Duality and hypoellipticity: one-dimensional case studies*

To visualize how the randomness of a Markov process X is spreading, one can consider subset-valued dual processes I constructed by intertwining. In the framework of onedimensional diffusions, we investigate the behavior of such dual processes I in the presence of hypoellipticity for X. The Pitman type property asserting that the measure of I is a time-changed Bessel 3 process is preserved, the effect of hypoellipticity is only found at the level of the time change. It enables to recover the density theorem of Hörmander in this simple degenerate setting, as well as to construct strong stationary times by introducing different dual processes.


Introduction
The technique of duality by intertwining associates to a Markov process X a dual Markov process, which here will be taking subsets of the state space of X as values, showing how randomness is spreading. In particular, this approach offers decompositions of the time-marginal laws of X that are useful to deduce that they admit a density with respect to a reference measure, at positive times. In our program to recover Hörmander's theorem by following this probabilistic way, we investigate here the effects of hypoellipticity on duality, by considering the simple one-dimensional framework.
In the next section, we will check that X is hypoelliptic of order n at 0 and that τ is a.s. infinite.
Let I stand for the set of nonempty closed intervals from r´8,`8s, which are either included into r´8, 0q or into r0,`8s and which are different from t´8u and t`8u. Denote S the set of singletons from I, i.e. S ttxu : x P Ru. Consider µ`an µ´the speed measures associated to X on R`and p´8, 0q (whose definition will be recalled in Sections 2 and 3, respectively). We define a Markov kernel Λ from I to R by @ ι P I, @ A P BpRq, Λpι, Aq , when ι is the singleton txu, µ´pιXAq µ´pιq , when ι P IzS is included into r´8, 0q, µ`pιXAq µ`pιq , when ι P IzS is included into r0,`8s, (1.2) where BpRq stands for the set of Borel subsets from R and δ x for the Dirac mass at x.
We will check later on that the above expression are well-defined, as the denominators are finite.
Our first goal is the following construction of a dual process I with respect to X, a solution of (1.1): Theorem 1.1. There exists a process I pIptqq tě0 taking values in I such that Ip0q " tXp0qu, (1.3) @ t ą 0, PrIptq P Ss " 0, (1.4) @ t ě 0, LpXptq|Ir0, tsq " ΛpIptq,¨q, (1.5) where the conditional law in the l.h.s. is with respect to the trajectory Ir0, ts pIpsqq sPr0,ts .
In particular, we have for any t ě 0, the decomposition LpXptqq " ż Λpι,¨q LpIptqqpdιq, and the r.h.s. is absolutely continuous with respect to the Lebesgue measure for t ą 0.
As implied by (1.4), I immediately grows into a segment with non-empty interior. But contrary to the elliptic case, where the dual process never return to S, I collapses into the singleton t0u at τ 0 , the time when X hits 0 (this happens in positive time when Xp0q is negative). The process I is continuous (for the Hausdorff topology on the compact subsets of r´8,`8s), except at τ 0 , when I may be non left-continuous. Point (1.4) in Theorem 1.1 will be deduced from the fact that the law of τ 0 has no atom outside 0. Note that without this requirement, the trivial dual process defined by Iptq tXptqu, for all t ě 0, would be suitable.

Duality and hypoellipticity
After τ 0 , the behavior of I depends on n: ‚ For n P Nzt1u, in finite time the process I hits r0,`8s and stays there afterward. ‚ For n " 1, the process I converges to r0,`8s in large time, but never reaches it (starting from a singleton).
This dichotomy is also valid when Xp0q is non-negative and will be reformulated in terms of strong stationary times in the next sections. But whatever n P N, Theorem 1.1 recovers, on this example, the density part of Hörmander's theorem, stating that for all t ą 0, the law of Xptq is absolutely continuous with respect to the Lebesgue measure.
This study can be extended to any hypoelliptic diffusion on R (or on an interval of R), but we found the circle case more instructive.
Let a and b be two smooth functions on T R{Z, such that a is non-negative, ? a is smooth and vanishes at most at a finite number of points, write N for their set. Assume that for any x P N, bpxq " 0. Consider on C 8 pTq the Markov generator L aB 2`b B, (1.7) and let X pXptqq tě0 be a corresponding diffusion process. The generator L is hypoelliptic and we are looking for the behavior in law of X for large times. Let us write N ty k : k P Z N u, where the representative points in r0, 1r satisfy 0 ď y 0 ă y 1 ă¨¨¨ă y N´1 ă 1 and where N P N (what follows is also trivially true in the classical elliptic case where N " 0). For k P Z N , let I k be the projection on T of the interval py k , y k`1 q (for l " N´1, it is the interval py N´1 , y 0`1 q), to which is added y k if bpy k q ą 0 and y k`1 if bpy k`1 q ă 0. Remark that pI k q kPZ N forms a partition of T. Denote for k P Z N , µ k the speed measure associated to the restriction of L to I k . Let I stand for the set of non-empty closed intervals from T which are included into one of the I k , for k P Z N and let S ttxu : x P Tu. Define a Markov kernel Λ from I to T by @ ι P I, @ A P BpTq, Λpι, Aq $ & % δ x pAq , when ι " txu P S, µ k pιXAq µ k pιq , when ι P IzS with ι Ă I k and k P Z N .
(1. 8) In Section 4, it will be checked that the last r.h.s. is well-defined, i.e. 0 ă µ k pιq ă`8 for ι P IzS with ι Ă I k and k P Z N . Theorem 1.1 extends to this context: Theorem 1.3. Let X be a diffusion on the circle whose generator is the hypoelliptic elliptic L given in (1.7). There exists a dual process I associated to X satisfying all the statements of Theorem 1.1, where I and Λ are defined as in (1.8).
The process I collapses into a singleton when X hits N. But our definition of the dual process I will not always be optimal, with respect to the construction of strong stationary times. We will see that sometimes it is better to let the dual process I collapses into a pair of points when X exits from the segments I k which are open, for k P Z N . The description of the evolution of the corresponding dual process is a little more involved and left to Section 4, as well as the definition of another Markov kernel (4.9) replacing (1.8) and Theorem 4.4, the extension of Theorem 1.3 in this situation. Nevertheless and similarly to the toy model case, we deduce from Theorem 4.4 the density part of Hörmander's theorem for the one-dimensional generator (1.7).
Another interest of the dual process I, associated to the Markov kernel (1.8) and constructed in Theorem 1.3, is to quantify the convergence to equilibrium of X, but only when b takes different signs over N, in which case I converges a.s. for large time. When b has a constant sign over N, the process I does not converge a.s. for large time. Indeed, writing I rY, Zs, one of the two processes Y or Z ends up being Markovian, with a behavior of the same nature as X, after the first time X goes through N. In this situation, X admits an invariant probability measure π absolutely continuous with respect to the Lebesgue measure. The support of π is T but its density vanishes on N. It is then natural to consider r I the set of non-empty closed intervals from T and to define a Markov kernel , when ι P r IzS.
(1.9) Theorem 1.3 is still valid when Λ is replaced by r Λ: Theorem 1.4. Let X be a diffusion on the circle whose generator is the hypoelliptic elliptic L given in (1.7), where b has a constant sign over N. There exists a dual process r I p r Iptqq tě0 associated to X taking values in r I and satisfying all the statements of Theorem 1.1, with I and Λ replaced by r I and r Λ defined in (1.9).
The dual process r I converges a.s. in finite time to the whole state space T, so we are able to construct a strong stationary time for X and to deduce the weak convergence of LpXptqq toward π for large times. The dual process r I never collapses to a singleton: in this situation the deduction of the density part of Hörmander's theorem is straightforward, since we have, whatever the initial condition, Pr@ t ą 0, r Iptq P r IzSs " 1.
The plan of the paper is as follows: the next two sections are respectively devoted to the restriction of the toy model to R`and to R´. In Section 4 we consider the circular hypoelliptic diffusion and its dual process mentioned in Theorem 1.3. Section 5 deals with the situation where b has a constant sign over N and in particular Theorem 1.4.
Finally Appendix A recalls and adapts some computations from [12] and [3] about the segment-valued dual processes.

On RT
he situation treated here is quite similar to that from [12]. This section serves as a reminder of some notions from the theory of duality by intertwining.
We begin by some general considerations about the diffusion X pXptqq tPr0,τ q whose evolution is described in (1.1). The generator L associated to X is the operator acting on C 8 pRq via where x¨,¨y stands for the bracket of semi-martingales. It follows that the generator can be rewritten under the Hörmander's form L " V 2 1`V0 , where V 0 and V 1 are the vector fields on R, seen as first order differential operators, whose coefficients are given by @ x P R, To see that L satisfies the Hörmander's condition (cf. Hörmander [8] or the pedagogical paper of Hairer [7]), define for all l P Z`, the set of vector fields V l through the iteration where r¨,¨s stands for the usual Lie bracket. For any x P R, let V l pxq tV pxq : V P V l u.
For any x P Rzt0u, we have V 0 pxq " t0u, so that L is elliptic on Rzt0u. At 0, the first l P Zs uch that V 0 p0q " t0u is l " n, so that L is hypoelliptic of order n at 0, as announced in the introduction.
Despite the above choice of R as state space, starting from R`, the process X lives in R`. Indeed, to check the status of the point 0 seen from R`, let us introduce the scale and speed functions associated to L: The interest of these functions is that on p0,`8q, we can write The corresponding scale and speed measures, also written σ`and µ`, are given by @ z ě y ą 0, σ`pry, zsq " ż z y σ`pxq dx, µ`pry, zsq " ż z y µ`pxq dx " v`pzq´v`pyq.
We get that Thus using Chapter 15 from Karlin and Taylor [10], it appears that 0 is an entrance boundary for the restriction of L on R`: when Xp0q is distributed on R`, the positions of the process X are in p0,`8q for any t P p0, τ q. The status of`8 can be investigated similarly. Since σ`pyq converges to expp´1{p2n1 qq as y goes to`8, it appears that for any x ą 0, σ`prx,`8qq "`8 and consequently ż`8 1 σ`prx,`8qq µ`pxqdx "`8.
Furthermore, we have as x ą 0 goes to`8, Thus when X starts from an initial distribution on R`, we deduce again from Chapter 15 of Karlin and Taylor [10] that`8 is a natural boundary if n " 1 and an entrance boundary if n P Nzt1u. In both cases,`8 cannot be reached, so that τ "`8 a.s.
Following the approach developed in [12], we would like to construct an intertwining dual to X. In this section, we restrict our attention to the case where X starts from R`.
Consider I` tpy, zq : y, z P r0,`8s, y ď zuztp`8,`8qu, I` tpy, zq P p0,`8q 2 : y ă zu (the interior of I`X R 2 ) and the diagonal S` tpy, yq : y P R`u Ă I`. As in the introduction, the element py, zq P I`should be interpreted as the compact interval ry, zs in R`\ t`8u and the elements of S`, as singletons. This is illustrated by the following definition of the Markov kernel Λ`from I`to R`: @ py, zq P I`, @ A P BpR`q, Λ`ppy, zq, Aq µ`pry,zsXAq µ`pry,zsq , otherwise.
Note that the above expression is well-defined, as we have µ`pr0,`8qq " v`p`8q´v`p0q " expˆż`8 Let L`be the diffusion generator onI`given by L` pz n B z´y n B y q 2`p ny 2n´1´1 qB y`p nz 2n´1´1 qB z (2.5) 2 y n µ`pyq`z n µ`pzq µ`pry, zsq pz n B z´y n B y q.
Complete this definition on t0uˆp0,`8q by L` pz n B z q 2`p nz 2n´1´1 qB z`2 z 2n µ`pzq µ`pr0, zsq B z , (2.6) on p0,`8qˆt`8u by L` py n B y q 2`p ny 2n´1´1 qB y´2 y 2n µ`pyq µ`pry,`8qq B y , and on p0,`8q P I`, L` 0, (2.8) namely p0,`8q (alias r0,`8s) is absorbing for L`. More precisely, L`is defined on D`, the set of continuous and bounded functions on I`which are smooth on each of the subsetsI`, t0uˆp0,`8q and p0,`8qˆt`8u. Since D`is an algebra, we define the carré du champs Γ L`a ssociated to L`via @ F, G P D`, Γ L`r F, Gs 1 2 pL`rF Gs´F L`rGs´GL`rF sq. (2.9) For instance onI`, we compute that @ py, zq PI`, Γ L`r F, Gspy, zq " pz n B z´y n B y qrF spz n B z´y n B y qrGs.
It is not difficult to check that for any f P C 8 b pR`q, the set of bounded smooth functions on R`, the mapping Λ`rf s is an element of D`.
The interest of Λ`and L`is the intertwining relation L`Λ`" Λ`L, in the sense that, @ py, zq P I`zS`, @ f P C 8 b pR`q, L`rΛ`rf sspy, zq " Λ`rLrf sspy, zq. (2.10) This can be checked by direct computation, as in Lemma 20 from [12]. Alternatively, as in [3], one can resort to an algebra A`of convenient observables, containing the mappings Λ`rf s for f P C 8 b pR`q, see Appendix A below with p0, 1q replaced by R`.
Following the arguments leading to Proposition 4 from [12], we get that the martingale problems associated to pD`, L`q are well-posed: Theorem 2.1. For any probability distribution m 0 on I`, there is a unique (in law) continuous Markov process I pY ptq, Zptqq tě0 whose initial distribution is m 0 and whose generator is L`in the sense of martingale problems: for any F P D`, the process M F pM F ptqq tě0 defined by @ t ě 0, M F ptq F pY ptq, Zptqq´F pY p0q, Zp0qq´ż t 0 L`rF spY psq, Zpsqq ds is a local martingale. Furthermore the diagonal S`is an entrance boundary for I: for any t ą 0, we have pY ptq, Zptqq R S`. x n µ`pxqδ x . (2.11) For any x 0 P R`, to get the singleton px 0 , x 0 q as a starting point, an approximation by px 0 , x 0` q, for small ą 0, is performed.
Stone-Weierstrass theorem enables us to see that the algebra A`of observables presented in Appendix A is dense in the space of continuous functions on I`zS`, endowed with the uniform convergence on compact subsets (but this is not true on I`, since the elementary observables vanish on S`, so that the composed observables from Ad oes not separate the elements of S`). We strongly believe the martingale problems associated to pA`, L`q are equally well-posed (cf. Section 4.4 of Ethier and Kurtz [5] for valuable information in this direction).
As a consequence, we have the following result (this sentence is slightly misleading, since a preliminary version of Corollary 2.3 plays an important role in the proof of Theorem 2.1, to be able to let the process I start from the singletons from S`, see [12]), for which we need to introduce some notations: (where µ`pBIpsqq " pY psqq n µ`pY psqq`pZpsqq n µ`pZpsqq, according to (2.11)), with the conventions that x n µ`pxq " 0 for x P t0,`8u, a priori ς`P p0,`8s, but we will see in Corollary 2.3 below that ς`is finite a.s. Let the time change pθ`ptqq tPr0,ς`s be defined by @ t P r0, ς`q, 2 ż θ`ptq 0 µ`pBIpsqq 2 ds " t, (2.13) and θ`pς`q lim tÑpς`q´θ`p tq. We are interested in the process R` pR`ptqq tě0 given by @ t ě 0, R`ptq µ`pIpθ`pt^ς`qqq. (2.14) Proposition 14 from [12] and its proof lead to the following result.
Corollary 2.3. The process R`is a Bessel process of dimension 3 starting from µ`pIp0qq and stopped when it hits µ`pp0,`8qq. In particular, ς`is finite a.s. and is the hitting time of µ`pp0,`8qq by R`. More precisely, we have (conditioning by the initial value Ip0q for the second point): ‚ for n P Nzt1u or Ip0q of the form py 0 ,`8q for some y 0 P r0,`8q, we have θ`pς`q ă`8 and the process I hits p0,`8q in finite time (a.s.) ‚ for n " 1 and Ip0q not of the form py 0 ,`8q for some y 0 P r0,`8q, we have θ`pς`q "`8 and the process I does not hit p0,`8q in finite time (a.s.).
‚ For n P Nzt1u, since 0 and`8 are entrance boundaries for X, we know from Theorem 1 in [12] that I hits p0,`8q in finite time, say τ. So the mapping R`Q θ Þ Ñ ş θ 0 µ`pBIpsqq 2 ds is increasing on r0, τq and constant on rτ,`8q. It follows that lim tÑpς`q´θ`p tq " τ.
‚ For n " 1 and Zp0q "`8, since`8 is not an entrance boundary, we know from Theorem 1 in [12] that Z does not hit`8 in finite time. Thus the mapping R`Q θ Þ Ñ ş θ 0 µ`pBIpsqq 2 ds is increasing and lim tÑpς`q´θ`p tq "`8. When Zp0q "`8, since 0 is an entrance boundary for X, the proof of Theorem 1 in [12] shows that Y hits 0 in finite time. At this hitting time, I hits p0,`8q and we are in the situation where θ`pς`q ă`8. Corollary 2.3 can be seen as an illustration of Theorem 1 from [12] for elliptic diffusions X defined on R (here p0,`8q), stating that the dual process hits the whole state space in finite time for all initial distributions if and only if both boundaries are of entrance type. But in the present context, we are not so much concerned with the behavior in large time as with the behavior in small time and with the influence of hypoellipticity. According to Corollary 2.3, the latter does not modify the Pitman-type property that the process of the volumes pµ`pIptqqq tě0 of the dual process is a stopped Bessel 3 process, up to a time change. The impact is to be found in the time change itself: Proposition 2.4. Fix py, zq P I`and consider the process I defined in Theorem 2.1 starting from py, zq. There are several behaviors for the time change θ`as t goes to 0`: ‚ If py, zq " p0, 0q, we have θ`ptq " t 2py n µ`pyq`z n µ`pzqq 2 .
Thus in the latter case, the volume µ`rIs begins by evolving very slowly (since the inverse function θ´1 ptq is negligible with respect to t, for t Ñ 0`) and the order n of hypoellipticity can be recovered through For multidimensional diffusions X, the hypoellipticity should also impact the germ of the shape of the dual process, see [3] for a first approach to the elliptic case.
When py, zq " p0, 0q, according to (2.6), the diffusion I is given by where pW ptqq tě0 is a standard Brownian motion. We compute that for all z ą 0, from which we deduce that a.s. Zptq " t for small t ą 0. Since for any t ą 0, we have µ`pzq " v`pzq, (2.13) can be rewritten under the form, Since for any ą 0, we can find (a random) t ą 0 sufficiently small so that for any s P p0, tq, p1´ qs ď Zpsq ď p1` qs, we are led to study the behavior for small θ ą 0 of ş θ 0 expp´αs 1´2n q ds, where α ą 0 is a constant (that will take the values 2p1´ q{p2n´1q and 2p1` q{p2n´1q). A usual integration by parts shows that for small θ ą 0, ż θ 0 expp´αs 1´2n q ds " θ 2n expp´αθ 1´2n q, and by consequence, These considerations show that for small t ą 0, 2 p2n´1qθ 2n´1 " lnp1{tq, and this leads to the announced result when py, zq " p0, 0q.
Due to (2.10), the arguments of Section 4 of [12] show that the processes X and I can be coupled in the following way: Theorem 2.5. Let m 0 be a probability distribution on I`and consider m 0 m 0 Λ`. There exists a coupling of X with initial distribution m 0 and of I with initial distribution m 0 such that for any t ě 0, Furthermore, the construction of I from X is adapted, in the sense that given the trajectory X, for any t ě 0, the conditional law of Ir0, ts depends only on Xr0, ts. Remark 2.6. Note that conversely, for any probability distribution m 0 on R`, we can find a law m 0 on I`such that m 0 " m 0 Λ`. It is sufficient for instance to take m 0 ş δ px,xq m 0 pdxq, as it was done in Theorem 1.1 (at least when LpX 0 q is supported by R`). But in general it is not the unique possible choice, e.g. when m 0 " Λ`ppy, zq,¨q, for some py, zq P I`zS`, just consider m 0 " δ py,zq .
As a classical consequence, going back to Diaconis and Fill [4] in the framework of finite Markov chains (see also [12] for one-dimensional diffusions), we obtain the existence of strong stationary times when n " 1. Recall that a strong stationary time τ for X is a finite stopping time (with respect to a possibly enlarged filtration for X) such that τ and Xpτq are independent and such that Xpτq is distributed according to the invariant distribution π, the probability distribution whose density is proportional to µ`(π exists due to (2.4)). ‚ for n P Nzt1u, whatever the initial distribution supported by R`, there exists a strong stationary time for X. ‚ for n " 1, for some initial distributions on R`(in particular for any initial Dirac measure), a strong stationary time does not exist for X.
Proof. When n P Nzt1u, the first time I hits p0,`8q is a strong stationary time for X, see for instance [12] for more details.
When n " 1, since`8 is not an entrance boundary for X, the proof of Theorem 1 in [12] shows that there is no strong stationary time τ for X, if the initial law of X is of the form Λpr0, x 0 s,¨q, for any x 0 P R`(because τ would be stochastically bounded below by the hitting time of r0,`8s by I starting from p0, x 0 q, which is infinite), see also Fill and Lyzinski [6]. In particular, there is no strong stationary time for X starting with Xp0q " 0. Let us extend this result to all initial Dirac measure. So let x 0 P R`be given and assume, by contradiction, there is a strong stationary time for X starting from x 0 . Then one would be able to construct a strong stationary time for X started from 0, by considering the first time X hits x 0 (which is a.s. finite) and by adding to it a strong stationary time for X starting from x 0 . This is in contradiction with our previous observation, so there is no strong stationary time for X starting from x 0 .
As at the end of the proof of Corollary 2.3, remark that if the initial distribution of X is of the form Λ`prx 0 ,`8s,¨q, for some x 0 P R`, then there exists a strong stationary time for X, consider again the first time I hits p0,`8q.
Here we are more interested in the following density result, which is the easy part of the Hörmander's theorem and corresponds to the last statement of Theorem 1.1 when LpX 0 q is supported by R`.
In particular, for any t ą 0, m t is absolutely continuous with respect to the Lebesgue measure on R`.
Proof. The above equality is obtained by taking the expectation in Theorem 2.5. From Theorem 2.1, for any t ą 0, the set of singletons S`is negligible with respect to m t . Furthermore for any ι P I`zS`, Λpι,¨q is absolutely continuous with respect to the Lebesgue measure on R`. We can thus conclude to the validity of the last statement of Corollary 2.8.

On RT
he situation of R´follows a pattern similar to the investigation of the previous section. Putting together the results on R´and R`will lead to Theorem 1.1.
On R´, it is then more convenient to consider These modified scale and speed functions, where the base point 1 has been replaced bý 1, lead to the corresponding scale and speed measures on R´, still denoted σ´and µ´.
We compute that Thus when X starts from an initial distribution supported by R´, 0 is an exit boundary (i.e. it is a.s. attained in finite time). Furthermore, depending on n " 1 or n P Nzt1u,´8 is an entrance or a natural boundary. As a summary, conditioning by the initial position, we have the following a.s. behavior for X: starting from Xp0q ă 0, the diffusion will reach 0 in finite time and instantaneously pass to p0,`8q, where X will next live forever. Of course, when Xp0q " 0 or Xp0q ą 0, the first stage or the first and second stages of this description has/have to be removed.
We now come to the construction of the dual process I when the initial distribution of X is supported by p´8, 0q.
Again, the element py, zq P I´should be interpreted as the compact interval ry, zs in r´8, 0q. Let Λ´be the Markov kernel from I´to p´8, 0q given by: µ´pry,zsXAq µ´pry,zsq , otherwise.
Note that the above expression is well-defined, as we have for any x P p´8, 0q, Let L´be the diffusion generator onI´given by L´ pz n B z´y n B y q 2`p ny 2n´1´1 qB y`p nz 2n´1´1 qB z (3.4) 2 y n µ´pyq`z n µ´pzq µ´pry, zsq pz n B z´y n B y q, and complete this definition on t´8uˆp´8, 0q by More precisely, L´is defined on DpL´q, the set of continuous functions on I´which are smooth on each of the subsetsI´and t´8uˆp´8, 0q. It is not difficult to check that for any f P C 8 b pp´8, 0qq, the mapping Λ´rf s is an element of DpL´q. As in the previous section, the interest of Λ´and L´is the intertwining relation L´Λ´" Λ´L, in the sense that, Again, this can be computed directly as in Lemma 20 of [12] or by introducing, as in [3] (see also Appendix A), an algebra A´Ă DpL´q and a measure µ´ ř xPp´8,0q x 2n µ´pxq, similarly to what was done in the previous section, replacing R`and µ`by p´8, 0q and µ´.
The martingale problems associated to pDpL´q, L´q are also well-posed: For any probability distribution m 0 on I´, there is a unique (in law) continuous Markov process I pY ptq, Zptqq tPr0,τ I q whose initial distribution is m 0 and whose generator is L´in the sense of martingale problems: for any F P DpL´q, the process M F pM F ptqq tě0 defined by @ t P r0, τ I q, L´rF spY psq, Zpsqq ds is a local martingale. The diagonal S´is an entrance boundary for I: for any t P p0, τ I q, we have pY ptq, Zptqq R S´. Furthermore, the explosion time τ I corresponds to the "hitting" time of 0 by Z, in the sense that lim tÑτ I´Z ptq " 0. Proof. The arguments are similar to those of Proposition 4 in [12], except that in this previous paper, the situation of an exit boundary was not considered. So let us sketch the necessary modifications. First consider the case where m 0 " δ ι0 , for some ι 0 P I´. Consider ą 0 such that ι 0 Ă r´8,´2 q. Let L be the generator acting like L on p´8,´ q and such that´ is an reflecting boundary (i.e. a Neumann condition is imposed at´ on the functions entering in the domain of L ). Use Proposition 4 in [12] to construct the corresponding generator L and an associated I´, -valued diffusion I pY , Z q, where I´, stands for the elements of I´included into r´8,´ s. The process Z is stopped at the time τ I it hits´ . Up to this stopping time τ I , I is the unique (in law) solution of the martingale problem associated to L starting from ι 0 . Due to the Dirichlet condition on Z , some functions from DpLq are missing to conclude that pI pt^τ I qq tě0 is a stopped solution of the martingale problem associated to L and starting from ι 0 . To go around this little difficulty, rather stop I when Z hits´2 . When ą 0 varies, all these processes are consistent, so we can apply Kolmogorov's extension theorem to get a process I as in the above theorem. Its uniqueness is shown similarly by stopping. For more general initial distribution m 0 , just condition by Ip0q, see for instance the book of Ethier and Kurtz [5]. Proof. This result would be obvious, if we already had Theorem 3.5 below at our disposal, since it provides a coupling such that Zptq ě Xptq for all t P r0, τ I q and we already know that X hits 0 in finite time.
To prove that τ I is a.s. finite, it remains to show that r τ is a.s. finite. Since r Z is a diffusion process, it is enough to check that 0 is an exit boundary and that´8 is not an exit boundary.
Via the introduction of the corresponding scale and speed functions, Chapter 15 of Karlin and Taylor [10] implies that 0 is an exit boundary and that´8 is an entrance boundary.
Proof. Proposition 14 from [12] shows that r0, ς´q Q t Þ Ñ R´ptq is a Bessel process of dimension 3 (stopped if ς´ă`8). So to get that R´is a Bessel process of dimension 3, we must show that the event E tς´ă`8u has probability 0.
which is a.s. finite according to Lemma 3.2. Let us begin by checking that on E, the trajectory pZptqq tPrp τ ,τ I s is Hölder of any order α P p0, 1{2q. Indeed, taking (3.8) into account, we have for any s ă t P rp τ , τ I q, Since M pM ptqq tě0 is a continuous martingale, up to enlarging the underlying probability space, we can find a standard Brownian motion Ă W p Ă W ptqq tě0 so that The trajectories of Ă W are a.s. of order α (see e.g. Chapter 1 of Revuz and Yor [13]), so the same is true for M , since the mapping R`Q t Þ Ñ ş τ I^t 0 Z 2n puq du is Lipschitzian (these statements hold a.s., i.e. the corresponding "constants" are random). The mapping R`Q t Þ Ñ ş τ I^t 0 nZ 2n´1 puq´1 du is also Lipschitzian, so according to (3.14), it remains to bound the term ş t s µ´pIpuqq µ´pIpuqq Z n puq du. This is done via Cauchy-Schwartz' inequality, for s, t P rp τ , τ I s:ˇˇˇˇż The quantity max uPrp τ ,τ I s |Z n puq| is finite by continuity of Z and max uPrp τ ,τ I s 1{µ´pIpuqq is finite due to the fact that the Bessel process of dimension 3 R´does not hit zero once it has left 0 (this the reason for the introduction of p τ ). Since on E, b ş τ I 0 µ´pIpuqq 2 du is also finite, we deduce the trajectory pZptqq tPrp τ ,τ I s is Hölder of order α. In particular, there exists a (random) constant C ą 0 such that for all @ s P rp τ , τ I s, |Z s | " |Z s´Zτ I | ď C |τ I´s | 1{4 .
We deduce that on E, in contradiction with the definition of E. Since all the above assertions are a.s., we get that E is negligible. Finally, the equality θ´p`8q " τ I is a consequence of the (strict) monotonicity of the mapping r0, τ I q Q θ Þ Ñ ş θ 0 µ´pBIpsqq 2 ds. Iptqq "`8. It suggests the following behavior for approximations: for ą 0, consider the elliptic generator L px 2n` qB 2`B (not to be mistaken with the reflecting generator introduced in the proof of Theorem 3.1). The associated speed function µ is defined by It is also possible to define dual processes pI ptqq tě0 with values in the set of closed intervals in the extended line r´8,`8s (except the singletons t´8u and t`8u). Assume that I p0q is a fixed element of I´. Then we guess that lim Ñ0`µ pI pτ qq "`8, where τ inftt ą 0 : 0 P I ptqu (or at least with τ inftt ą 0 : η P I ptqu, for all fixed η ą 0). Due to (3.6), the processes X and I can be coupled in the following way: Theorem 3.5. Let m 0 be a probability distribution on I´and consider m 0 m 0 Λ´. There exists a coupling of X with initial distribution m 0 and of I with initial distribution m 0 such that for any t ě 0, we have on tτ I ą tu, LpXptq|Ir0, tsq " Λ´pIptq,¨q. Furthermore, the construction of I from X is adapted.
With the above coupling, we get that τ I " τ 0 , the hitting time of 0 by X seen in the introduction: Proposition 3.6. In addition to (3.7), we have lim tÑτ I´X ptq " 0.
We have lim Ñ0`τ " τ 0 , thus by continuity of the the trajectories of X, a.s. lim Ñ0`X pτ q " Xpτ I q.
To get Xpτ I q " 0, it is sufficient to check that Xpτ q converges in probability toward 0 as goes to 0`. The relation (3.16) is also true when t is replaced by a stopping time for I (see Diaconis and Fill [4]), so we have LpXpτ q|Ir0, τ sq " Λ´pIpτ q,¨q.
In general, we cannot conclude that lim tÑpτ0q´Y ptq " 0 (convergence which should be sufficiently slow to be compatible with (3.15)), e.g. if we started with Y p0q "´8, then Y ptq "´8 for all t P r0, τ 0 q. Anyway, Proposition 3.6 enables to set pY pτ 0 q, Zpτ 0 qq p0, 0q while preserving the validity of (3.16). See also Remark 1.2, where Λ is just Λ´in (1.6).
Next we extend the process I after time τ 0 as in Theorem 2.1, starting from p0, 0q. Note that the Markov kernel Λ from I´\I`to R defined in (1.2), is obtained by imposing that Λ " Λ´on I´ˆBpRq and Λ " Λ`on I`ˆBpRq. Taking into account this observation, we can merge Theorems 2.5 and 3.5 and Corollary 2.8 into Theorem 1.1.
Remark 3.7. Corollary 2.7 is still valid, replacing R`by R. Indeed, the unique invariant measure remains π, the probability measure defined before Corollary 2.7. The first time I hits p0,`8q is a strong stationary time, as soon as it is finite.
To deduce the density part of Hörmander's theorem, stating that for any t ą 0, LpXptqq is absolutely continuous with respect to the Lebesgue measure on R, it remains to show the next result: Lemma 3.8. For any t ą 0, we have PrIptq " p0, 0qs " 0.
To prove that Prτ 0 " ts " 0, up to conditioning with respect to Xp0q, we can assume that Xp0q " x 0 for some x 0 P R. When x 0 ě 0, the previous section shows that PrXptq " 0s " 0 for all t ą 0. So assume that x 0 ă 0 and decompose τ 0 " r τ`p τ , with Due to the strong Markov property of X, r τ and p τ are independent. Thus to get that the law of τ 0 has no atom, it is sufficient to see that Lpr τ q has no atom. By contradiction, assume there exists s ą 0 such that Prr τ " ss ą 0. We would have PrXpsq " x 0 {2s ą 0. Couple X with I " pY, Zq starting from px 0 , x 0 q as in Theorem 3.5. Taking into account the equality τ 0 " τ I and (3.16), we have PrXpsq " x 0 {2s " PrXpsq " x 0 {2, τ 0 ą ss " PrXpsq " x 0 {2, τ I ą ss " ErEr1 Xpsq"x0{2 |Ir0, sss1 τ I ąs s " ErΛ´pIpsq, x 0 {2q1 τ I ąs s " 0, because for s P p0, τ I q, Λ´pIpsq,¨q is absolutely continuous with respect to the Lebesgue measure. This is the wanted contradiction.

On the circle
In the circle framework presented in the introduction, we begin by studying X and its dual I on each of the segments I k , with k P Z N . The global behavior of pX, Iq is deduced by putting together the obtained informations, similarly to what was done in the previous section.
Let I be one of the segments I k , for k P Z N . To simplify the notation, we see I as a subset of R and up to an affine transformation, we assume that the interior of I is p0, 1q (where the boundaries 0 and 1, may or not be the same in T). There are four possibilities for the status of the boundaries of I, that we investigate below. First we recall some classical definitions, valid in the four cases. To the restriction on I of the generator L defined in (1.7), we associate its scale and speed functions: The interest of these functions is that on p0, 1q, we can write The corresponding scale and speed measures, also written σ and µ, are given by @ z ě y P p0, 1q, σpry, zsq " ż z y σpxq dx, µpry, zsq " ż z y µpxq dx. µppu, 1qq σpuqdu.
The finiteness or not of Σp0q and N p0q determine the status of the boundary 0 with respect to the diffusion X associated to L, seen from I, and similarly for 1. To get these status of the boundaries, as well as their orders of ellipticity, we only need the asymptotic behavior of a and b near the boundaries. That is why we assumed ? a to be smooth, so that by considering expansions of ? a near the boundaries, we can come back to the computations made in Sections 2 and 3. Probably these computations can be extended to more general positive exponents n, in particular with n " 1{2 we would only need to assume that a is smooth. We refrained from this generality, just to avoid the emergence of singularities in the formulation of Hörmander's condition.
Define I the set of compact subsegments included in I and S the set of singletons from I. Consider the Markov kernel Λ from I to r0, 1s: @ ry, zs P I, Λpry, zs,¨q ‚ Case (C1): I " r0, 1s, namely bp0q ą 0 and bp1q ă 0, by considering the behavior of µ and σ near 0 and 1, we compute that Σp0q "`8, N p0q ă`8, Σp1q "`8 and N p1q ă`8, so that 0 and 1 are entrance boundaries for X. It follows that under the initial condition Xp0q " x 0 , where x 0 is fixed in r0, 1s, the process X stays forever in r0, 1s and, more precisely, in p0, 1q for positive times. Since lim xÑ0`µ pxq " 0 " lim xÑ1´µ pxq, the measure µ has a finite weight over I. It is also clear that µ is positive on p0, 1q. It justifies the above definition of Λ and enables to define π as the normalization of µ into a probability measure, which is just Λpr0, 1s,¨q.
As in Section 2 and in [12], it is possible to construct a I-valued dual process I pIptqq tě0 , so that Theorem 1.1 is valid. It follows that for any t ą 0, LpXptqq is absolutely continuous with respect to µ (or equivalently to the Lebesgue measure restricted to r0, 1s), because S is an entrance boundary for I. More precisely, note that µ satisfies (A.1) with c " 0, so according to Appendix A, I can be described in the following way. Writing I pY, Zq ppY ptq, Zptqq tě0 , the processes Y and Z are solutions, up to the time (finite a.s.) when either Y hits 0 or Z hits 1, of the s.d.e.
is finite a.s. and is a strong stationary time for X. Recall that the separation discrepancy between two probability measures m and π is defined in general via where dm{dπ is the Radon-Nikodym derivative of the absolutely continuous part of m with respect to π. We have the following bound, due to Diaconis and Fill [4] (in the case of finite Markov chains, but valid in general): @ t ě 0, }LpXptqq´π} tv ď spLpXptqq, πq ď Prτ ě ts, (4.6) where the norm in the l.h.s. is the total variation. In particular, Xptq converges in law toward π for large t ě 0.
‚ Case (C2): I " r0, 1q, namely bp0q ą 0 and bp1q ą 0, we get that Σp0q "`8, N p0q ă`8, Σp1q ă`8 and N p1q "`8, so that 0 is an entrance boundary and 1 an exit boundary for X. It follows that under the initial condition Xp0q " x 0 , where x 0 is fixed in r0, 1q, the process X ends up exiting r0, 1q by hitting 1 in finite time, say at τ inftt ě 0 : Xptq " 1u. We have lim xÑ0`µ pxq " 0 (but lim xÑ1´µ pxq "`8), so any compact segment included into I has a finite weight, which is positive if it is not reduced to a singleton. Thus the Markov kernel Λ is well-defined.
As in Section 3, it is possible to construct a I-valued dual process I prY ptq, Zptqsq tPr0,τ q , so that Theorem 3.5 is valid, see also Appendix A with c " 0. Up to the time τ , the processes Y and Z are solutions to (4.4) (or (4.5), after Y has hit 0, this may happen or not before Z hits 1). We have a.s. lim tÑτ´Z ptq " 1, and the natural way to extend I after time τ is to define Ipτ q " t1u and to let I start from there into the corresponding segment. Note that for any time t ě 0, we can write LpXpt^τ qq " Prτ ă tsLpXptq|τ ă tq`Prτ ě tsδ 1 , with LpXptq|τ ă tq " ż Λpι,¨q PrIptq P dι|τ ă ts, so that the conditional law in the l.h.s. is absolutely continuous with respect to the Lebesgue measure for t ą 0. As in Lemma 3.8, we show that for any given t ě 0, Prτ " ts " 0.
‚ Case (C4): I " p0, 1q, namely bp0q ă 0 and bp1q ą 0, we get that Σp0q ă`8, N p0q "`8, Σp1q ă`8 and N p1q "`8, so that 0 and 1 are exit boundaries for X. It follows that under the initial condition Xp0q " x 0 , where x 0 is fixed in p0, 1q, the process X ends up exiting p0, 1q by hitting 0 or 1 in finite time, say τ X inftt ě 0 : Xptq P t0, 1uu. Since µ as function is continuous and positive, any compact segment included into I has a finite weight, which is positive if it is not reduced to a singleton. Again the Markov kernel Λ is well-defined.
As in Section 3, it is possible to construct a I-valued dual process I prY ptq, Zptqsq tPr0,τ I q , where τ I ą 0 is the explosion time, so that Theorem 3.5 is valid, see also Appendix A with c " 0. Up to the time τ I , the processes Y and Z are still solutions to (4.4). A priori the explosion time τ I is such that τ I ď τ X , but the arguments of Proposition 3.6 show that lim tÑτ I´Y ptq " 0 or lim tÑτ I´Z ptq " 1, and τ I " τ X . When lim tÑτ I´Y ptq " 0 and lim tÑτ I´Z ptq ă 1, it is safe to set Ipτ I q " t0u. In this situation we have Xpτ I q " 0, according to the proof of Proposition 3.6. We can thus let pX, Iq start from p0, t0uq into the segment containing 0. Symmetrically when lim tÑτ I´Y ptq ą 0 and lim tÑτ I´Z ptq " 1, we set Ipτ I q " t1u and we have Xpτ I q " 1, so we can let pX, Iq start from p1, t1uq into the segment containing 1.
Consider now the case where lim tÑτ I´Y ptq " 0 and lim tÑτ I´Z ptq " 1. When furthermore we have lim tÑτ I´Λ pIptq,¨q " δ 0 or lim tÑτ I´Λ pIptq,¨q " δ 1 , again we can respectively define Ipτ I q " t0u and Ipτ I q " t1u. But what should we do when the limit of ΛpIptq,¨q, as t goes to τ I´, charges both 0 and 1, or worse, if this limit does not exist? In fact, we believe the former alternative is always true (killing even the possibility of (4.7)): Whether this assertion is true or wrong, it is always possible to look at Xpτ I q, which is either 0 or 1, and to set Ipτ I q " tXpτ I qu. This idea was also used in Copros [2], in the context of denumerable Markov processes. Immediately after τ I , X and I will evolve in the segment containing tXpτ I qu. This choice leads to a dual process I satisfying Theorem 1.1.

Remark 4.2.
One does not need to wait that X pass through 0 or 1 for making an observation of X and subsequently concentrate I to a singleton: at any stopping time ζ for X, one can decide to change the value of I and impose that Ipζq " tX ζ u. This quantum physics sounding property does not impact condition (1.5), but of course it may destroy condition (1.4), for instance if ζ is the minimum of a positive deterministic time with τ X . Note that the observation may also be imperfect: assume that p0, 1q is decomposed into a measurable partition \ sPS A s , where S is a denumerable index set, and that we observe that Xpζq P A s , then we can replace Ipζq by Ipζq X A s . In general we are looking for the largest dual processes, so the above observation/concentration procedure should be avoided, see Example 4.3 below.
More precisely, let us come back to the circle setting described before Theorem 1.3. Consider the segments I k , for k P Z N , as the vertices of an oriented graph whose edges are as follows: there is an edge from I k to I k`1 , if y k`1 P I k`1 and an edge from I k`1 to I k , if y k`1 P I k . Except when the segments are all of type (C2), or all of type (C3), following the oriented edges, one goes from segments of type (C4) or springs to segments of type (C1) or sinks, after possibly visiting a successive sequence of segments of type (C2), turning anti-clockwise, or a successive sequence of segments of type (C3), turning clockwise. In particular, it appears that the number of springs is the number of sinks. Inside each segment, the dual process is constructed according to its type. From the above considerations, we get all the requirements on the dual process I presented in Theorem 1.1.
Note that the segments are all of type (C2) (respectively (C3)) if and only if b is positive (resp. negative) on N. Thus assuming the drift b does not take a fixed sign on N, whatever the starting point, X ends up into a sink in finite time, since the exit times from segments of type (C2), (C3) and (C4) are all a.s. finite. In this situation, for large times, the process X converges in law, the process I converges a.s. and the limit law of X is ErΛpIp`8q,¨qs, where Ip`8q lim tÑ`8 Iptq (convergence taking place in finite time). Except when there is only one sink (in which case it is possible to construct a strong stationary time, since there is a unique invariant probability measure, namely the normalizations of the speed measure on the sink), the limit law of X depends on its initial condition. E.g. starting from a spring, the process X have positive probabilities (depending on the exact initial position in the spring) to exit it from the right or from the left and with the same probabilities, I collapse on the right or on the left boundary. After that, I will converge toward the closest sink following the above edges. The limit law of X is then a convex combinaison (with the previous probabilities) of the normalizations of the corresponding speed measures.
When b has a fixed sign on N, the process I does not converge a.s. since it appears that Ipt n q " tXpt n qu for all n P N, where pt n q nPN is the unbounded increasing sequence of times t ě 0 such that Xptq P N. More precisely, assume for instance that b is positive on N, after the first time X hits N, according to (4.5), we have, according to Appendix A with c " 0, dZptq "ˆa 1 pZptqq´bpZptqq`2 µ Kptq pZptqq µ Kptq pry Kptq , Zptqsq apZptqq˙dt`a2apZptqq dW ptq, where Kptq is the unique index k P Z N such that Zptq P I k (furthermore, we have Y ptq " y Kptq ). Thus it appears that Z becomes a Markov processes, whose behavior is quite similar to that of X (they even coincide at each time X pass through N). The dual process I is not very helpful to understand the convergence in law of X. Indeed, as announced in the introduction, another dual process r I should be considered to go in this direction. It will be done in the following section.
Let us now present an example showing the above dual is not optimal with respect to the construction of a strong stationary time. apxq cos 2 pxq, bpxq " sinpxq.
dZptq "˜a 1 pZptqq´bpZptqq`2 a apZptqqµ 1 pZptqq`aap´Zptqqµ 1 p´Zptqq µ 1 pr´Zptq,´π{2s \ rπ{2, Zptqsq a apZptqq¸dt a 2apZptqq dW ptq (4.8) "˜a 1 pZptqq´bpZptqq`2 a apZptqqµ 1 pZptqq µ 1 prπ{2, Zptqsq a apZptqq¸dt`a2apZptqq dW ptq, where µ 1 is the speed measure associated to I 1 and pW ptqq tě0 is a standard Brownian motion. For the second equality, we used the symmetry of I 1 and of L (with respect to the real axis, when T is seen as the unit circle in C). When Z hits π, I hits rπ{2, 3π{2s and the corresponding hitting time is a strong stationary time for X.
Consider now the case where we set Ipτ q " tXpτ qu and assume for instance that Xpτ q " π{2. For t ě τ , our construction for Theorem 1.3 leads to a dual of the form Iptq " rπ{2, Zptqs, where Z takes values in rπ{2, 3π{2s and solves the s.d.e. (4.8). The dual process will be absorbed at I 1 when Z hits 3π{2 and this provides a strong stationary larger than the previous one, since Z must go through π before hitting 3π{2.
As just seen, starting from 0, this example can be brought back to the case of a diffusion on a segment starting from its boundary. This situation is well-understood (see Fill and Lyzinski [6]) and the strong stationary time constructed in the former case is in fact sharp, namely stochastically smaller than any other strong stationary time.
For the remaining part of this section, let us assume that Conjecture 4.1 is true. To construct a dual process J pJptqq tě0 able to collapse on pairs of points, we modify the definitions given in the introduction in the following way. Let I 1 stand for the set of non-empty closed intervals from T which are included into one of the I k , for k P Z N and I 2 the set of pairs pι 1 , ι 2 q, where ι 1 , ι 2 P I 1 are disjoints. Now set I I 1 \ I 2 and S " S 1 \ S 2 , with S 1 ttxu : x P Tu and S 2 tpy k , y l q : k " l P Z N u. For any α P r0, 1s, define a Markov kernel Λ α from I to T by @ ι P I, Λ α pι,¨q αδ y k p¨q`p1´αqδ y l p¨q , when ι " py k , y l q P S 2 , µ k pιX¨q µ k pιq , when ι P I 1 zS 1 and ι Ă I k , , when ι " pι 1 , ι 2 q P I 2 zS 2 , ι 1 Ă I k and ι 2 Ă I l . (4.9) Then Theorem 1.3 can be extended into: There exists a process J pJptqq tě0 taking values in I, whose construction is adapted with respect to X, such that Jp0q " tXp0qu, where αpIp0qq P r0, 1s only depends on Ip0q (or equivalently on Xp0q). In particular, when LpXp0qq " δ x0 for some x 0 P T, we have for any t ě 0, the decomposition LpXptqq " ż Λ αptx0uq pι,¨q LpJptqqpdιq.
Proof. When Xp0q does not belong to a spring, the dual process J is the same as I in Theorem 1.3 and the introduction of I 2 and S 2 are not necessary. When Xp0q " x 0 belongs to a spring, say I k , let τ its exit time from I k and αptx 0 uq P x0 rXpτ q P I k´1 s. Before τ , J is constructed as I in Theorem 1.3, but at τ , we impose Jpτ q pty k u, ty k`1 uq.

Conjecture 4.1 enables us to see that
LpXpτ q|Jr0, τ sq " Λ αpIp0qq pJpτ q,¨q, from which we can keep up constructing J after the time τ , by setting @ t P r0, r τ q, Jpτ`tq pI 1 ptq, I 2 ptqq, r τ inftt ě 0 : I 1 ptq X I 2 ptq " Hu, where I 1 and I 2 are the same as in Theorem 1.3, starting with I 1 p0q " ty k u and I 2 p0q " ty k`1 u, and directed by the same Brownian motion pW ptqq tě0 in (4.4) and (4.5). When r τ ă`8 (as in Example 4.3, where it corresponds to the time Z hits π), we set Jpr τ`tq I 1 pr τ q Y I 2 pr τ q for all t ě 0.
As in Remark 1.2, the probability measure-valued Markov process pΛ αptx0uq pJptq,¨q is continuous and seems the right object to consider as a dual.
The main advantage of Theorem 4.4 over Theorem 1.3, i.e. of the Markov kernel given in (4.9) over the Markov kernel (1.8), is that it enables to extend the construction of strong time stationary times τ, in the sense that the position X τ is distributed according to an invariant probability measure (maybe non longer the unique invariant probability measure as before). This is possible when Xp0q starts from a fixed position x 0 P T and when b does not take a fixed sign on N. Indeed, in this case the dual process ends up being absorbed in a state Jp8q depending only on x 0 , which is either a closed segment from tI k : k P Z N u or a disjoint union of two such segments. Since Λ αptx0uq pJp8q,¨q is an invariant probability measure for X depending only on x 0 , classical arguments from Diaconis and Fill [4] then show that the absorbing time for J is a strong stationary time.

The turning diffusion
Here we consider more precisely the circle situation where b has a fixed sign on N , to show Theorem 1.4 and to deduce the convergence of X in law for large time.
Up to conjugacy with respect to T Q x Þ Ñ´x P T, it is sufficient to study the case where b ą 0 on N . We begin by investigating the invariant measure for the generator L given in (1.7). For k P Z N , recall that µ k is the speed measure of the restriction of L on I k . It is defined up to a positive factor by @ x P I k , µ k pxq 1 apxq exp˜ż where z k is a chosen point belonging to I k and where a segment ru, vs Ă T will always be understood as the path going from u to v anti-clockwise. For any family of non-negative numbers pp k q kPZ N , the measure µ ř kPZ N p k µ k satisfies µrLrf ss " 0 for any smooth f with compact support in TzN. But this is not sufficient for µ to be a invariant measure.
Furthermore we are here looking for an invariant probability measure and it can be easily check that µpTq "`8, except in the trivial case where all the p k , for k P Z N , are equal to zero. In fact, for fixed k P Z N , the restriction of L to smooth functions with compact support in I k is symmetric in L 2 pµ k q but the problem at hand is really non reversible since the diffusion X has a strong tendency to turn anti-clockwise around T. Lemma A.1 in Appendix A suggests to rather look for the solutions η k of the equation given on the interior of I k by where c k is a constant. When a did not vanish on ty k , y k`1 u, it is not difficult to check that the general solution of this equation is, for any x PI k , η k pxq " 1 apxq˜p where p k and q k are two constants such that p k`qk " c k . If we want this expression to converge when a does vanish on ty k , y k`1 u and b is positive on ty k , y k`1 u, we must take p k " 0. It leads us to consider @ x PI k , η k pxq " ż rx,y k`1 s exp˜´ż rx,vs b a pvq dv¸.
We compute that lim xÑy k`η k pxq " 0 " lim xÑy k`1´η k pxq, so define η k py k q 0 η k py k`1 q. Since we have paη k q 1 " bη k´1 , we deduce from the decomposition (A.12) with c " 1 that for any f P C 8 pry k , y k`1 sq, we have η k rLrf ss " raη k f 1 s y k`1 y k´r f s y k`1 y k "´pf py k`1 q´f py k qq.
Define the function η on T by imposing that η coincide with η k on I k for all k P Z N . Also denote η the measure admitting η as density with respect to the Lebesgue measure and remark that this density is continuous (and vanish on N ), so that ηpTq ă`8. Furthermore we have for any f P C 8 pTq, ηrLrf ss "´ÿ kPZ N f py k`1 q´f py k q " 0, namely η is invariant for L. The probability π appearing in (1.9) is just the normalization of η into a probability measure.
Let us now describe the evolution of the dual process r I p r Y , r Zq. Assume that Xp0q " x 0 P I k " ry k , y k`1 q, for some k P Z N . Following (A.2) and (A.6), we begin by defining p r Y ptq, r Zptqq tPr0,τ1q as the solution of the s.d.e.
for t P p0, τ 1 q, where τ 1 is the first time either r Y hits y k or r Z hits y k`1 . First, assume that r Y pτ 1 q " y k . We extend the process p r Y , r Zq after time τ 1 by letting r Y ptq " y k , for all t ě τ 1 , and by solving for r Z the s.d.e., for t P rτ 1 , τ 2 q, where τ 2 is the first time after τ 1 that r Z hits y k`1 . This time is a.s. finite, because y k`1 is an exit boundary for r Z (as well as for X) on ry k , y k`1 q. Next for t P rτ 2 , τ 3 q, we ask that r Z solves again the s.d.e. (5.1), where τ 3 is the first time after τ 2 that r Z hits y k`2 . This time is a.s. finite, because y k`2 is an exit boundary for r Z on ry k`1 , y k`2 q. We keep solving this equation until r Z ends up hitting y k , say at time τ, which is also a.s. finite. After τ, we take r I to be equal to T. Since the generator of r I p r Y , r Zq is intertwined with L through r Λ, we construct a coupling of r I with the diffusion X, associated with the generator L, so that r Ip0q " tXp0qu, @ t ě 0, LpXptq| r Ir0, tsq " r Λp r Iptq,¨q.
Then we get that τ is a strong stationary time for X. It follows that X converges toward π in separation and in total variation in large time, due to the general bounds (4.6). As pointed out by the referee, since by compactness the above hitting times can be bounded uniformly with respect to the starting point, these convergences are uniform with respect to the starting point, implying an exponential convergence.

A About segment-valued dual processes
Putting together considerations from [12] and [3], we present here some computations that were used throughout the paper.
On p0, 1q, consider a generator L aB 2`b B, where a ą 0 and b are smooth functions on p0, 1q. Let η ą 0 be a smooth function on p0, 1q satisfying paηq 1 " bη´c, where c is a constant. Then the measure (still denoted η) admitting η with respect to the Lebesgue measure λ on p0, 1q is invariant for L in the following sense: Lemma A.1. For any f P C 8 c p0, 1q, the space of smooth functions with compact support inside p0, 1q, we have ηrLrf ss " 0. Furthermore, η is reversible with respect to L, in the sense that for all f, g P C 8 c p0, 1q, ηrgLrf ss " ηrf Lrgss, if and only if c " 0.
Proof. These results are immediate consequences of the following integration by parts: for all f, g P C 8 c p0, 1q, ηrgLrf ss " When g " 1 (the mapping always taking the value 1, here on p0, 1q), the r.h.s. is equal to showing the first assertion of the above lemma. Concerning the second one, the reversibility is equivalent to By another integration by parts, the r.h.s. is equal to´c ş gf 1 dλ, so we must have c ş gf 1 dλ " 0, fo all f, g P C 8 c p0, 1q and this is true if and only if c " 0.
We compute that for any f P C 8 c p0, 1q and py, zq P , B y F f py, zq "´f pyqηpyq, It follows from Itô's formula that dF f pY ptq, Zptqq y F f pY ptq, ZptqqapY ptqqdt " pf ηqpZptqqdZptq´pf ηqpY ptqqdY ptq`pf ηq 1 pZptqqapZptqqdt´pf ηq 1 pY ptqqapY ptqqdt " dM f ptq`ApY ptq, Zptqqdt, and where Apy, zq pf ηqpzq´a 1 pzq´βpzq`2Hpy, zq a apzq¯´pf ηqpyq´a 1 pyq´βpyq´2Hpy, zq a apyqp f ηq 1 pzqapzq´pf ηq 1 pyqapyq " pf aηq 1 pzq´pf aηq 1 pyq´pβf ηqpzq`pβf ηqpyq`2Hpy, zqG f py, zq " pf 1 aηqpzq´pf 1 aηqpyq`pf ppaηq 1´β ηqqpzq´pf ppaηq 1´β ηqqpyq`2Hpy, zqG f py, zq. Apy, zq " F Lrf s py, zq`2Hpy, zqG f py, zq. (A.7) Remark A.2. Let us make the link with the formulation adopted in [3] in the context of Riemannian geometry in dimension strictly larger than 1. Endow p0, 1q with the Riemannian metric given by 1{a (so that the norm of the usual unit vector 1 above x P p0, 1q is 1{ a apxq, or equivalently,˘aapxq are the unit vectors above x in the new Riemannian metric). Let d be the corresponding distance and for any A Ă p0, 1q and ą 0, let A tx P p0, 1q : dpx, Aq ď u, the -enlargement of A. Then we have for any f P C 8 c p0, 1q and py, zq P , (η will only serve to measure the boundaries Bry, zs " ty, zu of segments ry, zs, with py, zq P , we used the symbol ş in (A.8) instead of a sum over the two elements of Bry, zs to adopt the same notation as in higher dimensional Riemannian geometry). It appears that where x¨,¨y and ∇ are relative to the considered Riemannian metric and where ν is the "unit exterior normal vector" on Bry, zs. The function U lnpdη{dγq is the logarithm of the Radon-Nikodym density of η with respect to the Riemannian measure γ, which admits itself the density 1{ ? a with respect to the usual Lebesgue measure. Thus we have dη{dγ " η, where by a traditional abuse of notation, we also interpret η as the function ?
aη. In the usual definitions in higher dimensional Riemannian geometry (see e.g. Proposition 1.2.1 of Mantegazza [11]), the r.h.s. of (A.10) should contain a supplementary term ş f ρ dη where ρ would be the "mean" curvature on the boundary Bry, zs with respect to the unit exterior normal vectors. Thus we recover that in dimension 1, the mean curvature of a boundary of dimension 0 vanishes: ρ " 0. To see the coherence of (A.2) with the formulation of [3] in the context of diffusions in Riemannian manifolds of dimension larger or equal to 2, we should check that a 1´β "´x∇U´b H , νy ν´ρν, (A.11) where b H is such that the Helmoltz-Hodge decomposition b " ∇U`b H holds (note the change of sign with respect to (A.11)), i.e. Remark that in general the Helmoltz-Hodge decomposition b " ∇U`b H is different from the decomposition b " paηq 1 {η`c{η, which enables to write where 1 η BpaηBq is symmetric in L 2 pµq and c η B is skew-symmetric in L 2 pµq. Let pD, Lq be the generator L of pY, Zq in the sense of (local) martingale problems. It follows from (A.4) and (A.7) that L acts on the elementary observable F f , with f P C 8 c p0, 1q, by LrF f spy, zq " F Lrf s py, zq`2Hpy, zqG f py, zq.
Furthermore, the carré du champs Γ L associated to L is such that the bracket of the martingale M f defined in (A.5) satisfies @ t P r0, τ q, @ M f D t " 2 ż t 0 Γ L rf, f spY psq, Zpsqq ds.
It follows by polarization that all f, g P C 8 c p0, 1q, @ py, zq P , Γ L rF f , F g spy, zq " G f py, zqG g py, zq.
Since pY, Zq is a diffusion (namely a Markov process with continuous trajectories), the generator L and the carré du champs Γ L extend in the following way (see e.g. the book of Bakry, Gentil and Ledoux [1]). Consider the algebra A consisting of the composed observables of the form F fpF f1 , ..., F fn q, where n P Z`, f 1 , ..., f n P C 8 c p0, 1q and f : R Ñ R is a C 8 mapping, with R an open subset of R n containing the image of by pF f1 , ..., F fn q. Then A is included into D and since L is a differential operator of order 2 without terms of order 0, we have for any F fpF f1 , ..., F fn q and G gpF g1 , ..., F gm q belonging to A, LrFs " ÿ jPJnK B j fpF f1 , ..., F fn qLrF fj s`ÿ k,lPJnK B k, l fpF f1 , ..., F fn qΓ L rF f k , F f l s, Γ L rF, Gs " ÿ lPJnK, kPJmK B l fpF f1 , ..., F fn qB k gpF g1 , ..., F gm qΓ L rF f l , F g k s (where JnK t1, 2, ..., nu).
Note that for any f P C 8 c p0, 1q, we have Λrf s " F f {F 1 , so Λrf s P A and the above formulas lead without difficulty to the intertwining relation @ py, zq P , @ f P C 8 b p0, 1q, LrΛrf sspy, zq " ΛrLrf sspy, zq.
(A.13) Furthermore, by considering observables of the form fpF 1 q, where f P C 8 pRq, it appears that pηprY ptq, Zptqsqq tPr0,τ q is a (possibly stopped) time-changed Bessel process of dimension 3. This property enables us to let the process pY, Zq start from the singleton py 0 , y 0 q, by passing to the limit as z 0 goes to y 0`a nd to see that the set of the singletons is an entering boundary for pY, Zq, see Section 2 from [12]. Under the assumption that LpY 0 , Z 0 qΛ " LpX 0 q, proceeding as in Section 4 from [12], we construct a coupling of the diffusion X associated to the generator L with the process pY, Zq, so that @ T ě 0, LpX t |pY t , Z t q tPr0,T s q " ΛppY T , Z T q,¨q. (A.14) Alternatively, conditioning furthermore by the initial condition X 0 , we can also couple X and pY, Zq so that Y 0 " X 0 " Y 0 , in addition to (A.14).
When ηpr0, xsq ă`8 for (one or all) x P p0, 1q, in the above considerations Y can be fixed equal to 0 (and symmetrically, Z can be fixed equal to 1, when ηprx, 1sq ă`8 for x P p0, 1q). In particular, we can impose this restriction once Y has hit 0 (or Z has hit 1).
Then the natural extensions of the previous results hold.