an operator K defined on A is a derivation when The following result is well-known, see e.g. the book of Bakry, Gentil and Ledoux [1]: Proposition 2.5.When L is a diffusion generator, Ľ is a derivation operator.
Assume now that V is a differential manifold and that A is the space of smooth functions.A Markov generator L : A Ñ A is of diffusion type if and only if it is a second order operator without zero order term.When the second order part of L does not vanish identically (which is just asking for L not being a derivation), we deduce from Proposition 2.5 that it is natural to ask for χ " 1 in (2.7), if we don't want p L to contain the same second order terms as L, up to a factor.A derivation operator K comes from a vector field if and only if K transforms R-valued functions into R-valued functions.Furthermore this condition is equivalent to the fact that K is a Markov generator (leading to the deterministic dynamical system obtained by following the corresponding vector field).Thus in this context, the existence of the decomposition (2.2) is equivalent to the existence of ζ P T such that ζ Ľ comes from a vector field.When this is satisfied, we can take χ " 1 and p L " ζ Ľ.

The free motion Schrödinger equation on the line
Consider the case where V := R is endowed with the Laplacian operator L := B 2 on the smooth functions, namely on A := C 8 pRq.The corresponding carré du champ is the usual one: Γrf, gs " 2pBf qpBgq. (3.1) Take W := R `.To any z P W , we associate the eigenvalue λ z " ´z2 , and a corresponding eigenvector ϕ z defined by Note that the eigenvector ϕ z is only algebraic, in the sense that it satisfies the relation Lrϕ z s " λ z ϕ z everywhere on R, but ϕ z does not belong to L 2 space of the Lebesgue measure.For z P W zt0u, z and ´z parametrize the same eigenvalue ´z2 and their eigenvectors ϕ z and ϕ ´z " ϕ z are linearly independent.It appears that for any f P A, As in the end of the previous section, it leads us to take χ z " 1, ζ z " i and p L z " 2zB, the generator of the (deterministic) free motion at speed 2z.With the notation of Lemma 2.3, ξ " i does not depend on z P W . Remark 3.1.As an illustration of Remark 2.1, we could have chosen ξ " ´i, since the corresponding operator p L is the generator of the free motion at speed ´2z.
For any z P W , the operators L " B 2 and p L z commute and Remark 2.4 holds, with B := L 2 pµq, where µ is the usual Lebesgue measure on R, and with C := C `and p C := R `.The assumptions of Section 2 are thus satisfied and we can apply Lemma 2.3.Consider a test function f on R. For any pt, y, zq P V and s P R `, we have P s Rrf spt, y, zq " RP is rf spt, y, zq.
(3.3) Thus, we have solved the free Schrödinger equation on the line, i.e. we have found an expression for the solution u : R `ˆR Q ps, yq Þ Ñ P is rf spyq P C of " up0, ¨q " f @ ps, yq P R `ˆR, B s ups, yq " iB 2 q ups, yq At least if we are able to identify directly the kernel R, i.e. without just inverting (3.3)!It follows from (2.5) that for any test function f on R, @ pt, y, zq P V, Rrf spt, y, zq " expp´itλ z q exppizyq P it rexppiz¨qf spyq. (3.4) It may seem that we have not made much progress, since we still have to compute P it rϕf spyq.Indeed, let us complete this task, just as an illustration since our goal is to by-pass such computations.
Recall that for any s ą 0 and y P R, we have the following expression for the heat kernel: By using the holomorphic extension of ?¨on Czp´8, 0s, the above formula is also valid for s P Czp´8, 0s, for appropriate test functions (e.g. when f is continuous and with compact support).It follows that @ pt, y, zq P V, Rrf spt, y, zq " ˙.
Nevertheless, the main advantage of (3.4) is to suggest the introduction of appropriate "concentrating" mappings to avoid direct computations.More precisely, for any h ą 0 and py 0 , z 0 q P R 2 , consider the function f h,y0,z0 given by @ x P R, f h,y0,z0 pxq := expp´iz 0 px ´y0 q ´px ´y0 q 2 {p2hqq{pπhq 1{4 . (3.6) Extending as usual the action of P h to probability measures, the function f h,y0,z0 can be written as f h,y0,z0 " ?2pπhq 1{4 ϕ z0 py 0 q ϕ z0 P h{2 rδ y0 s. (3.7) We also have |f h,y0,z0 | 2 " P h{4 rδ y0 s and this relation explains the choice of the normalization in (3.6).We deduce that for small h ą 0, |f h,y0,z0 | 2 is an approximation of the Dirac mass δ y0 , since it is the Gaussian density of mean y 0 and variance h{4.
Namely, we have for any t ě 0 and x P R, Normalizing the time t into ht, we get Finally we deduce from (3.3) that for f h,y0,z0 given by (3.6), 1 `2it exp ˆ´iz 0 px `2z 0 ht ´y0 q `ihtz 2 0 ´px `2z 0 ht ´y0 q 2 2hp1 `2itq In particular, we get, which is the Gaussian density of mean y 0 ´2z 0 ht and variance h ? 1 `4t 2 {2.This result is well-known and corresponds to the quantification of the free motion on R with speed ´2z 0 , the mapping |P iht rf h,y0,z0 s| 2 being strongly concentrated around y 0 ´2z 0 ht for h ą 0 small.Of course, we could have computed directly P iht rf h,y0,z0 spyq, starting from the formulas (3.5) and (3.6), but the apparition of the free motion R `Q t Þ Ñ y 0 ´2z 0 t would (maybe) have been more mysterious.Furthermore, the intertwining relation (3.3) imposed the form of the concentrating mappings f h,y0,z0 .
Indeed, let us translate the above observations into the framework presented in the introduction, to show Theorem 1.5.On V := R ˆR, consider the unitary algebra A of continuous functions f such that for any fixed z P R, f p¨, zq P A, namely f is globally continuous and smooth in the first variable.Define on A the operator L given by @ f P A, @ py, zq P V, Lrf spy, zq := p L z rf p¨, zqspyq " 2zB y f py, zq.
It is the generator of the free motion: for any initial point py 0 , z 0 q P V, the motion generated by L is given by @ s ě 0, pY psq, Zpsqq := py 0 `2z 0 s, z 0 q.
Since we have

L. MICLO
Hypotheses (H1) and (H2) are satisfied.In the latter, the concentration is even strong: for any k P Z `, any T ě 0 and any r ą 0, Hypothesis (H3) is a direct consequence of (3.3), so Theorem 1.5 is shown.
Remark 3.2.In traditional semi-classical analysis, for h ą 0 and py 0 , z 0 q P R 2 , the functions g h,y0,z0 := f h,y0,z0{h are often preferred to f h,y0,z0 and lead to the following formulas valid for x P R, and It amounts to consider the process pXphsqq sě0 " phs, Y phsq, Zphsqq sě0 starting from p0, y, z{hq, and for this purpose, it is important that z is not confined to a compact set in the supremum in (H2).In this way, we recover the classical motion @ s ě 0, Y phsq " y `2zs.
This high frequency normalization (leading to comparison of quantum times ht, in the above lhs, to classical times t, in the above rhs) does not seem so natural in our Markov process context, specially when the fibers W are not vector spaces, as in the next sections.
Another manner to obtain the classical motion is to consider a small frequency normalization and long times: consider the process pXps{hqq sě0 " ps{h, Y ps{hq, Zps{hqq sě0 starting from p0, y, hzq, we get @ s ě 0, Y ps{hq " y `2zs.
Unfortunately, this normalization is not very useful, since the concentration property (H2) holds only for times of order h.
We will see in the next sections to which extent this approach can be extended to discrete settings.

The free motion on Z
We consider now V := Z.There are two natural difference operators on Z: B `and B ´, acting on A, the space of all bounded mappings from Z to C, via @ f P A, @ x P Z, " B `f pxq " f px `1q ´f pxq B ´f pxq " f px ´1q ´f pxq.
It is immediate to check that @ f P A, @ x P Z, B `B´f pxq " B ´B`f pxq " ´pB ``B ´qf pxq " 2f pxq ´f px `1q ´f px ´1q.
Take W := r0, 2πq, seen as the set of angles of elements from T, and for any z P W , consider the function ϕ z defined by It is an algebraic eigenvector of L associated to the eigenvalue λ z " 2pcospzq ´1q, but note that it does not belong to l 2 pZq.For z P p0, πq, z and z `π parametrize the same eigenvalue 2pcospzq ´1q and their eigenvectors ϕ z and ϕ ´z " ϕ z are linearly independent.We compute that for any test function f P A, @ x P Z, Ľz rf spxq " expp´izxqΓrexppiz¨q, f spxq " pexppizq ´1qpf px `1q ´f pxqq `pexpp´izq ´1qpf px ´1q ´f pxqq, namely Ľz " pexppizq ´1qB ``pexpp´izq ´1qB " pexpp´izq ´1qL `pexppizq ´1 ´pexpp´izq ´1qqB " pexpp´izq ´1qL `2i sinpzqB `.
So according to (2.7), for z P r0, πs, we can take χ z " expp´izq, ζ z " i and p L z " 2 sinpzqB `, which is the generator of the Markov process always jumping toward the right, with intensity 2 sinpzq.These choices lead to ξ z " i expp´izq.´, which leads to ξ z " ´i exppizq, conjugate to its previous value.
For z P pπ, 2πq, we proceed similarly, except that we rather take χ z " exppizq, ζ z " i, p L z " ´2 sinpzqB ´and ξ z " i exppizq.To simplify the presentation, from now on, we restrict W to be r0, πs.The missing part pπ, 2πq of the fibers can be treated similarly and enable to reverse the direction of the "free motion".
Due to the commutation properties mentioned at the beginning of this section, L and p L z commute.Note also that endowing A with the supremum norm transforms it into a Banach algebra and that L is bounded on A. The assumptions of Section 2 are satisfied and we can apply Lemma 2.3.Starting from p0, y, zq P V, the operator L generates the process ps, Y psq, zq sě0 , where pY psqq sě0 is a Markov process starting from y and whose generator is p L z .More precisely, pY psqq sě0 " py `N p2 sinpzqsqq sě0 , where pN psqq sě0 is a standard Poisson process starting from 0 and of intensity 1.It follows that for any test function f P A, (3.3) has to be replaced by @ y P Z, @ z P R, @ s ě 0, ErRrf sps, Y psq, zqs " P ξzs rf spyq, where the operator R is given by (2.5): @ pt, y, zq P V, @ f P A, Rrf spt, y, zq " expp´2ipcospzq ´1qtq exppizyq P it rexppiz¨qf spyq.(4.4)Thus, we have found a probabilistic representation of a modified free Schrödinger equation on the discrete line, i.e. of the solution u : R `ˆZ Q ps, yq Þ Ñ P ξzs rf spyq P C of " up0, ¨q " f @ ps, yq P R `ˆZ, B s ups, yq " ξ z Lrusps, yq.
Contrary to the previous section, ξ z P iR only for z P t0, πu, which corresponds to degenerate situations, since p L z " 0 and so Y psq " y for all s ě 0. This was to be predicted from Remark 1.1, asking in this situation for pY psqq sě0 to be a deterministic process.For z " 0, (4.3) and (4.4) are both equivalent to Rrf spt, y, 0q " P it rf spyq, for all t ě 0 and y P Z.For z " π, (4.3) and (4.4) are respectively equivalent to @ t ě 0, @ y P Z, Rrf spt, y, πq " P ´it rf spyq, and @ t ě 0, @ y P Z, Rrf spt, y, πq " expp4itq p´1q y P it rϕ π f spyq.
In view of Remark 1.4, another intriguing case is when ξ z P t˘1u.Here, it corresponds to z " π{2 and we get ξ π{2 " 1, ϕ π{2 pxq " i x for all x P Z and λ π{2 " ´2.In this situation we have @ t ě 0, @ y P Z, Rrf spt, y, π{2q " expp2itq i y P it rϕ π{2 f spyq, and (4.3) leads to a strange formula: for any f P A and y P Z, @ s ě 0, E " expp2isq i y`N p2sq P is rϕ π{2 f spy `N p2sqq  " P s rf spyq.
Let us now see how some features of the treatment of the free motion on R presented in the previous section can be adapted to the the present discrete setting of Z.
Following the strategy described in (2.6), define for any h ą 0 and py, zq P Z ˆr0, πs, where the Kronecker delta appears in the rhs.Note that the function f h,y,z is so concentrated "around" y that the parameters h and z do not play a role.Next we consider, for any h ą 0, pt, y, zq P V and x P V , where the term ˝ph 2 q is uniform over y, x P Z.In particular, for x " y, we get g t,y pyq " 1 ´2ih ´3h 2 `˝ph 2 q, because Lrδ y spyq " Lpy, yq " ´2 Lpy, y 1 qLpy 1 , yq " Lpy, yq 2 `Lpy, y `1qLpy `1, yq `Lpy, y ´1qLpy ´1, yq " 6.

L. MICLO
Hypothesis (H3) is a direct consequence of (4.3) applied with f replaced by f h,y,z , since @ h ą 0, @ py, zq P Z ˆr0, πs, @ x P Z, F h p0, y, z; xq " f h,y,z pxq, and since under P p0,y,zq , @ s ě 0, Xpsq " ps, y `N p2 sinpzqsq, zq.(4.8) Remark 4.3.In the spirit of Remark 3.2, it is now only possible to consider the not very convincing small frequency normalization: replacing in (4.8) s by s{h and z by zh (assuming z P r0, π{hs), as h goes to 0 `, the base component converges toward the process py `N p2zsqq sě0 , which is a Poisson process whose rate 2z can be as high as wanted (for h small).
Remark 4.4.It is tempting to play with the objects at hand, to see how the results are affected by their modifications.For instance, we could replace the operator r L defined in (2.2) by p1 ´ q r L ` L, where P p0, 1q.Then we have It follows that ξ p q " ξp χ `p1 ´ qq{a .In the setting of the present section, we get for z P r0, πs, ξ p q z " ξ z p exppizq `p1 ´ qq{ | exppizq `p1 ´ q| , and this complex number can be "more imaginary" than ξ z .Remark 1.4 may then let us believe that is advantageous to consider such transformations with P p0, 1q.But it is wrong, because computing the corresponding operator R p q and the functions pF p q h q hą0 (via (2.6) with the functions f h,y,z given by (4.5)), we get that (H2) is not satisfied, the concentration being only of order 0.

The free motions on tori
After proving Theorems 1.7, 1.8 and 1.9 in their respective torus settings, we will discuss generally about the tensorization of multi-dimensional quantification.
The case of V " Z n , for a given n P Nzt1u, is very similar to the situation of Z described in the previous section.The difference operators B ´and B `are extended to act on Z n , which is endowed with the discrete Laplacian L := B ´B`" B `B´" B ´`B `(when n " 2, we have furthermore B ´" B `and L " 2B `).The underlying Banach algebra A n is just the usual algebra of all C-valued functions defined on Z n .The carré du champs of L is still given by (4.1), where x takes values in Z n .Let W := W n := 2π n rr0, n ´1ss and consider for any z P W n , the function @ x P Z n , ϕ z pxq := exppizxq, ( which is an eigenvector of L associated to the eigenvalue λ z := 2pcospzq ´1q.All the computations and observations of the previous section are still valid, once Z has been replaced by Z n and r0, 2πq by W n .In particular, for z P W n X r0, πs " p2π{nqrr0, tn{2uss (where t¨u stands for the integer part), we can choose p L z " 2 sinpzqB `, with χ z " expp´izq, ζ z " i and ξ z " i expp´izq.Again we can apply Lemma 2.3: starting from p0, y, zq P V " R `ˆZ n ˆWn , the operator L generates the process ps, Y psq, zq sě0 , where pY psqq sě0 is a Markov process starting from y and whose generator is p L z .Namely, pY psqq sě0 " py `N p2 sinpzqsqrnsq sě0 , where pN psqq sě0 is a standard Poisson process starting from 0 and of intensity 1, and where rns means modulo n.In conformity with (1.1), L := p L z is the generator of the process pY psq, zq sě0 .Going through the same constructions of R, pf h,y,z q hą0,yPZn,zPWn and pF h q hą0 , given respectively in (4.4), (4.5) and (2.6) (see also (4.6)), we conclude to the validity of Theorem 1.7.
Remark 5.1.Contrary to Remarks 3.2 and 4.3, neither the high frequency nor the fruitless small frequency normalizations are possible for the above quantification, since W n is finite.
The case of V " R{p2πZq, has similarities with both the situations of R and Z n .We consider the Laplacian operator L " B 2 on the algebra A of smooth functions defined on R{p2πZq.Its carré du champs is given by (3.1).Take W " Z, to any z P Z, we associate the eigenvalue λ z " ´z2 and a corresponding eigenvector ϕ z is defined by @ x P R{p2πZq, ϕ z pxq := exppizxq.
As in Section 3, for any z P Z, we take χ z " 1, ζ z " i, ξ z " i and p L z " 2zB.Lemma 2.3 can be applied: starting from p0, y, zq P V " R `ˆR{p2πZq ˆZ, the operator L generates the process pXpsqq sě0 " ps, y `2zs, zq sě0 .In conformity with (1.1), L := p L z is the generator of the process py `2zs, zq sě0 .Going through the same constructions of R, pf h,y,z q hą0,yPR{p2πZq,zPZ and pF h q hą0 , given respectively in (3.4), (3.7) (in both equations, pP t q tě0 is now the heat semi-group generated by L on R{p2πZq) and (2.6), we conclude to the validity of Theorem 1.9.Remark 5.2.Similarly to the first part of Remark 3.2, it is possible to consider a high frequency normalization for the above quantification.More precisely, for given z P R, consider H z := th ą 0 : z{h P Zu.We get that for h P H z , the base component of the process pXphsqq sě0 " phs, Y phsq, Zphsqq sě0 starting from p0, y, z{hq is equal to py `2zsq sě0 , the classical free motion on R{p2πZq.
We now come to the situation of the free motions on finite multidimensional tori.With the notations introduced before Theorem 1.8, consider the Laplacian operator L := ř jPrrmss L pjq , on the space A of all C-valued functions defined on V .For any z := pz 1 , z 2 , ..., z m q P W , λ z := 2 ř jPrrmss pcospz j q ´1q is an eigenvector of L associated to the eigenfunction ϕ z given by @ x := px 1 , x 2 , ..., x m q P V, ϕ z pxq := exp ¨i ÿ jPrrmss z j x j ‚.
Considering the associated Doob transform r L z defined as in (2.1), (2.2) must be replaced by r L z " ÿ jPrrmss χ zj L pjq `ip L z (5.2) where @ j P rrmss, χ zj := expp´iz j q

Remark 4 . 1 .
As another illustration of Remark 2.1, we could also have considered the decomposition Ľz " pexppizq ´1qL ´2i sinpzqB L. MICLOwhen V is a finite set and A is the space of all C-valued functions on V .For a more general set of hypotheses, see Remark 2.4.The interest of the previous operators is: Lemma 2.2.For any η, η 1 P C, we have the intertwining relation p P η 1 r P ´ζpη`η 1 q " r P ´ζη P ξη 1 , where ξ := ´ζχ.
y, z; xq := Rrf h,y,z spt, x, zq Let µ the counting measure on Z.The validity of (H1) and (H2) is provided by Lemma 4.2.For any t ě 0 and y P Z, Proof.Since the operator L is self-adjoint in L 2 pµq, the operator P ih is unitary in L 2 pµq.It follows that where @ t ě 0, y P Z, @ x P Z, g t,y pxq := P it rδ y spxq.ih " exppihLq on the Banach algebra A, for any y, x P Z, we have the following expansion for h ą 0 small, P ih rδ y spxq " δ y pxq `ihLrδ y spxq `pihq 2 2 L 2 rδ y spxq `˝ph 2 q,