Marc Arnaudon

Abdoulaye Koléhé

Laurent Coulibaly-Pasquier

Miclo

Koléhè Coulibaly-Pasquier

Laurent Miclo

On the separation cut-off phenomenon for Brownian motions on high dimensional spheres

Keywords: Spherical Brownian motions, strong stationary times, separation discrepancy, hitting times, small noise one-dimensional diffusions primary: 58J35, secondary: 58J65 60J60 35K08 37A25

 to deduce the wanted cut-off phenomenon.

Introduction

Consider the Brownian motion X pXptqq tě0 on the sphere S n`1 Ă R n`2 of dimension n `1 ě 1, time-accelerated by a factor 2, so the generator of X is the Laplacian (and not the Laplacian divided by 2). Starting from a point, the time marginal laws of X spread over S n`1 and approach the uniform distribution in large times. A traditional question is to estimate corresponding speeds of convergence, or mixing times, especially for large n. The answer depends on the way the difference between the time marginal and the uniform distribution is measured. Saloff-Coste [START_REF] Saloff-Coste | Precise estimates on the rate at which certain diffusions tend to equilibrium[END_REF] has proven that for the total variation, the mixing time is equivalent to lnpnq{p2nq and furthermore a cut-off phenomenon occurs (see also Méliot [9] for extensions). Due to reversibility and cut-off, general arguments, see (1.5) in Hermon, Lacoin and Peres [START_REF] Hermon | Total variation and separation cutoffs are not equivalent and neither one implies the other[END_REF], imply that for the separation discrepancy the mixing time asymptotically belongs to the interval rlnpnq{p2nq, lnpnq{ns. The convergence of X to the uniform distribution can be brought back to a one-dimensional question, by considering its radial part (with respect to the starting point), since its "angular part" is at once at equilibrium by symmetry. Onedimensional diffusions are quite close to birth and death processes, so we can expect from the results of Diaconis and Saloff-Coste [START_REF] Diaconis | Separation cut-offs for birth and death chains[END_REF] and Ding, Lubetzky and Peres [START_REF] Ding | Total variation cutoff in birth-and-death chains[END_REF] that a cut-off phenomenon equally occurs in the separation sense. Our goal here is to check this is indeed the case and that this abrupt convergence occurs at times round lnpnq{n. Our proof is based on two ingredients: (1) the resort to the strong stationary times for X presented in [START_REF] Arnaudon | Construction of set-valued dual processes on manifolds[END_REF] and (2) quantitative estimates on the hitting times for one-dimensional diffusion processes, obtained via elementary calculus (and a very restricted dose of stochastic calculus). This alternative point of view on cut-off differs from the traditional approach based on spectral analysis and could be extended to other situations where less spectral information is available.

Without loss of generality, we can assume that X starts from x 0 p1, 0, 0, ..., 0q P S n`1 Ă R n`2 . It was seen in [START_REF] Coulibaly | On the evolution by duality of domains on manifolds[END_REF] that X can be intertwined with a process D pDptqq tě0 taking values in the closed balls of S n`1 centered at x 0 , starting at tx 0 u and absorbed in finite time τ n in the whole set S n`1 . In [START_REF] Arnaudon | Construction of set-valued dual processes on manifolds[END_REF], several couplings of X and D were constructed (two of them are recalled in Corollary 4 below), so that for any time t ě 0, the conditional law of Xptq knowing the trajectory Dpr0, tsq pDpsqq sPr0,ts is the normalized uniform law over Dptq, which will be denoted ΛpDptq, ¨q in the sequel. Furthermore, D is progressively measurable with respect to X, in the sense that for any t ě 0, Dpr0, tsq depends on X only through Xpr0, tsq. Due to these couplings and to general arguments from Diaconis and Fill [START_REF] Diaconis | Strong stationary times via a new form of duality[END_REF], τ n is a strong stationary time for X, meaning that τ n and Xpτ n q are independent and Xpτ n q is uniformly distributed over S n`1 . As a consequence we have @ t ě 0, spLpXptqq, U n`1 q ď Prτ n ě ts where the l.h.s. is the separation discrepancy between the law of Xptq and the uniform distribution U n`1 over S n`1 .

Recall that the separation discrepancy between two probability measures µ and ν defined on the same measurable space is given by spµ, νq " ess sup where dµ{dν is the Radon-Nikodym density of µ with respect to ν.

Remark 1 Note that for any t P r0, τ n q, the "opposite pole" p´1, 0, 0, ..., 0q does not belong to the support of ΛpDptq, ¨q. It follows from an extension of Remark 2.39 of Diaconis and Fill [START_REF] Diaconis | Strong stationary times via a new form of duality[END_REF] that τ n is even a sharp strong stationary time for X, meaning that @ t ě 0, spLpXptqq, U n`1 q " Prτ n ě ts Thus the understanding of the convergence in separation of X toward U n`1 amounts to understanding the distribution of τ n . From the bibliographical survey given above, it can be expected that τ n is of order lnpnq{n.

In confirmation of the above observation, a first purpose of this note is to prove the following result.

Theorem 2

We have for all n large,

Erτ n s " lnpnq n
Let us go further by showing a cut-off phenomenon, namely that in the scale lnpnq{n, the random variable τ n is in fact close to its mean Erτ n s. This property can be written under several forms, see e.g. the review of Diaconis [START_REF] Diaconis | The cutoff phenomenon in finite Markov chains[END_REF] or the book [START_REF] Levin | Markov chains and mixing times[END_REF] of Levin, Peres and Wilmer (both in the context of finite Markov chains). We consider the following simple formulation: Theorem 3 For any r ą 0, we have

lim nÑ8 P " τ n ą p1 `rq lnpnq n  " 0 lim nÑ8 P " τ n ă p1 ´rq lnpnq n  " 0
For any t ě 0, denote Rptq the Riemannian radius of Dptq in S n`1 , so that Rp0q " 0 and

τ n " inftt ě 0 : Rptq " πu (1)
It was seen in [START_REF] Coulibaly | On the evolution by duality of domains on manifolds[END_REF] that R pRptqq tě0 is solution to the stochastic differential equation

@ t P p0, τ n q, dRptq " ? 2dBptq `bn pRptqqdt (2)
where pBptqq tě0 is a standard Brownian motion in R and the mapping b n is given by @ r P p0, πq, b n prq 2 sin n prq ş r 0 sin n puq du

´n cosprq sinprq (3)
It is not difficult to check (see e.g. the bound (32) which is an equivalent as x Ñ 0 `) that as r goes to 0 `bn prq "

n `2 r
and this is sufficient to insure that 0 is an entrance boundary for R, so that starting from 0, it will never return to 0 at positive times.

In the following corollary we explicit two intertwinings, which were constructed in [1] Theorems 3.5 and 4.1.

Corollary 4 Let pX t q tě0 be a Brownian motion in S n`1 started at x 0 . For x P S n`1 ztpx 0 , ´x0 u, denote by N pxq the unit vector at x normal to the circle with radius ρpx 0 , xq where ρ is the distance in the sphere, pointing towards x 0 : N pxq " ´∇ρpx 0 , ¨qpxq.

(1) Full coupling. Let D 1 ptq be the ball in S n`1 centered at x 0 with radius R 1 ptq solution started at 0 to the Itô equation dR 1 ptq " ´?2xN pX t q, dX t qy `n r2 cotpρpx 0 , X t qq ´cotpR 1 ptqqs dt This evolution equation is considered up to the hitting time τ p1q n of π by R 1 ptq.

(2) Full decoupling, reflection of D on X. Let D 2 ptq be the ball in S n`1 centered at x 0 with radius R 2 ptq solution started at 0 to the Itô equation dR 2 ptq " ´?2dW t `2dL R 2 t pρpx 0 , ¨qqpXq ´n cotpR 2 ptqq dt where pW t q tě0 is a real-valued Brownian motion independent of pX t q tě0 and L R 2 t rρpx 0 , Xqs is the local time at 0 of the process R 2 ´ρpx 0 , Xq. These considerations are valid up to the hitting time τ p2q n of π by R 2 ptq. Let Dptq be the ball in S n`1 centered at x 0 with radius Rptq, defined in (2), and let τ n be the stopping time defined in [START_REF] Arnaudon | Construction of set-valued dual processes on manifolds[END_REF].

Then we have:

(1) for i " 1, 2 X τ piq n is uniformly distributed in S n`1 ,
(2) the pairs pτ p1q n , pD 1 ptqq tPr0,τ p1q n s q, pτ p2q n , pD 2 ptqq tPr0,τ p2q n s q and pτ n , pDptqq tPr0,τns q have the same law. In particular τ p1q n and τ p2q n satisfy Theorems 2 and 3.

Heuristically speaking, the mapping b n is of order n (see Lemma 7, nevertheless mitigated by Proposition 8), thus renormalizing time by a factor 1{n, we end up with a small noise diffusion, so large deviation estimates could lead to the desired result.

Indeed, in the next section we will show that lnpnq{n is an equivalent of the time needed to go from 0 to π for the dynamical system obtained by removing the Brownian motion in [START_REF] Coulibaly | On the evolution by duality of domains on manifolds[END_REF]. But instead of subsequently resorting to the large deviation theory, which cannot be directly applied here due to the existence of two scales 1{n and 1{

? n, we present in Section 3 an alternative direct perturbative argument to estimate hitting times, leading to curious optimization problems over avatars of the drift. The latter are approximatively solved in Section 4, leading to the proofs of Theorems 2 and 3. The last section justifies the resort to avatars, by showing that the cut-off phenomenon cannot be deduced by only working with the initial drift.

Corresponding dynamic systems

In the spirit of the small noise approach alluded to above, we give here a heuristic justification of the lnpnq{n term by forgetting the Brownian motion in [START_REF] Coulibaly | On the evolution by duality of domains on manifolds[END_REF]. Nevertheless the following computations are not disconnected from our main goal, as they will be re-used later on.

The dynamical system associated to (2) is defined by

" x 0 " 0 9 x t " b n px t q (4)
up to the time T n it hits π (Proposition 8 below will imply in particular that px t q tPr0,Tns is increasing and that T n is finite). The goal of this section is to show the following behavior for this hitting time:

Theorem 5 For large n we have

T n " lnpnq n
This bound can serve as an "explanation" for the quantity lnpnq{n as Theorem 2 will be obtained via perturbative arguments around this result.

The proof of Theorem 5 consists of the two matching lower and upper bounds separately presented in the next subsections. In both cases, b n will be replaced by more manageable drifts.

The upper bound

Our goal here is to show one "half" of Theorem 5, the most interesting one if we were in a sampling context, since it serves as a guarantee for convergence. The previous bound has the drawback to vanish at x " π{2, which is problematic for the hitting time of π. So we need another lower bound for b n :

Proposition 8 There exists a constant r c ą 0 such that for all n large enough,

@ x P p0, πq, b n pxq ě r c ? n
Fix some A ą 0 and note that for x P p0, πq outside rπ{2 ´A{ ? n, π{2 `A{ ? ns, we have

| cotpxq| ě | cospxq| ě cos ˆπ 2 ´A ? n " A ? n (5)
It follows from Lemma 7 that to prove Proposition 8, it sufficient to investigate the behavior of b n pxq on rπ{2 ´A{ ? n, π{2 `A{ ? ns. We begin with the point π{2: By integration by part, it appears that this quantity satisfies,

@ n ě 2, ι n " n ´1 n ι n´2
from which we get that for n large

ι n " c π 2n (6)
and we deduce the wanted equivalent.

For the other points x P rπ{2 ´A{ ? n, π{2 `A{ ? ns (with n ą 4A 2 {π 2 ), we are to systematically consider the change of variable

a ? n ´x ´π 2 ¯P r´A, As (7)
We need the following ingredients.

Lemma 10 With the parametrization (7), we get for large n, uniformly over a P r´A, As, cospxq " ´a ? n

sin n pxq " e ´a2 {2
I n pxq " hpaq ? n where @ x P r0, πs, I n pxq

ż x 0 sin n puq du @ a P R, hpaq ż a ´8 e ´u2 {2 du
Proof Writing

x " π 2 `a ? n the first equivalent is obtained via an immediate expansion around π{2.

For the second equivalent, note that

sin n pxq " ´a1 ´cos 2 pxq ¯n " exp ˆn 2 ln ˆ1 ´cos 2 ˆπ 2 `a ? n ˙˙" exp ˆ´n 2 cos 2 ˆπ 2 `a ? n ˙" e ´a2 {2
For the last equivalent, write

I n pxq " ż π{2 0 sin n pyq dy `ż x π{2
sin n pyq dy From the previous computation, especially its uniformity, we deduce

ż x π{2 sin n pyq dy " ż a 0 e ´v2 {2 dv ? n
From Lemma 9 we have for large n,

ż π{2 0 sin n pyq dy " c π 2n " 1 ? n ż 0 ´8 e ´v2 {2 dv
and thus finally the wanted equivalent.

Recalling the definition of b n given in (3), we deduce from Lemma 10 that uniformly for a P r´A, As, b n pxq " ? nβpaq with

@ a P R, βpaq 2 e ´a2 {2 hpaq `a (8)
This mapping will be precisely investigated in Section 4, but for the moment just note that by continuity we can choose A ą 0 sufficiently small so that

@ a P r´A, As, βpaq ě βp0q 2 " c 2 π
Proposition 8 then follows from this bound and ( 5), for any given r c P p0, a 2{π ^Aq.

The previous lower bounds on b n lead us to introduce a new function r b n on p0, πq via

@ x P p0, πq, r b n pxq # r c ? n , if x P rπ{2 ´A{ ? n, π{2 `A{ ? ns n| cotpxq| , otherwise
Our interest in r b n is its simplicity and the fact that

b n ě r b n
Thus if we replace (4) by

" r x 0 " 0 9 r x t " r b n pr x t q (9)
defined up to the time r T n it hits π, we get

@ n P N, T n ď r T n
Proposition 6 is an immediate consequence of this bound and Lemma 11 For n large, we have

r T n " lnpnq n Proof We decompose r T n into r T p1q n `r T p2q n `r T p3q n where r T p1q n inf " t ě 0 : r x t " π 2 ´A ? n * r T p2q n inf " t ě 0 : r x r T p1q n `t " π 2 `A ? n * r T p3q n inftt ě 0 : r x r T p1q n `r T p2q n `t " πu
and we analyse each of these times separately.

' For t P r0, r T p1q n q, we rewrite the second equation of (9) as

sinpr x t q cospr x t q 9 r x t " n i.
" π{2 `A ? n ´pπ{2 ´A ? n q r c ? n " 2 A ? n r c ? n " 2A r cn ' For t P p r T p2q n `r T p2q n , r T p1q n `r T p2q n `r T
p3q n q we rewrite the second equation of (9) as ´sinpr x t q cospr x t q 9 r x t " n which can be treated as before to show that

r T p3q n " lnpnq 2n
Putting together these estimates, we deduce the desired result.

The lower bound

Our goal here is to show the second "half" of Theorem 5:

Proposition 12 We have lim inf nÑ8 n lnpnq T n ě 1
As in the previous section, we are to replace b n by a simpler drift b n ď p b n , whose corresponding hitting time p T n of π will furnish a time satisfying p T n ď T n . We start by remarking that the arguments that have led to Proposition 8 imply equally:

Lemma 13 For any A ą 0, we can find a constant p c A ą 0 such that for all n large enough,

@ a P r´A, As, b n ˆπ 2 `a ? n ˙ď p c A ? n Fix A ą 0.
Here is an analogue of Lemma 7.

Lemma 14 There exists a quantity pAq ą 0 such that for all n sufficiently large, depending on A,

@ x P p0,
1 ď ˆ1 `1 A ˙cospx A q
For any x P p0, x A s, we have cospxq ě cospx A q and thus

b n pxq ď ˆ2pn `1q n ˆ1 `1 A ˙´1 ˙n cotpxq ď ˆ1 `3 A ˙n cotpxq
for n large enough. Denote η n 1{ ? n and assume that n is sufficiently large so that η n ď x A . For x P rx A , π{2 Á{ ? ns, we have In conjunction with Proposition 6, this bound ends the proof of Theorem 5.

I n pxq ě ż x x´ηn sin n puq du ě 1 cospx ´ηn q ż x x´ηn cospuq sin n puq du " 1 cospx ´ηn q " sin n`1 puq n `1  x x´ηn " 1 cospx ´ηn q " sin n`1 pxq n `1 ´sin n`1 px ´ηn q n `1  " cospxq cospx ´ηn q « 1 ´ˆsinpx ´ηn q sinpxq ˙n`1 ff sin n`

Perturbative arguments for absorption

We present here general and very simple perturbative arguments for the expectation and the concentration of a hitting time.

Consider a diffusion on r0, πs of the form

dXptq " ? 2dBptq `1 ϕ 1 pXptqq dt (12)
where ϕ : r0, πs Ñ R `is twice continuously differentiable and increasing on r0, πs and such that 0 is an entrance boundary (insured by lim inf xÑ0 `x{ϕ 1 pxq ě 1), and where pBptqq tě0 is a standard Brownian motion. We start with X 0 " 0 and the above diffusion is defined up to the hitting time τ of π. By the above assumptions τ is a.s. finite and our first objective here is to give a simple upper bound of Erτ s in terms of ϕ. Thus integrating between 0 and τ , we get

ϕpX τ q ´ϕp0q " ż τ 0 ϕ 1 pXptqq dBptq `ż τ 0 1 `ϕ2 pXptqq dt (13)
Taking the expectation, we deduce These two results will be the unique insertion into the field of stochastic calculus needed to deduce Theorem 2. They will be reinforced by Lemmas 17 and 18 below to get Theorem 3.

ϕpπq ´ϕp0q " E "ż τ 0 1 `ϕ2 pXptqq dt  ě ˆ1 `min
We would like to apply them with ϕ 1 " 1{b n , but as we will see at the end of next section, this is not a good idea.

It is better to first slightly improve the bounds of Lemmas 15 and 16. Consider

Ψ `pϕq # ψ P C 2 pr0, πs, R `q : ψ 1 ě ϕ 1 ,
To evaluate the r.h.s. seems an interesting optimisation problem. We will not investigate it here in general, but we will see that for our particular problem it leads to the right equivalent (while only considering ψ " ϕ P Ψ does not).

Similarly, introduce

Ψ ´pϕq # ψ P C 2 pr0, πs, R `q : ψ 1 ď ϕ 1 ,
Both ( 15) and ( 16) will enable us to get the equivalent given in Theorem 2 for the expectation of the strong stationary time τ n , since we will exhibit appropriate avatars whose second derivatives will be smaller and smaller in terms of the parameter n.

By going a little further, it is possible to deduce the cut-off phenomenon of Theorem 3: instead of using that the expectation of a martingale is zero, as in Lemmas 15 and 16, we can evaluate its variance via its bracket. It leads to the following result for the hitting time τ of π by the diffusion (12) starting from 0.



Let us evaluate the last expectation as we have done for Erτ s. Denote γ the function on r0, πs satisfying γp0q " 0 and @ x P r0, πs, γ 1 pxq pϕ 1 pxqq 3 so that, taking into account that γ 2 " 3pϕ 1 q 2 ϕ 2 ,

pϕ 1 q 2 " γ 2 `γ1 {ϕ 1 ´3pϕ 1 q 2 ϕ 2 ď γ 2 `γ1 {ϕ 1 ´3 ˆmin r0,πs ϕ 2 ˙pϕ 1 q 2
It follows that

ˆ1 `3 ˆmin r0,πs ϕ 2 ˙˙E "ż τ 0 pϕ 1 pXpsqqq 2 ds  ď E "ż τ 0 rγ 2 `γ1 {ϕ 1 spXpsqq ds  " E " γpX τ q ´γpX 0 q ´ż τ 0 γ 1 pXpsqq dBpsq  " γpπq
The wanted result follows.

The same arguments show:

Lemma 18 Assume that ϕp0q " 0 and min r0,πs

ϕ 2 ą ´1{3
Then we have for any r ą 0,

P " τ ă ϕpπq 1 `max r0,πs ϕ 2 p1 ´rq  ď 1 r 2 ϕ 2 pπqp1 `3 min r0,πs ϕ 2 q ż π 0 pϕ 1 puqq 3 du
The comparison with diffusions of the form (14) leads to the following extensions of the two previous lemmas: for any ψ P Ψ `pϕq, such that min r0,πs ψ 2 ą ´1 3 , P " τ ą ψpπq ´ψp0q 1 `min r0,πs ψ 2 p1 `rq  ď 1 r 2 pψpπq ´ψp0qq 2 p1 `3 min r0,πs ψ 2 q ż π 0 pψ 1 puqq 3 du (17)

and for any ψ P Ψ ´pϕq, such that min r0,πs ψ 2 ą ´1 3 , P " τ ă ψpπq ´ψp0q 1 `max r0,πs ψ 2 p1 ´rq  ď 1 r 2 pψpπq ´ψp0qq 2 p1 `3 min r0,πs ψ 2 q ż π 0 pψ 1 puqq 3 du (18)

Construction of appropriate avatars

We come back to the diffusion defined in [START_REF] Coulibaly | On the evolution by duality of domains on manifolds[END_REF]. We would like to apply the bounds of the previous section with ϕ 1 n " 1{b n , for given n P N. It leads us to construct appropriate avatars ψ n P Ψ `pϕ n q and ψ n,´P Ψ ´pϕ n q, whose corresponding bounds will imply Theorems 2 and 3.

As suggested by the computations of Section 2, it is important to understand the behavior of b n at the scale 1{

? n: we fix A ą 0 and consider the change of variable x " π{2 `a{ ? n for a P r´A, As. Here is a first result about the mapping β defined in (8):

Lemma 19 There exists a unique a 0 P R such that β 1 pa 0 q " 0. Furthermore, we have a 0 ą 0.

Proof

We compute @ a P R,

β 1 paq " ´2 ae ´a2 {2 hpaq ´2 e ´a2 h 2 paq `1
Denote X e ´a2 {2 {hpaq, so that β 1 paq " 0 is equivalent to the equality 2aX `2X 2 ´1 " 0 Furthermore we compute

@ a P R, β 2 paq " ´2Xr1 ´a2 ´3aX ´2X 2 s
It follows that if a P R is such that β 1 paq " 0, then

β 2 paq " 2aXpa `Xq (19)
We examine separately two cases:

' If a ą 0, then β 2 paq ą 0, namely the critical point a is a local minimum. ' If a " 0, we verify directly that

β 1 p0q " ´2 1 h 2 p0q `1 " ´4 π `1 ă 0
' If a ă 0, let us show that a `X ą 0. Indeed, for u ă a ă 0, we have 1{u ą 1{a and thus hpaq "

ż a ´8 u u e ´u2 {2 du ă 1 a ż a ´8 ue ´u2 {2 du " ´1 a e ´a2 {2
(20)

implying a `X ą 0. We deduce from (19) that β 2 paq ă 0, i.e. the critical point a is a local maximum.

Since two different local minima (respectively maxima) are necessarily separated by a local maximum (resp. minimum), we deduce there is at most one point a in p0, `8q (resp. p´8, 0q) satisfying β 1 paq " 0.

Note that as a goes to `8 we have βpaq " a and that as a goes to ´8,

βpaq " ´a

This relation comes from the fact that ( 20) is well known to be an equivalent for hpaq as a Ñ ´8 (this is proven by an integration by parts). It follows that coming from ´8 and going to `8, β cannot have first a local maximum. Since β must have at least one local minimum, it appears finally that β has a unique critical point a 0 , which is a local minimum. We also infer that a 0 ą 0.

Fix ε 0 ą 0 sufficiently small so that the following quantities are finite for any ε P p0, ε 0 q: a `pεq infta ą a 0 : β 1 paq{β 2 paq " εu a ´pεq supta ă a 0 : β 1 paq{β 2 paq " ´εu (the existence of such an ε 0 ą 0 is a consequence of β 2 pa 0 q ą 0, as seen in the above proof). Consider the fonction f n given by

@ x P r0, πsztπ{2u, f n pxq | tanpxq| n
We have for large n and for any given a ‰ 0,

f

Proof

Fix any M ą 2βpa 0 q. Taking into account that lim εÑ0 `a`p εq " a 0 for ε ą 0 sufficiently small, we have 1 βpa `pεqq ´εpM ´a`p εqq ą 1 2βpa 0 q ą φpM q It follows there exists m P pM, 1{pa `pεqεq `a`p εqq such that We deduce |θ 1 paq| ď maxp1{m 2 ´pεq, 1{m 2 `pεq, εq. To conclude to the desired bound, note that at m `pεq, we have ´ε ď φ 1 pm `pεqq ď 0 since after m `pεq, φ is above the line of slope ´ε passing through φpm `pεqq. Thus we get 1{m 2 `pεq ď ε. Similarly we have 1{m 2 ´pεq ď ε and the announced result follows.

Let us check that for ε ą 0 small enough, θ remains above 1{β.

Lemma 22 There exists ε 1 P p0, ε 0 q such that for any ε P p0, ε 1 q, we have θ ě 1{β.

Proof

To simplify the notation, let us write q 1{β and let us work on ra 0 , `8q, similar arguments are valid on p´8, a 0 s.

For ε P p0, ε 0 q, define c `pεq min " m ą a `pεq : 1 βpa `pεqq ´εpm ´a`p εqq " φpmq * On ra 0 , `8q, it is clear from the definition of θ that θ ě q, except maybe on ra `pεq, c `pεqs (note that on pc `pεq, m `pεqq, θ ě φ ě q).

We have already seen that lim εÑ0 `a`p εq " a 0 and we have

lim εÑ0 `c`p εq " c `p0q (21)
where c `p0q " 1{qpa 0 q is the unique positive solution a of φpaq " qpa 0 q. We compute that @ a P R,

q 1 paq " 1 2 ´p1 `a2 2 qq 2 paq (22)
from which, we get @ a P R, q 2 paq " ´aq 2 paq ´2p1 `a2

2 qqpaqq 1 paq (23)
Thus we can find ε 2 ą 0 such that @ a P ra 0 , a 0 `ε2 s, q 2 paq ď q 2 pa 0 q 2 " ´a0 q 2 pa 0 q 2 ă 0

Let ε 3 ą 0 be such that for ε P p0, ε 3 q, we have a `pεq P pa 0 , a 0 `ε2 {2q. By the strict concavity of q on ra 0 , a 0 `ε2 s, the affinity of θ on ra `pεq, m `pεqs and the fact that θ 1 pa `pεqq " q 1 pa `pεqq, we deduce that for ε P p0, ε 3 q, @ a P ra `pεq, m `pεq ^pa 0 `ε2 qs, θpaq ě qpaq Furthermore, up to reducing ε 3 ą 0, we can assume that m `pεq ą a 0 `ε2 . It remains to consider the situation on the segment ra 0 `ε2 , c `pεqs.

Taking into account (21) and the fact that the slope of θ tends to zero as ε Ñ 0 `, to show that θ ě q on ra 0 `ε2 , c `pεqs (for ε P p0, ε 1 q for some ε 1 P p0, ε 3 q), it is sufficient to show that q 1 ă 0 on pa 0 , `8q.

By contradiction, assume there exists a 1 ą a 0 such that q 1 pa 1 q " 0. From (23), we deduce that q 2 pa 1 q " ´a1 q 2 pa 1 q ă 0 From the fact that q 1 pa 0 q " 0 and q 2 pa 0 q " ´a0 q 2 pa 0 q ă 0, there must exist a 2 P pa 0 , a 1 q with q 1 pa 2 q " 0 and q 2 pa 2 q ě 0. This is in contradiction with the fact that q 2 pa 2 q " ´a2 q 2 pa 2 q ă 0.

Fix ε P p0, ε 1 q and take A ą 0 large enough, so that ´A ă m ´pεq and A ą m `pεq. For n ě A 2 , define the mapping ξ n on r0, πs satisfying ξ n p0q " 0 and

@ x P p0, πq, ξ 1 n pxq # 1 ? n θ paq , if a P r´A, As f n pxq , otherwise (24)
(recall that a " ? npx ´π{2q). The function ξ n may not be strictly differentiable at π{2 ´A{ ? n and π{2 `A{ ? n (the above formulas giving the right derivative at ´A and the left derivative at A), nor twice differentiable at π{2 ´m´p εq{ ? n and π{2 `m`p εq{ ? n. But outside these four points, ξ n is twice differentiable. Convoluting ξ n with an approximation of the Dirac mass at 0 and taking into account Lemma 21, we construct an increasing function ψ n twice differentiable on p0, πq such that for n large enough,

b n ě p1 ´εq 1 ψ 1 n (25) sup p0,πq |ψ 2 n | ď εp1 `εq (26)
Furthermore, the computations of Lemma 11 show that for large n, ξ n pπq " lnpnq n thus for n large enough,

ψ n pπq ´ψn p0q ď p1 `εq lnpnq n (27)
Taking into account that for ε ą 0 small enough, we have for n large enough, ψ n,` ψ n {p1 ´ q P Ψ `pϕ n q, we deduce from (15)

lim sup nÑ8 n lnpnq Erτ n s ď 1 `ε 1 ´ε ´εp1 `εq
(where τ n is the strong stationary time defined in (1)) and letting ε go to zero, we conclude to the bound

lim sup nÑ8 n lnpnq Erτ n s ď 1 (28)
To get a reverse bound, it is sufficient to apply (16) with appropriate avatars ψ n,´P Ψ ´pϕ n q. Inspired by the computations of Section 2.2, we first take A ą 0 sufficiently large and consider the quantity pAq ą 0 defined there. Up to choosing A even larger, the above arguments are still valid, except that (25) and ( 27 In conjunction with (28), this ends the proof of Theorem 2.

To end this section, let us show Theorem 3. We begin by its first convergence, where r ą 0 is fixed from now on. For ε ą 0 sufficiently small, consider again the mapping ψ n,`P Ψ `pϕ n q defined above. According to (17), we have for any r ą 0, converging toward 0 for large n.

P " τ n ą ψ n,
Similarly we have

n 2 ln 2 pnq 1 n 2 ż A ´A θ 3 paq da " 1 ln 2 pnq ż A
´A θ 3 paq da converging toward 0 for large n and ending the proof of (31).

The proof of the second convergence of Theorem 3 follows a similar pattern, via (18) applied to ψ n,´P Ψ ´pϕ n q.

Indeed, r P p0, 1q being fixed, we first find A ą 0 sufficiently large and ε ą 0 sufficiently small so that for all large enough n, ψ n,´p πq ´ψn,´p 0q 1 `min r0,πs ψ 2 This bound implies the second convergence of Theorem 3 via the analogue of Lemma 23, where ψ n,`i s replaced by ψ n,´, and which is proven in exactly the same way.

We also deduce the following consequences from the proof of Lemma 23:

Corollary 24 For any x P S n`1 , let X x pX x t q tě0 be the Brownian motion on the sphere S n`1 (time-accelerated by a factor 2), starting with X x 0 " x. There exist C ą 0 and n 0 P N such that for all r ą 0 and for all n ě n 0 ,

› › › › L ˆXx p1`rq lnpnq n ˙´µ S n`1 › › › › tv ď C r 2 ln 2 pnq @ y P S n`1 , P pn`1q p1`rq lnpnq n px, yq ě ˆ1 ´C r 2 ln 2 pnq ˙1 volpS n`1 q
where } ¨}tv stands for the total variation norm, LpX x t q is the law of X x t , µ S n`1 is the uniform measure in S n`1 , and P pn`1q t p¨, ¨q is the heat kernel density at time t ą 0 associated to the Laplacian on S n`1 .

Proof

From the computations in the proof of Lemma 23, there exist a constant C depending on the quantity maxt ş A ´A θ 3 paq da, 1 A 2 u, and n 0 P N such that for all n ě n 0 ,

P " τ n ą p1 `rq lnpnq n  ď C r 2 ln 2 pnq
The first conclusion follows, since

› › › › L ˆXx p1`rq lnpnq n ˙´µ S n`1 › › › › tv ď s ˆL ˆXx p1`rq lnpnq n ˙, µ S n`1 ˙ď P " τ n ą p1 `rq lnpnq n 
The second conclusion follows by definition of the separation discrepancy, since for all y P S n`1 and t ą 0, 1 ´P pn`1q t px, yqvolpS n`1 q ď spLpX x t q, µ S n`1 q

At the other boundary π, we get ϕ We now restrict our attention to the case where x P p0, π{2q.

Lemma 27 We have @ x P p0, π{2q, N pxq ě 0

Proof

We compute that for any x P p0, πq, N 1 pxq " 2n sinpxqJ n pxq ´2n cospxqJ Ȃssume there exists some x 0 P p0, π{2q such that N px 0 q " 0, namely 2 ´2n cospx 0 qJ n px 0 q ´nJ 2 n px 0 q " 0 then we get

´1 `n cospx 0 qJ n px 0 q `pn `1qJ 2 n px 0 q " ´1 `n cospx 0 qJ n px 0 q `n `1 n p2 ´2n cospx 0 qJ n px 0 qq " n `2 n ´pn `2q cospx 0 qJ n px 0 q ě n `2 n ´n `2 n `1 ą 0
From this observation we get N 1 px 0 q ą 0 and in conjunction with the fact that N p0q " n `2 pn `1q 2 ą 0 we deduce that N remains non-negative on p0, π{2q.

It follows that min r0,π{2s ϕ 2 n ě 0 (34)

On rπ{2, πs

We study here the minimum of ϕ 2 n on rπ{2, πs. Let us change the notations of the previous section and rather write for x P p0, πq, As in Section 2, fix some A ą 0, and for n ą A 2 , we consider the parametrization x " π{2 `a{ ? n with a P r0, As. Taking into account Lemma 10, introduce the functions ν and δ defined on R `via @ a ě 0, # νpaq 2e ´a2 `2ae ´a2 {2 hpaq ´h2 paq δpaq 2e ´a2 {2 `ahpaq

We get for large n, uniformly over a P r0, As, N pxq " νpaq Dpxq " δpaq (except that if νpaq " 0 the first equivalence must be replaced by lim nÑ8 N pxq " 0).

Taking into account that our equivalences are up to a factor of the form 1 `Oa p1{ ? nq where the bounding factor in the Landau notation O a is uniform over a P r0, As, we deduce that uniformly over a P r0, As,

lim nÑ8 ϕ 2 n pxq " χpaq νpaq δ 2 paq
Here are the variation of the function χ:

Lemma 28 There exists a 0 ą 0 such that χ is decreasing on r0, a 0 s and increasing on ra 0 , `8q. This a 0 is the unique solution of p2 `a2 0 qe ´a2 0 `a0 p3 `a2 0 qe ´a2 0 {2 hpa 0 q ´h2 pa 0 q " 0 (35)

Proof

We have that for any a ą 0, χ 1 paq " ν 1 paqδpaq ´2νpaqδ 1 paq δ 3 paq and we want to show that there exists a unique a 0 ą 0 such that χ 1 pa 0 q " 0 and that furthermore χ 1 is negative on p0, a 0 q and positive on pa 0 , `8q. We compute Our goal amounts to find a unique a 0 ą 0 such that ξpa 0 q " 0 and that furthermore ξ is positive on p0, a 0 q and negative on pa 0 , `8q. Let us differentiate: for any a ą 0, implying in particular the first statement of Proposition 25.

ν
ξ

Lemma 9

 9 For large n, we have

r0,πs ϕ 2

 2 ˙Erτ s which implies the desired bound. The above arguments equally lead to a reverse bound: Lemma 16 Assume that max r0,πs ϕ 2 ą ´1 Then we have Erτ s ě ϕpπq ´ϕp0q 1 `max r0,πs ϕ 2

Lemma 17 τ ą ϕpπq 1 `

 171 Assume that ϕp0q " 0 and min r0,πs ϕ 2 ą ´1{3 Then we have for any r ą 0, P " min r0,πs ϕ 2 p1 `rq  ď 1 r 2 ϕ 2 pπqp1 `3 min r0,πs ϕ 2 q

n pxq " φpaq ? n with φpaq 1 |a|Figure 1 :

 11 Figure 1: The mappings φ and 1{β are respectively in blue and red. The half-tangents with slope ´ε and ε are in green.

2

 2 pψ n,´p πq ´ψn,´p 0qq 2 p1 `3 min r0,πs ψ 2 n,´q ż π 0 pψ 1 n,´p uqq 3 du

ϕ 2 n pxq " N pxq D 2

 2 pxq where N pxq 2 sin 2pn`1q pxq ´2n cospxq sin n`1 pxqI n pxq ´nI 2 n pxq Dpxq 2 sin n`1 pxq ´n cospxqI n pxq

  e.

		´d dt	lnpcospr x t qq " n
	Integrating between 0 and r T n we get p1q
	n r T p1q n	" lnpcosp0qq ´ln ˆcos	ˆπ 2	´A ? n	˙"
		´ln ˆcos	ˆπ 2	´A ? n	˙Ḟor
	large n, we have
		cos	ˆπ 2	´A ? n	˙" A ? n
	and it follows that
	´ln ˆcos	ˆπ 2	´A ? n	˙˙"	lnpnq 2
	and by consequence
				r T p1q n	"	lnpnq 2n
	' For t P p r T n , r p1q T n `r T p1q n q, (9) writes p2q
					9 r x t " r c ?	n
	and we get
		r T p2q n

  .h.s. converges toward e ´pA`1q 2 {2 e A 2 {2 " e ´pA`1{2q for large n. It follows that for n sufficiently large,

	We also have
	max	"ˆs inpx ´ηn q sinpxq	˙n : x P px A , π{2 ´A{ ?	nq *	"	ˆsinpπ{2 ´A{ sinpπ{2 ´A{ ? n ´ηn q nq ?	˙n
	and the rI n pxq ě	A	A `2 p1 ´e´A q	sin n`1 pxq pn `1q cospxq
	and we deduce that for x P rx A , π{2 ´A{	? nqs,
					b n pxq ď	ˆ2 A Ap1 ´e´A q `2	´1˙n	cotpxq
								" p1 ` ´pAqqn cotpxq
	with
								´pAq	2	2 `Ae
							#	p c A	?	n	, if x P rπ{2 ´A{ ?	n, π{2 `A{ ?	ns
								p1 ` pAqqn| cotpxq| , otherwise
	satisfying
									b n ď p b n
	Replacing (4) by
								"	p x 0 " 0 9 p x t " p b n pp x t q	(11)
	defined up to the time p T n it hits π, we get
								@ n P N,	T n ě p T n
	The proof of Lemma 11 shows
								lim nÑ8	n lnpnq	p T n " 1 ` pAq
	We deduce that for any A ą 0,	lim inf nÑ8	n lnpnq	T	1 pxq pn `1q cospxq
	Note that
		min	"	cospxq cospx ´ηn q	: x P px A , π{2 ´A{	? nq	*	"	cospπ{2 ´A{ ? cospπ{2 ´A{ ? n ´ηn q nq

and the r.h.s. converges toward A{pA `1q for large n.

´A

Ap1 ´e´A q

The wanted bound follows with pAq ´pAq _ `pAq, satisfying

[START_REF] Saloff-Coste | Precise estimates on the rate at which certain diffusions tend to equilibrium[END_REF]

. The two previous upper bounds on b n lead us to introduce a new function p b n on p0, πq via @ x P p0, πq, p b n pxq n ě 1 ` pAq and letting A go to `8, we deduce lim inf nÑ8 n lnpnq T n ě 1

  minFor any ψ P Ψ `pϕq, which should be seen as an avatar of ϕ, consider the diffusion starting with Y p0q " 0 and satisfying The definition of Ψ insures that 0 is an entrance boundary and that τ ď σ. We deduce the upper bound

							+
				r0,πs	ψ 2 ą ´1 and lim sup xÑ0 `ψ1 pxq{x ď 1
	dY ptq "	?	2dBptq	`1 ψ 1 pY ptqq	dt	(14)
	up to the hitting time σ of π.
	Erτ s ď	ψpπq ´ψp0q 1 `min r0,πs ψ 2
	and finally
	Erτ s ď	inf ψPΨ `pϕq	ψpπq ´ψp0q 1 `min r0,πs ψ 2

  pψ n,`p πq ´ψn,`p 0qq 2 p1 `3 min r0,πs ψ 2

	and thus
			n 2 ln 2 pnq	1 n 3	ż p0,πqzrπ{2´A{ ? n,π{2`A{ ?	ns	| tanpuq| 3 du ď	1 pA lnpnqq 2
		`p πq ´ψn,`p 0q 1 `min r0,πs ψ 2 n,`p 1 `r{2q		ď	4 r 2 pψ n,`p πq ´ψn,`p 0qq 2 p1 `3 min r0,πs ψ 2 n,`q	ż π 0	pψ 1 n,`p uqq 3 du
	Up to choosing ε ą 0 even smaller, (26) and (27) insure that for all n sufficiently large, we have
						ψ n,`p πq ´ψn,`p 0q 1 `min r0,πs ψ 2 n,`p 1 `r{2q ă p1 `rq	lnpnq n
	implying
	P	"	τ n ą p1 `rq	lnpnq n		ď	4 r 2 n,`q	ż π 0	pψ 1 n,`p uqq 3 du
	Thus the first convergence of Theorem 3 is a consequence of (29) and
	Lemma 23 We have
									lim nÑ8	n 2 ln 2 pnq	ż π 0	pψ 1 n,`p uqq 3 du " 0
	Proof
	The above convergence is equivalent to
									lim nÑ8	n 2 ln 2 pnq	ż π 0	pψ 1 n puqq 3 du " 0	(30)
	Since differentiation and convolution commute and convolution is a contraction in L p , for p ě 1
	(recall that ψ 1 n ą 0), (30) is itself implied by
									lim nÑ8	n 2 ln 2 pnq	ż π 0	pξ 1 n puqq 3 du " 0	(31)
	Coming back to Definition (24), we write
	ż π				ż
	0	pξ 1 n puqq 3 du "	p0,πqzrπ{2´A{ ? n,π{2`A{	? ns	pξ 1 n puqq 3 du	`żrπ{2´A{ ? n,π{2`A{	ns ?	pξ 1 n puqq 3 du
			"	1 n 3		ż p0,πqzrπ{2´A{	? n,π{2`A{ ? ns	| tanpuq| 3 du	`1 n 2	ż A ´A θ 3 paq da
	Note that the first term of the r.h.s. si equal to
					2 n 3		ż π{2´A{ 0	? n	tan 3 puq du "	2 n 3	ż π{2 A{ ? n	cot 3 puq du
										ď	2 n 3	ż π{2 A{ ? n	1 u 3 du
										"	1 n 3	"	´1 u 2	 π{2 A{ ? n
										ď	1 n 3	n A 2 "	1 pAnq 2

  pxq `cospxqI n pxq 2 sin n`1 pxq ´n cospxqI n pxq ´sinpxqI n pxqr2pn `1q cospxq sin n pxq `n sinpxqI n pxq ´n cospxq sin n pxqs p2 sin n`1 pxq ´n cospxqI n pxqq 2 " 2 sin 2pn`1q pxq ´2n cospxq sin n`1 pxqI n pxq ´nI 2 n pxq p2 sin n`1 pxq ´n cospxqI n pxqq 2

		1 n pxq "	´In pπq sinpxq nI n pπq
			Ñ 0
	We compute for any x P p0, πq,
	ϕ 2 n pxq " sin n`1 Let us rewrite ϕ 2 n pxq " N pxq{D 2 pxq with
	N pxq	2 ´2n cospxqJ n pxq ´nJ 2 n pxq
	Dpxq	2 ´n cospxqJ n pxq
	with
			J n pxq		I n pxq sin n`1 pxq
	Here are some observations on this function
	Lemma 26 We have
	J 1 n pxq "	1 sinpxq	p1 ´pn `1q cospxqJ n pxqq
	and in particular J n is increasing on p0, πq.
	Proof
	Indeed, we compute
	J 1 n pxq "	sin n pxq sin n`1 pxq ´pn `1q cospxq sin n pxqI n pxq sin 2pn`1q pxq
	"	1 sinpxq	p1 ´pn `1q cospxqJ n pxqq
	From inequality (32), we get that for any x P p0, πq, J 1 n pxq ě 0 on p0, πq.
	Since the first bound in (32) is an equivalent for small x, we also get
			lim xÑ0 `Jn pxq "	n	1 `1
	and thus
	lim xÑ0 `N pxq " 2	´2n n `1	´n	1 pn `1q 2
			"	2pn `1q 2 ´2npn `1q pn `1q 2	´n
			"	n pn `1q 2 `2

  1 paq " ap1 ´a2 qe It appears that ξ is increasing on p0, 1q and decreasing on p1, `8q. Since we deduce the desired result on ξ.ProofFix A ą a 0 and for n ą A 2 , consider x n such that cospx n q " ´A{ ? n. For any x P rx n , πs, we have From Lemma 28 and since A ą a 0 , we have By choosing furthermore A ą a 0 such that 1{A 2 ă |χpa 0 q|, we deduce

					ξp0q " 2	´c π 2	ą 0
		lim aÑ`8	ξpaq " ´2π ă 0
	Note that
					lim aÑ`8	χpaq " 0
	so that χpa 0 q ă 0. As a consequence we get
	Proposition 29 We have
		lim nÑ8	inf xPrπ{2,πs	ϕ 2 n pxq " χpa 0 q
		ϕ 2 n pxq ě	´nI 2 n pxq pn cospxqI n pxqq 2
						"	´1 n cos 2 pxq
						ě	´1 n cos 2 px n q
						"	´1 A 2
	We get
		inf xPrxn,πs	ϕ 2 n pxq ě	´1 A 2
		lim nÑ8	inf xPrπ{2,xns	ϕ 2 n pxq " χpa 0 q	(37)
	inf xPrπ{2,πs	ϕ 2 n pxq "	inf xPrπ{2,xns	ϕ 2 n pxq ^inf xPrxn,πs	ϕ 2 n pxq
					"	inf xPrπ{2,xns	ϕ 2 n pxq
	for n large enough. The announced result now follows from (37).
	Taking into account (34), we get
		lim nÑ8	inf xPr0,πs	ϕ 2 n pxq " χpa 0 q
						´a2	`p1 ´a4 qe ´a2 {2 hpaq

Funding from the grant ANR-17-EURE-0010 is aknowledged.

5 On the necessity of avatars

Our goal here is to see the bound given in (15) can be strictly better than Lemma 15.

Indeed, it will a consequence of the following result. For any n P N, consider the function ϕ n defined on r0, πs satisfying ϕ n p0q " 0 and

sinpxqI n pxq 2 sin n`1 pxq ´n cospxqI n pxq From the computations of Section 2, we have for large n, ϕ n pπq "

lnpnq n

But we have:

Proposition 25 The limit lim nÑ8 min r0,πs ϕ 2 n exists and its value belongs to r´5{11, ´1{7s.

Thus Lemma 15 alone would not have permitted us to prove Theorem 2.

On the two following subsections, we respectively investigate ϕ 2 n on r0, π{2s and rπ{2, πs.

On r0, π{2s

Before investigating the minimum of ϕ 2 n on r0, π{2s, we start with some considerations valid on r0, πs. For any x P r0, πs, we have

We deduce that the denominator of ϕ 1 n pxq satisfies

which stays positive on p0, πq. Furthermore, as x tends to 0 `, we have

Let us show its last statement. Extracting h 2 pa 0 q from (35):

and replacing first in νpa 0 q: νpa 0 q " 2e ´a2 0 `2a 0 e ´a2 0 {2 hpa 0 q ´h2 pa 0 q " 2e ´a2 0 `2a 0 e ´a2 0 {2 hpa 0 q ´rp2 `a2 0 qe ´a2 0 `a0 p3 `a2 0 qe ´a2 0 {2 hpa 0 qs " ´a2 0 e ´a2 0 ´a0 p1 `a2 0 qe ´a2 0 {2 hpa 0 q and next in δ 2 pa 0 q:

Recalling (36), it is immediate to compute that ξp1q ą 0 while ξp2q ă 0, implying that a 0 P p1, 2q. It follows that



Remarking that the mapping a Þ Ñ a 2 {p4 `2a 2 `a4 q is increasing on r1, ? 2s and decreasing on r ?

2, 2s, we furthermore get



The wanted bounds follow: