William Oçafrain

Convergence to quasi-stationarity through Poincaré inequalities and Bakry-Émery criteria

Keywords: Absorbed Markov processes, quasi-stationary distribution, Poincaré inequality, Bakry-Émery condition, 1-Wasserstein distance, multi-dimensional diffusion processes. 2010 Mathematics Subject Classification. 60B10, 60F99, 60J25, 60J50, 39B62, 60J60

This paper aims to provide some tools coming from functional inequalities to deal with quasistationarity for absorbed Markov processes. First, it is shown how a Poincaré inequality related to a suitable Doob transform entails exponential convergence of conditioned distributions to a quasi-stationary distribution in total variation and in 1-Wasserstein distance. A special attention is paid to multi-dimensional diffusion processes, for which the aforementioned Poincaré inequality is implied by an easier-to-check Bakry-Émery condition depending on the right eigenvector for the sub-Markovian generator, which is not always known. Under additional assumptions on the potential, it is possible to bypass this lack of knowledge showing that exponential quasi-ergodicity is entailed by the classical Bakry-Émery condition.

Notation

For a general metric space (F, d):

• M 1 (F ) : Set of the probability measures defined on F .

• P p (F ) : Set of the probability measures defined on F such that F d(x 0 , x) p µ(dx) < +∞, where x 0 ∈ F is arbitrary.

• B(F ) : Set of the measurable bounded functions defined on F .

• B 1 (F ) : Set of the measurable bounded functions defined on F such that ||f || ∞ ≤ 1.

• L 2 (µ) : Set of the functions such that F |f | 2 dµ < +∞, endowed with the norm

• L 2 (µ) : f → F |f | 2 dµ.
• For any µ ∈ M 1 (F ) and f ∈ B(F ), µ(f ) := F f (x)µ(dx).

• For two probability measures µ and ν, the notation µ ν means that there exists a density function f such that

µ(•) = • f (x)ν(dx),
and this density function will be denoted by dµ dν . • For any positive measure µ and any measurable function f such that µ(f ) < +∞, denote f • µ the probability measure defined by

f • µ(dx) := f (x)µ(dx) µ(f ) . ( 1
)
1 Introduction

Consider a time-homogeneous Markov process (X t ) t≥0 defined on a metric state space (E ∪ {∂}, d), where the element ∂ ∈ E is a cemetery point for the process X, which means that

X t = ∂, ∀t ≥ τ ∂ ,
where τ ∂ := inf{t ≥ 0 : X t = ∂} is the hitting time of ∂. We associate to the process (X t ) t≥0 a family of probability measures (P x ) x∈E such that, for any x ∈ E, P x (X 0 = x) = 1. For any µ ∈ M 1 (E ∪{∂}), denote P µ := E P x µ(dx). Then, under P µ , the law of X 0 is µ. Finally, the expectations E x and E µ are respectively associated to P x and P µ . Moreover, assume that, for any x ∈ E, P x [τ ∂ < +∞] = 1, and P x [τ ∂ > t] > 0, ∀t ≥ 0.

A natural notion to study considering absorbed Markov processes is the notion of quasi-stationarity, dealing with the weak convergence of the probability measures

P µ (X t ∈ •|τ ∂ > t)
when t goes to infinity. It is well-known that, if such a convergence holds for a given initial law µ, then the limiting probability measure α satisfies

P α (X t ∈ •|τ ∂ > t) = α, ∀t ≥ 0.
Such a probability measure is called a quasi-stationary distribution and can be understood as an invariant measure for the semi-flow (φ t ) t≥0 defined by

φ t : M 1 (E) → M 1 (E) µ → P µ (X t ∈ •|τ ∂ > t), ∀t ≥ 0.
For a general overview on this theory, we refer the reader to [START_REF] Collet | Quasi-stationary distributions[END_REF][START_REF] Méléard | Quasi-stationary distributions and population processes[END_REF][START_REF] Van Doorn | Quasi-stationary distributions for discrete-state models[END_REF], where it is shown that, defining the sub-Markovian semi-group (P t ) t≥0 as

P t f (x) := E x (f (X t )1 τ ∂ >t ), ∀t ≥ 0, ∀f ∈ B(E), ∀x ∈ E, (2)
α ∈ M 1 (E) is a quasi-stationary distribution if and only if there exists λ 0 > 0 such that αP t := P α (X t ∈ •, τ ∂ > t) = e -λ 0 t α, ∀t ≥ 0.

In other words, quasi-stationary distributions are left eigenvectors for the operators P t , associated to the eigenvalues e -λ 0 t . Hence, quasi-stationarity can be dealt with through spectral methods, and existence and uniqueness of quasi-stationary distributions has been shown by this way for several processes, such as discrete-time Markov chains [START_REF] Darroch | On quasi-stationary distributions in absorbing discrete-time finite Markov chains[END_REF][START_REF] Seneta | On quasi-stationary distributions in discrete-time Markov chains with a denumerable infinity of states[END_REF], birth-death processes [START_REF] James | Quasi-stationary distributions of birth-and-death processes[END_REF][START_REF] Kijima | Some results for quasi-stationary distributions of birth-death processes[END_REF][START_REF] Van Doorn | Quasi-stationary distributions and convergence to quasi-stationarity of birth-death processes[END_REF] and diffusion processes [START_REF] Cattiaux | Quasi-stationary distributions and diffusion models in population dynamics[END_REF][START_REF] Kolb | Quasilimiting behavior for one-dimensional diffusions with killing[END_REF][START_REF] Littin | Uniqueness of quasistationary distributions and discrete spectra when ∞ is an entrance boundary and 0 is singular[END_REF][START_REF] Martínez | Quasi-stationary distributions for a Brownian motion with drift and associated limit laws[END_REF][START_REF] Steinsaltz | Quasistationary distributions for one-dimensional diffusions with killing[END_REF]. More recently, other methods were developed in order to study quasi-stationarity. These methods aim to obtain exponential convergence towards quasi-stationary distributions for some processes and are based on well-known probabilistic tools coming from the framework without absorption, such as Doeblin's condition or Lyapunov functions (see [START_REF] Sean | Markov chains and stochastic stability[END_REF] for an overview on these tools). In particular, in [START_REF] Champagnat | Exponential convergence to quasi-stationary distribution and Q-process[END_REF], necessary and sufficient conditions for the uniform-in-law exponential convergence in total variation are provided, where we recall that the total variation distance of two probability measures µ, ν is defined by µ -ν T V := sup

f ∈B 1 (E) |µ(f ) -ν(f )|.
Since, other papers showed exponential convergences in total variation under weaker assumptions, allowing convergences in total variation holding non-uniformly in the initial measure. In particular, we refer the reader to [START_REF] Champagnat | General criteria for the study of quasi-stationarity[END_REF][START_REF] Velleret | Unique quasi-stationary distribution, with a possibly stabilizing extinction[END_REF] for the study of absorbed Markov processes, and [START_REF] Bansaye | A non-conservative Harris' ergodic theorem[END_REF][START_REF] Champagnat | Practical criteria for R-positive recurrence of unbounded semigroups[END_REF][START_REF] Ferré | More on the long time stability of Feynman-Kac semigroups[END_REF] for the study of general renormalized Feynman-Kac semi-groups.

For non-absorbed Markov processes, the rate of convergence towards invariant measures can also be studied through functional inequalities, such as Poincaré inequalities. A probability measure π is said to satisfy a Poincaré inequality if there exists a constant C > 0 such that, for any f ∈ D(E),

Var π (f ) ≤ -C E f Lf dπ, (3)
where Var π (f ) := E (f -π(f )) 2 dπ, L is a generator which cancels π, and D(E) is the set of the measurable functions such that E(f, f ) := -E f Lf dπ is well-defined. We refer the reader to [START_REF] Bakry | Analysis and geometry of Markov diffusion operators[END_REF][START_REF] Royer | An initiation to logarithmic Sobolev inequalities[END_REF] to go further about Poincaré inequalities. The inequality [START_REF] Bakry | Analysis and geometry of Markov diffusion operators[END_REF] is actually equivalent to the exponential decay of the χ 2 -divergence between the semi-group µe tL and π, the χ 2 -divergence being defined as follows :

χ 2 (µ|ν) :=    E dµ dν -1 2 dν if µ ν +∞ otherwise.
In particular, this implies an exponential decay of the total variation distance between µe tL and π when the quantity χ 2 (µ|π) is finite.

In the literature, some papers dealing with the use of Poincaré inequalities for quasi-stationarity have been already written, in particular for Markov processes living on discrete state spaces ( [START_REF] Chazottes | Sharp asymptotics for the quasi-stationary distribution of birth-and-death processes[END_REF][START_REF] Diaconis | Analytic-geometric methods for finite Markov chains with applications to quasi-stationarity[END_REF][START_REF] Diaconis | On quantitative convergence to quasi-stationarity[END_REF]). However, the proofs provided by these papers strongly rely on the discrete aspect of the state space, and are therefore hardly applicable for processes living on continuous state space, such as diffusions processes. Our aim will be therefore to show how to use such inequalities to get exponential convergence towards quasi-stationarity for such processes. In particular, the convergence in total variation will be studied, as well as the convergence in 1-Wasserstein distance, which is defined as

W 1 (µ, ν) := inf (X,Y )∈Π(µ,ν) E[d(X, Y )], ∀µ, ν ∈ P 1 (E),
where Π(µ, ν) is the set of all the couplings (X, Y ) such that the law of X (respectively Y ) is µ (respectively ν). We refer to Theorem 3 in Section 2 for the general statement and Corollary 1 for the convergence in 1-Wasserstein distance.

In the third and last section, we will be more particularly interested in quasi-stationarity for diffusion processes (X t ) t≥0 living on a domain D ⊂ R d , absorbed at the boundary ∂D, and satisfying on D the stochastic differential equation

dX t = √ 2dB t -∇V (X t )dt, X t ∈ D, (4)
where (B t ) t≥0 is a d-dimensional Brownian motion and V is a C 2 -function on R d . In the non-absorbed framework, it is well-known that the reversible probability measure

γ(dx) := Z -1 e -V (x) dx
(Z is the renormalization constant) satisfies a Poincaré inequality when the condition

Hess V ≥ κId (5)
is satisfied for a given κ > 0. This last result is a consequence from the one shown by Bakry and Émery in [START_REF] Bakry | Diffusions hypercontractives[END_REF] and the condition ( 5) is usually called the Bakry-Émery condition or curvaturedimension condition. In particular, under (5), the diffusion process (X t ) t≥0 converges towards γ in total variation and in 1-Wasserstein distance. Our goal is therefore to recover this property of convergence in the quasi-stationary framework through a condition similar to [START_REF] Bansaye | Diffusions from infinity[END_REF]. More precisely, the following result is obtained in Section 3 :

Theorem 1.

• Assume that there exists η positive on D, vanishing on ∂D such that γ(η 2 ) < +∞ and there exists λ 0 > 0 such that

∆η(x) -∇V (x) • ∇f (x) = -λ 0 η(x), ∀x ∈ D,
• and assume that there exists κ > 0 such that Hess[V -2 log(η)] ≥ κId.

Then there exists C > 0 such that, for any µ ∈ M 1 (D) and t ≥ 0,

P µ [X t ∈ •|τ ∂ > t] -η • γ T V ≤ Cχ 2 (η • µ|η 2 • γ)e -κt , (6)
where we recall that the notation f • µ is defined previously in (1) in Notation. Moreover, if D (1 + |x|) 2 e -V (x) dx < +∞, the inequality (6) holds in 1-Wasserstein distance for t large enough.

A more specific study will focus on multi-dimensional diffusion processes living on D = (0, +∞) d and absorbed when one component is 0. In this particular case, and assuming moreover that V can be expressed as

V (x 1 , . . . , x d ) = d i=1 V i (x i ), ∀ (x 1 , . . . , x d ) ∈ D,
where, for all i, V i are C 2 -functions, one has the following result :

Theorem 2. If Hess V ≥ κId,
then there exists a quasi-stationary distribution

α = α 1 ⊗ • • • ⊗ α d ∈ M 1 (D)
and C d > 0 (depending on the dimension d) such that, for any µ ∈ M 1 (D) and t large enough,

W 1 (P µ [X t ∈ •|τ ∂ > t], α) ≤ C d χ 2 (η • µ|η • α)e -κt ,
where η := dα dγ . If moreover µ = µ 1 ⊗ • • • ⊗ µ d , there exists a constant C > 0, which does not depend on d, such that, for t large enough,

W 1 (P µ [X t ∈ •|τ ∂ > t], α) ≤ C d i=1 χ 2 (η i • µ i |η i • α i ) e -κt ,
where η i := dα i dγ i .
This theorem is further referenced as Theorem 6 in Subsection 3.3.3. A particular attention will be paid on processes coming down from infinity, for which it will be shown that the rate of convergence κ provided by the Bakry-Émery condition (5) can actually be bettered (see Theorems 5 and 8).

2 Exponential convergence to quasi-stationarity through a Poincaré inequality

Main result

Let (X t ) t≥0 be a Markov process absorbed at a cemetery point ∂, and let (P t ) t≥0 be the sub-Markovian semi-group defined in [START_REF] Bakry | Diffusions hypercontractives[END_REF]. Denote by

D(L) := f : lim t→0 P t f (x) -f (x) t exists for any x , (7)
and define the generator L as

Lf (x) := lim t→0 P t f (x) -f (x) t , ∀x ∈ E, ∀f ∈ D(L). (8)
Now, let us state the following theorem :

Theorem 3. Assume that (P1) there exists a quasi-stationary distribution α ∈ M 1 (E) satisfying αP t = e -λ 0 t α, ∀t ≥ 0, with λ 0 > 0, and an eigenfunction η positive on E such that α(η) = 1 and

P t η(x) = e -λ 0 t η(x), ∀x ∈ E, ∀t ≥ 0; (P2) there exists C P ∈ (0, +∞) such that Var η•α (f ) ≤ -C P E f (λ 0 f η + L(f η))dα , (9)
for any measurable function f such that

λ 0 E f 2 d(η • α) + E f L(f η)dα is well-defined;
(P3) and there exists a function ψ : E → [1, +∞) such that α(ψ) < +∞, and α(ψ 2 /η) < +∞.

Then, for any µ ∈ M 1 (E), there exists t µ such that, for any t ≥ t µ ,

sup |f |≤ψ |E µ [f (X t )|τ ∂ > t] -α(f )| ≤ C ψ χ 2 (η • µ|η • α)e -t C P , (10)
where

C ψ = (a + bα(ψ))α(ψ 2 /η) 1/2 ,
with some positive constants a, b.

Before proving this theorem, let us do some important remarks:

Remark 1. (P1) is satisfied under the Lyapunov conditions (E) and (F) presented in [START_REF] Champagnat | General criteria for the study of quasi-stationarity[END_REF], or under the conditional Doeblin's conditions (A) in [START_REF] Champagnat | Exponential convergence to quasi-stationary distribution and Q-process[END_REF]. In particular, the results presented further in Subsection 3.3, dealing with quasi-stationarity for processes following stochastic differential equations like

dX t = √ 2dB t -∇V (X t )dt,
rely a lot on these assumptions, allowing to state easy-to-check criteria only based on the potential V (such as [START_REF] Kijima | Some results for quasi-stationary distributions of birth-death processes[END_REF] in Remark 8). For other criteria related to other types of processes (such as diffusion processes with multiplicative noise, birth-and-death processes, Markov chains,...), we refer the reader to [START_REF] Champagnat | General criteria for the study of quasi-stationarity[END_REF]. It is possible to state even less restrictive assumptions than [10, Assumptions (F)], also based on Lyapunov functions, entailing the existence of a quasi-stationary distribution (see for example [START_REF] Collet | Quasi-stationary distributions for structured birth and death processes with mutations[END_REF]Theorem 4.2.] or [START_REF] Benaïm | Degenerate processes killed at the boundary of a domain[END_REF]Theorem 7]), but without ensuring the existence of an eigenfunction η. To obtain such an eigenfunction, it is quite usual to use Krein-Rutman's theorem, once the compacity of the operators (P t ) t≥0 or related operators is known. Finally, note that the existence of α and η does not ensure in general that α(η) < +∞. For example, the one-dimensional Brownian motion with drift X t = B t -rt (r > 0) absorbed at 0 admits α(dx) = r 2 xe -rx dx and η(x) = xe rx , so that α(η) = +∞.

Remark 2. It is also important to note that the Poincaré constant C P fundamentally depends on the survival state space E. We refer the reader to the subsection 3.2, in particular the example of a Brownian motion living in the hypercube C N : (-N, N ) d ⊂ R d , for which the Poincaré constant increases as N 2 when N increases. Remark 3. For several processes, it is quite usual to have χ 2 (η • µ|η • α) = +∞ when the initial law is a Dirac measure δ x . In the most of the cases (see for instance the two examples provided in Subsection 3.2), considering a state x ∈ E, there exists a time t 0 > 0 such that

χ 2 (η • φ t 0 (δ x )|η • α) < +∞.
Hence, using the property of semi-flow of (φ t ) t≥0 (i.e. φ t+s = φ t • φ s for all s, t ≥ 0), the previous theorem implies that there exists t φt 0 (δx) such that, for any t ≥ t 0 + t φt 0 (δx) , sup

|f |≤ψ |E x [f (X t )|τ ∂ > t] -α(f )| = sup |f |≤ψ E φt 0 (δx) [f (X t-t 0 )|τ ∂ > t -t 0 ] -α(f ) ≤ C ψ χ 2 (η • φ t 0 (δ x )|η • α)e - (t-t 0 ) C P .
In other terms, the set of all the measures such that there exists t 0 ≥ 0 such that χ 2 (η •φ t 0 (µ)|η •α) < +∞ is included in the domain of attraction of α, denoted by D(α), that is the set of the initial measures such that the weak convergence P µ [X t ∈ •|τ ∂ > t] -→ t→∞ α holds. We refer the reader to Subsection 2.2 for a deepening on the study of domain of attraction.

Remark 4. Because of the condition ψ ≥ 1, the distance sup |f |≤ψ |E µ [f (X t )|τ ∂ > t] -α(f )|
is actually stronger than the total variation distance. In particular, Theorem 3 implies that there exists C > 0 such that, for any µ ∈ M 1 (E) and t ≥ 0,

P µ [X t ∈ •|τ ∂ > t] -α T V ≤ Cχ 2 (η • µ|η • α)e -t C P .
Hence, this theorem allows to obtain a result analogous to the ones obtained by Champagnat and Villemonais in [START_REF] Champagnat | Exponential convergence to quasi-stationary distribution and Q-process[END_REF][START_REF] Champagnat | General criteria for the study of quasi-stationarity[END_REF]. However, contrary to their results, the upper bound could be small if the initial measure is close enough to the quasi-stationary distribution α (it is even equal to 0 for µ = α). Moreover, Theorem 3 allows to obtain a convergence in 1-Wasserstein distance, as stated by the following corollary : Corollary 1. If the assumptions (P 3) holds for

ψ : x → 1 + d(x, x 0 )
for a given x 0 ∈ E, then, for any µ ∈ M 1 (E), there exists t µ such that, for any t ≥ t µ ,

W 1 (P µ [X t ∈ •|τ ∂ > t], α) ≤ C ψ χ 2 (η • µ|η • α)e -t C P .
Proof. By the dual formula for the 1-Wasserstein distance (for example see [START_REF] Villani | Optimal transport, volume 338 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences[END_REF]), for any probability measures µ and ν in P 1 (E), one has

W 1 (µ, ν) = sup f 1-Lip |µ(f ) -ν(f )| = sup f ∈C |µ(f ) -ν(f )|, (11)
where

C := {f 1 -Lip : f (x 0 ) = 1}.
Thus, any function f belonging to C satisfies

|f (x)| ≤ 1 + d(x, x 0 ) = ψ(x), ∀x ∈ E.
Hence, by Theorem 3, there exists t µ such that, for any t ≥ t µ ,

W 1 (P µ [X t ∈ •|τ ∂ > t], α) ≤ sup |f |≤ψ |E µ [f (X t )|τ ∂ > t] -α(f )| ≤ C ψ χ 2 (η • µ|η • α)e -t C P .
Now, let us tackle the proof of Theorem 3 : 10) is trivially satisfied. So, from now on, we will only consider initial measure such that

Proof of Theorem 3. First, remark that if χ 2 (η • µ|η • α) = +∞, the inequality (
χ 2 (η • µ|η • α) < +∞.
The proof is divided into two steps.

First step: When α(ψ 2 /η)χ 2 2 (η • µ|η • α) < 0.9 1 . Let µ ∈ M 1 (E) satisfying α(ψ 2 /η)χ 2 2 (η • µ|η • α) < 0.9. Denote by ( Pt ) t≥0 the Markovian semi-group defined by Pt f (x) := e λ 0 t P t [f η](x) η(x) ,
where we recall that λ 0 and η are such that, for any x ∈ E and t ≥ 0,

P t η(x) = e -λ 0 t η(x).
Then, since α is a quasi-stationary distribution for (X t ) t≥0 , the probability measure β(dx) := η(x)α(dx) is an invariant measure for ( Pt ) t≥0 . Moreover, denoting by L the generator of ( Pt ) t≥0 , then, for any measurable f such that f η ∈ D(L) and for any x ∈ E,

Lf (x) = λ 0 f (x) + L(f η)(x) η(x) .
This equality comes from the equality Lf (x) := d Ptf (x) dt t=0 . Then, the Poincaré inequality ( 9) can be written as follows :

Var β (f ) ≤ -C P E f Lf dβ.
In other words, the inequality ( 9) is the Poincaré inequality for the Markovian semi-group ( Pt ) t≥0 . Then it is well-known that it is equivalent to : for any probability measure ν on E and t ≥ 0,

χ 2 2 (ν Pt |β) ≤ e -2t C P χ 2 2 (ν|β). (12)
Now, let us define, for any

f ∈ B(E), t ≥ 0 and x ∈ E, Q t [f ](x) := Pt [f /η](x) = e λ 0 t η(x) P t [f ](x).
Since α(ψ 2 /η) < ∞ by the third assumption, one has, for any measurable function such that |f | ≤ ψ,

||f /η|| L 2 (β) ≤ α(ψ 2 /η) < ∞.
In particular, for any measurable function f such that |f | ≤ ψ, t ≥ 0 and ν ∈ M 1 (E)

|νQ t f -α(f )| 2 = ν Pt [f /η] -β[f /η] 2 ≤ α(ψ 2 /η)e -2t C P χ 2 2 (µ|β),
where the following equality is used :

∀ν 1 , ν 2 ∈ M 1 (E), χ 2 2 (ν 1 |ν 2 ) = sup f L 2 (ν 2 ) ≤1 |ν 1 (f ) -ν 2 (f )| 2 .
As a result, for any t ≥ 0 and any

ν ∈ M 1 (E), sup |f |≤ψ |νQ t [f ] -α(f )| ≤ [α(ψ 2 /η)χ 2 2 (ν|β)] 1/2 e -t C P .
Now note that, for any t ≥ 0 and any measurable function f ,

E µ [f (X t )|τ ∂ > t] = E P t [f ](x)µ(dx) E P t [1 E ](x)µ(dx) = E e λ 0 t η(x) P t [f ](x)η(x)µ(dx) E e λ 0 t η(x) P t [1 E ](x)η(x)µ(dx) = E Q t [f ](x)η(x)µ(dx) E Q t [1 E ](x)η(x)µ(dx) = (η • µ)Q t [f ] (η • µ)Q t [1 E ] , (13)
As a result, since α(ψ 2 /η)χ 2 2 (η • µ|β) < 0.9, for any t ≥ 0,

α(f ) -[α(ψ 2 /η)χ 2 2 (η • µ|β)] 1/2 e -t C P 1 + [α(ψ 2 /η)χ 2 2 (η • µ|β)] 1/2 e -t C P ≤ E µ (f (X t )|τ ∂ > t) ≤ α(f ) + [α(ψ 2 /η)χ 2 2 (η • µ|β)] 1/2 e -t C P 1 -[α(ψ 2 /η)χ 2 2 (η • µ|β)] 1/2 e -t C P . ( 14
) For any t ≥ 0, α(f ) + [α(ψ 2 /η)χ 2 2 (η • µ|β)] 1/2 e -t C P 1 -[α(ψ 2 /η)χ 2 2 (η • µ|β)] 1/2 e -t C P = α(f ) + [α(ψ 2 /η)χ 2 2 (η • µ|β)] 1/2 e -t C P 1 + [α(ψ 2 /η)χ 2 2 (η • µ|β)] 1/2 e -t C P 1 -[α(ψ 2 /η)χ 2 2 (η • µ|β)] 1/2 e -t C P ≤ α(f ) + [α(ψ 2 /η)χ 2 2 (η • µ|β)] 1/2 e -t C P + (α(ψ) + 1) [α(ψ 2 /η)χ 2 2 (η • µ|β)] 1/2 e -t C P 1 -[α(ψ 2 /η)χ 2 2 (η • µ|β)] 1/2 e -t C P ≤ α(f ) + 1 + α(ψ) + 1 1 -[α(ψ 2 /η)χ 2 2 (η • µ|β)] 1/2 [α(ψ 2 /η)χ 2 2 (η • µ|β)] 1/2 e -t C P .
In a same way, one can prove that, for any t ≥ 0,

α(f ) -(2 + α(ψ))[α(ψ 2 /η)χ 2 2 (η • µ|β)] 1/2 e -t C P ≤ α(f ) -[α(ψ 2 /η)χ 2 2 (η • µ|β)] 1/2 e -t C P 1 + [α(ψ 2 /η)χ 2 2 (η • µ|β)] 1/2 e -t C P .
As a result, using ( 14), for any t ≥ 0,

sup |f |≤ψ |E µ (f (X t )|τ ∂ > t) -α(f )| ≤ 1 + α(ψ) + 1 1 -[α(ψ 2 /η)χ 2 2 (η • µ|β)] 1/2 ∨ (2 + α(ψ)) [α(ψ 2 /η)χ 2 2 (η • µ|β)] 1/2 e -t C P ≤ (a + bα(ψ))(α(ψ 2 /η)χ 2 2 (η • µ|β)) 1/2 e -t C P , setting a := 1 + 1 1 - √ 0.9 , and b := 1 1 - √ 0.9 .
Second step : Conclusion.

Now let µ ∈ M 1 (E) such that χ 2 (η • µ|β) < +∞. Recalling the notation φ t (µ) := P µ [X t ∈ •|τ ∂ > t],
one has the following lemma, whose the proof is postponed after the end of this proof.

Lemma 1. For any t ≥ 0 and µ ∈ M 1 (E),

η • φ t (µ) = (η • µ) Pt
Then, using Lemma 1 and the inequality [START_REF] Champagnat | Practical criteria for R-positive recurrence of unbounded semigroups[END_REF], one has for any t ≥ 0,

χ 2 2 (η • φ t (µ)|β) = χ 2 2 ((η • µ) Pt |β) ≤ e -2t C P χ 2 2 (η • µ|β).
In particular, there exists t µ ≥ 0 such that, for any t ≥ t µ ,

α(ψ 2 /η)χ 2 2 (η • φ t (µ)|β) < 0.9.
Hence, applying what we obtained at the first step, one has, for any t ≥ t µ , sup

|f |≤ψ |E µ [f (X t )|τ ∂ > t] -α(f )| ≤ (a + bα(ψ))[α(ψ 2 /η)χ 2 2 (η • φ tµ (µ)|β)] 1/2 e - t-tµ C P ≤ (a + bα(ψ))[α(ψ 2 /η)χ 2 2 (η • µ|β)] 1/2 e -t C P ,
which concludes the proof. Now, let us prove Lemma 1.

Proof of Lemma 1. For any t ≥ 0, µ ∈ M 1 (E) and for any measurable function f ,

η • φ t (µ)(f ) = φ t (µ)(f η) φ t (µ)(η) = µP t [f η] µP t [η] = e λ 0 t µP t [f η] µ(η) = 1 µ(η) E Pt [f ](x)η(x)µ(dx) = (η • µ) Pt [f ],
where the equality P t [η](x) = e -λ 0 t η(x), for any t ≥ 0 and x ∈ E, was used.

Remark 5. By the tensorization property of Poincaré inequalities (see [3, Proposition 4.3.1]), the Poincaré constant C P does not depend on the dimension when the state space is a tensorial space. As a result, contrary to the technics using Lyapunov functions or minorization properties, the previous theorem provides in such cases a rate of convergence which does not explode in high dimension (as soon as the state space E is the product space of one-dimensional spaces E i ).

Remark 6. In the same manner, subgeometrical convergences to quasi-stationarity can be proved replacing the conditions (P2) and (LS2) by weaker functional inequalities, such as Nash inequalities or weak Poincaré inequalities (see [START_REF] Thomas | L 2 rates of convergence for attractive reversible nearest particle systems: The critical case[END_REF][START_REF] Röckner | Weak Poincaré inequalities and L 2 -convergence rates of Markov semigroups[END_REF]). This method does not allow however to cover all the processes having this property of subgeometrical convergence (see for example [START_REF] Oçafrain | Polynomial rate of convergence to the Yaglom limit for Brownian motion with drift[END_REF] where the Doob transform is not ergodic).

Remark 7. As stated in Corollary 1, the previous method using the Doob transform Pt allows to get convergence in 1-Wasserstein distance through a Poincaré inequality. A natural question is therefore if one can use the logarithmic Sobolev inequality

E f 2 log f 2 f L 2 (β) dβ ≤ -C LS E f Lf dβ (with C LS > 0)
to deal with the convergence in p-Wasserstein distance, which is defined by

W p (µ, ν) := inf (X,Y )∈Π(µ,ν) E[d(X, Y ) p ] 1/p , ∀µ, ν ∈ P p (E).
By the same methodology and using that, for any µ, ν ∈ M 1 (E),

H(µ|ν) = sup f ∈B(E) {µ(f ) -log(ν(e f ))},
where H(µ|ν) := E log dµ dν dν (when µ ν) is the entropy, one obtains the one-sided estimate

E µ [f (X t )|τ ∂ > t] ≤ log(β(e f /η )) + CH(η • µ|β)e -t C LS .
This estimate is unfortunately not sharp enough, since log(β(e f /η )) ≥ α(f ), and the convergence in W p for general p still remains an open question.

On the domain of attraction of α

This subsection follows Remark 3. In this remark, it was pointed that a consequence of Theorem 3 is the following inclusion:

{µ ∈ M 1 (E) : ∃t 0 ≥ 0, χ 2 (η • φ t 0 (µ)|β) < +∞} ⊂ D(α),
where D(α) is the domain of attraction of α, that is the set of initial measures such that the convergence of

P µ [X t ∈ •|τ ∂ > t] to α holds.
The aim of this subsection is to go a bit further on this point. Let us first state and prove the following proposition, which gives an expression of χ 2 (η • µ|η • α) revealing the density function dµ dα :

Proposition 1. For any µ ∈ M 1 (E) such that dµ dα exists,

χ 2 2 (η • µ|η • α) = 1 µ(η) 2 E dµ dα (y) × η(y)µ(dy) -1.
In particular, χ 2 (η • µ|η • α) < +∞ if and only if µ(η) > 0 and E dµ dα ηdµ < +∞.

Proof. First of all, recalling that

η • µ(dx) = η(x)µ(dx) µ(η) , η • α(dx) = η(x)α(dx), (15)
dµ dα exists if and only if

d(η•µ) d(η•α) exists. Then, by definition of χ 2 , χ 2 2 (η • µ|η • α) = E d(η • µ) d(η • α) -1 2 d(η • α) = E d(η • µ) d(η • α) 2 d(η • α) -1.
Using again [START_REF] Collet | Quasi-stationary distributions[END_REF],

E d(η • µ) d(η • α) 2 d(η • α) = E d(η • µ) d(η • α) d(η • µ) = 1 µ(η) 2 E dµ dα (y) × η(y)µ(dy).
Thus,

χ 2 2 (η • µ|η • α) = 1 µ(η) 2 E dµ dα (y) × η(y)µ(dy) -1,
which concludes the proof.

Proposition 1 entails the following corollary:

Corollary 2. Let µ ∈ M 1 (E) such that µ(η) > 0. If there exists t 0 ≥ 0 such that dφt 0 (µ) dα
exists and

E dφ t 0 (µ) dα ηdφ t 0 (µ) < +∞, then χ 2 (η • φ t 0 (µ)|η • α) < +∞.
Proof. In order to use Proposition 1, we have just to ensure that

φ t 0 (µ)(η) > 0.
However, one has

φ t 0 (µ)(η) = µP t 0 η µP t 0 1 E = e -λ 0 t 0 µ(η) P µ [τ ∂ > t 0 ] > 0,
which concludes the proof.

As a consequence, Corollary 2 entails, for a process satisfying (P 1) -(P 3), that if µ(η) > 0 and there exists t 0 ≥ 0 such that E dφt 0 (µ)

dα ηdφ t 0 (µ) < +∞, then P µ [X t ∈ •|τ ∂ > t]
converges to α exponentially fast at rate 1/C P . This property will be then used in Subsection 3.2 to show, in both examples presented in this subsection, that there is an exponential decay at rate 1/C P when the initial measure is a Dirac measure, even if such a measure does not admit any density function with respect to α.

Also, Proposition 1 entails another interesting corollary, when η is known to be upper-bounded:

Corollary 3. Let µ ∈ M 1 (E). If η is upper-bounded, µ(η) > 0 and χ 2 (µ|α) < +∞, then χ 2 (η • µ|η • µ) < +∞.
Proof. If η is upper-bounded, for any µ ∈ M 1 (E) absolutely continuous with respect to α,

E dµ dα ηdµ ≤ η ∞ 1 + χ 2 2 (µ|α) .
3 Bakry-Émery condition and quasi-stationarity : application to diffusion processes

In a practical way, Theorem 3 is hardly useable because the expressions of the quasi-stationary distribution α and the eigenfunction η are scarcely explicitly known, so the conditions (P2)-(P3) cannot be checked. In this section, diffusion processes will be only dealt with and easy-to-check assumptions will be given.

In all what follows, the space R d will be endowed with the L 1 -distance

d(x, y) := d i=1 |x i -y i | (16)
for any x = (x i ) i=1,...,d and y = (y i ) i=1,...,d . In particular, this distance will be implicitly used for the definition of W 1 . Let D ⊂ R d be an open subset of R d and ∂D its boundary. Let (X t ) t≥0 be the absorbed diffusion process following

dX t = √ 2dB t -∇V (X t )dt, X t ∈ D, (17)
with a d-dimensional Brownian motion (B t ) t≥0 and V ∈ C 2 (R d ), and absorbed when t ≥ τ ∂ , where

τ ∂ := inf{t ≥ 0 : X t ∈ ∂D}.
In order to keep the same notation as the ones in Section 2, let (P t ) t≥0 the sub-Markovian semigroup defined in (2), L the sub-Markovian generator defined in [START_REF] James | Quasi-stationary distributions of birth-and-death processes[END_REF] and D(L) the associated domain of definition defined in [START_REF] Cattiaux | Quasi-stationary distributions and diffusion models in population dynamics[END_REF]. In particular, any function f ∈ C 2 (D) with compact support in D belongs to D(L), and for such a function and x ∈ D,

Lf (x) = ∆f (x) -∇V (x) • ∇f (x).
Denote by γ(dx) := e -V (x) dx.

γ is therefore one reversible measure for L. Note that γ is not necessarily defined as a probability measure. In all what follows, it will be assumed that γ(D) < +∞.

Proof of Theorem 1

In this subsection, we will prove Theorem 1 stated earlier in the introduction, that we recall below :

Theorem 4. Let (X t ) t≥0 following [START_REF] Diaconis | Analytic-geometric methods for finite Markov chains with applications to quasi-stationarity[END_REF] and such that γ(D) < +∞.

(BE1) Assume that there exists a nonnegative function η ∈ D(L) defined on D ∪ ∂D, positive on D and vanishing on ∂D, such that γ(η 2 ) < +∞ and there exists λ 0 > 0 such that ∆η -∇V • ∇η = -λ 0 η.

(BE2) Assume moreover that there exists κ > 0 such that

Hess[V -2 log(η)] ≥ κId. ( 18
)
Then γ(η) < +∞, the probability measure α := η • γ is a quasi-stationary distribution for (X t ) t≥0 and (i) there exists a constant C > 0 such that, for any µ ∈ M 1 (D) and t ≥ 0,

P µ (X t ∈ •|τ ∂ > t) -α T V ≤ C γ(D)γ(η 2 ) γ(η) χ 2 (η • µ|η • α)e -κt , (19)
(ii) If moreover there exists x 0 ∈ D such that

D (1 + d(x, x 0 )) 2 e -V (x) dx < +∞,
then for any µ ∈ M 1 (D), there exists t µ such that for any t ≥ t µ ,

W 1 (P µ [X t ∈ •|τ ∂ > t], α) ≤ C(γ, η)χ 2 (η • µ|η • α)e -κt , (20)
where

C(γ, η) := a + b D (1 + d(x, x 0 ))η(x)γ(dx) γ(η) γ(η 2 ) D (1 + d(x, x 0 )) 2 γ(dx) γ(η) .
Remark 8. Following Remark 1, it is possible to state general assumptions on the potential V to entail (BE1). For example, when D = (0, +∞) d , the condition lim |x|→∞ ∇V (x) = +∞ [START_REF] Kijima | Some results for quasi-stationary distributions of birth-death processes[END_REF] entails Assumption (F) in [START_REF] Champagnat | General criteria for the study of quasi-stationarity[END_REF] (see this paper for the proof), so entails (P1) and (BE1).

Proof. First of all, remark that the property γ(η) < +∞ comes from the Cauchy-Schwarz inequality and the fact that γ(η 2 ) ∨ γ(D) < +∞ by assumptions.

To prove that α := η • γ is a quasi-stationary distribution for X, the proof of [15, Theorem 1.1] will be adapted to general Kolmogorov diffusion processes. Denote by

L := ∆ -∇V • ∇.
By Itô's formula, for any f twice continuously differentiable with compact support in D,

e λ 0 t f (X t∧τ ∂ ) = f (X 0 ) + t∧τ ∂ 0 (Lf (X s∧τ ∂ ) + λ 0 f (X s∧τ ∂ ))ds + M t ,
where M is a martingale. Hence, for any x ∈ D and t ≥ 0,

e λ 0 t E x [f (X t∧τ ∂ )] = f (x) + E x t∧τ ∂ 0 (Lf (X s∧τ ∂ ) + λ 0 f (X s∧τ ∂ ))ds . ( 22
)
Since L is symmetric with respect to γ (i.e.

R d gLhdγ = R d Lghdγ, ∀g, h ∈ L 2 (γ)), by Fubini's theorem, for any s ≥ 0, E η•γ [Lf (X s∧τ ∂ ) + λ 0 f (X s∧τ ∂ )] = 0.
Hence, integrating [START_REF] Kolb | Quasilimiting behavior for one-dimensional diffusions with killing[END_REF] over η • γ and using Fubini's theorem,

e λ 0 t E η•γ [f (X t∧τ ∂ )] = η • γ(f ).
As a result, it is shown that, for any f twice continuously differentiable with compact support in D,

(η • γ)P t f = e -λ 0 t η • γ(f ).
Since η vanishes at the boundary of D, the probability measure α = η • γ is a quasi-stationary distribution, associated to λ 0 . Moreover, remark that the measure γ(dx) = e -V (x) dx is a reversible measure for the semi-group (P t ) t≥0 , which means that, for any f, g ∈ B(D),

D (P t f )gdγ = D f (P t g)dγ, ∀t ≥ 0.
Then, for any t ≥ 0 and f ∈ B(D),

D (P t η)f dγ = D η(P t f )dγ = γ(η)αP t f = e -λ 0 t γ(η)α(f ) = e -λ 0 t D ηf dγ.
Thus, η is also an eigenfunction for (P t ) t≥0 , associated to λ 0 . Now, consider again the Doob transform ( Pt ) t≥0 defined by

Pt f (x) := e λ 0 t P t [η × f ](x) η(x) , ∀x ∈ D, ∀f ∈ B(D).
Then the generator of the semi-group ( Pt ) t≥0 , denoted by L, endowed with its domain D( L), is

Lf (x) = ∆f (x) -∇ [V -2 log(η)] (x) • ∇f (x), ∀x ∈ D, ∀f ∈ D( L).
The condition ( 18) is therefore the Bakry-Émery condtion for the generator L. This implies therefore (see for example [3, Proposition 4.8.1]) that the invariant measure β := η • α = η 2 • γ for the semigroup ( Pt ) t≥0 satisfies a Poincaré inequality with

C P = 1 κ , that is Var β (f ) ≤ - 1 κ D f Lf dβ,
which is (P2).

In order to deal with the total variation distance, it is enough to take ψ = 1. For such a choice of ψ, one has

η • γ ψ 2 η/α(η) = η • γ(η) × η • γ(1/η) = γ(D)γ(η 2 ) γ(η) 2 < ∞.
Hence the condition (P3) of the Theorem 3 is satisfied for ψ = 1. So, by Theorem 3, one has

P µ (X t ∈ •|τ ∂ > t) -η • γ T V = sup |f |≤1 |E µ [f (X t )|τ ∂ > t] -η • γ(f )| ≤ C γ(D)γ(η 2 ) γ(η) χ 2 (η • µ|η 2 • γ)e -κt .
The point (ii) of Theorem 4 is a straightforward consequence of Corollary 1.

Remark 9. Exponential decays like ( 19) and ( 20) hold also under weaker assumptions than [START_REF] Diaconis | On quantitative convergence to quasi-stationarity[END_REF], such as the two followings:

• There exists c > 0 and R ≥ 0 such that for |x| > R,

x • ∇[V -2 log(η)](x) ≥ c|x|. (23)
• There exists a ∈ (0, 1), c > 0 and R ≥ 0 such that for |x| > R,

a|∇[V -2 log(η)](x)| 2 -∆[V -2 log(η)](x) > c.
These two conditions actually appear in [1, Corollary 1.6] and imply (P2). In particular, the first condition is satisfied when V -2 log(η) is convex. It will be shown later that, for diffusion processes on (0, +∞) d , the convexity of V implies the one of V -2 log(η) for a particular eigenfunction η (which is not unique a priori), so (23) is satisfied.

Two examples

In this subsection, two examples whose the eigenfunctions η can be explicitly computed are studied through Theorem 4: a scaled Brownian motion living in a hypercube, and an Ornstein-Uhlenbeck process living on (0, +∞) d . In particular, several spectral properties will be claimed throughout this subsection. We refer the reader to the Appendix, at the end of the paper, for a few proofs on these spectral properties.

Brownian motion in a hypercube

Concerning quasi-stationarity for Brownian motion living in the interior of a general compact set in R d and absorbed at its boundary, we refer the reader to [START_REF] Collet | Quasi-stationary distributions[END_REF]Theorem 1.1]. Consider the open set D = C N := (-N, N ) d , with N ∈ N, and V = 0, that is to say (X t ) t≥0 = ( √ 2B t ) t≥0 . Then the function η Bm defined by

η Bm (x 1 , . . . , x d ) := d i=1 cos π 2N x i , ∀(x 1 , . . . , x n ) ∈ C N , (24)
is an eigenfunction of ∆ with respect to the eigenvalue -λ 0 , where

λ 0 = dπ 2 4N 2 .
A proof of this claim is written in Appendix, at the end of the paper. Moreover, η Bm is positive on C N , vanishing at ∂C N and

γ(η 2 Bm ) = C N η 2 Bm (x)dx = N -N cos 2 π 2N x d = N d .
Thus (BE1) is satisfied. Now, for any (x 1 , . . . , x d ) ∈ C N and i, j = 1, . . . , d,

(Hess log(η Bm (x 1 , . . . , x d ))) i,j = -π 2N 2 1 + tan 2 π 2N x i if i = j 0 otherwise
Hence, the Bakry-Émery condition ( 18) in (BE2) holds for κ = π 2 2N 2 . Then, Theorem 4 entails that the probability measure

α Bm (dx) = η Bm • γ(dx) = η Bm (x)dx γ(η Bm ) = π 4N d d i=1 cos π 2N x i dx
is a quasi-stationary distribution for (X t ) t≥0 and So, there exists C > 0 such that, for any initial measure µ ∈ M 1 (C N ) and t ≥ 0,

||P µ (X t ∈ •|τ ∂ > t) -α Bm || T V ≤ C γ(D)γ(η 2 Bm ) γ(η Bm ) χ 2 (η Bm • µ|η Bm • α Bm ) exp - π 2 2N 2 t . (25)
Now, one has

γ(η Bm ) = N -N cos π 2N x dx d = 4N π d , and
γ(D) = (2N ) d .
As a result, (25) becomes: for any µ ∈ M 1 (C N ) and t ≥ 0,

||P µ (X t ∈ •|τ ∂ > t) -α Bm || T V ≤ C π 2 √ 2 d χ 2 (η Bm • µ|η Bm • α Bm ) exp - π 2 2N 2 t .
Note however that this Bakry-Émery coefficient κ is not optimal. In particular, denoting β Bm := η Bm • α Bm , if we use directly Theorem 3, the Poincaré constant C P is equal to

1/C P = inf f ∈L 2 (β Bm ),β Bm (f )=0 -E f Lf dβ Bm E f 2 dβ Bm . ( 26
)
By this formula, one can compute (see Appendix to see the computation) that

1 C P = 3π 2 4N 2 > κ.
Thus, by Theorem 3, for any µ ∈ M 1 (E) and t ≥ 0,

P µ [X t ∈ •|τ ∂ > t] -α Bm T V ≤ C π 2 √ 2 d χ 2 (η Bm • µ|β Bm ) exp - 3π 2 4N 2 t .
Note then, as mentioned in Remark 2, the Poincaré constant C P and the Bakry-Emery constant κ depend on the domain D through the size N .

Concerning the 1-Wasserstein distance, one can remark that the exponential decay in total variation distance [START_REF] Lladser | Domain of attraction of the quasi-stationary distributions for the Ornstein-Uhlenbeck process[END_REF] implies the one in W 1 . As a matter of fact, since we are studying a process living on the compact set (-N, N ) d , one has, for any µ, ν ∈ M 1 (C N ),

W 1 (µ, ν) ≤ dN µ -ν T V .
This inequality allows actually to get a better estimate for the decay in 1-Wasserstein distance than the one provided by Corollary 1.

Finally, note that, if the initial measure µ admits a density function with respect to Lebesgue's measure which vanishes at ∂C N , µ(η Bm ) > 0 and, since α Bm (dx) = η Bm (x)dx γ(η Bm ) ,

C N dµ dα Bm η Bm dµ = γ(η Bm ) C N dµ dx (x) 2 dx < +∞, so that χ 2 (η Bm • µ|β Bm ) < +∞ according to Proposition 1. If µ = δ x , one can show that the probability measure P µ [X 1 ∈ •|τ ∂ > 1]
admits a density function with respect to Lebesgue's measure which vanishes at ∂C N , so one has also

χ 2 (η Bm • P µ [X 1 ∈ •|τ ∂ > 1]|β Bm ) < +∞.
By Remark 3, this shows that the convergence of

P µ [X t ∈ •|τ ∂ > t]
to α Bm at rate 3π 2 4N 2 holds for any initial distributions µ.

Ornstein-Uhlenbeck process

For this example, consider D = (0, +∞) d and V (x) = λ 2 d i=1 x 2 i , with λ > 0. Then, (X t ) t≥0 is a d-dimensional Ornstein-Uhlenbeck process satisfying the following stochastic differential equation

dX t = √ 2dB t -λX t dt, X t ∈ (0, +∞) d . (27)
A positive eigenfunction of ∆ -∇V • ∇ is

η OU (x 1 , . . . , x n ) := d i=1 x i , ∀(x 1 , . . . , x d ) ∈ D, (28)
associated to the eigenvalue -λd (the computation is in Appendix). Noting that, for this example,

γ(dx) = d i=1 e -λx 2 i 2 dx, one has furthermore γ(η 2 OU ) = ∞ 0 x 2 e -λx 2 2 dx d = π 2λ 3 d .
Thus, (BE1) in Theorem 4 is satisfied.

For any x = (x 1 , . . . , x d ) and i, j = 1, . . . , d,

[Hess(V -2 log(η OU ))(x)] i,j = λ + 2 x 2 i if i = j 0 otherwise
So the Bakry-Émery condition ( 18) is satisfied for κ = λ. Hence, by Theorem 4, the probability measure

α OU (dx) := η OU • γ(dx) = η OU (x)γ(dx) γ(η OU ) = λ d d i=1
x i e -λx i 2 dx is a quasi-stationary distribution for (X t ) t≥0 there exists C d > 0 such that, for any µ and t large enough,

W 1 (P µ [X t ∈ •|τ ∂ > t], α OU ) ≤ C d χ 2 (η OU • µ|η OU • α OU ) exp (-λt) .
Note that the rate of convergence does not depend on the dimension d, but the constant C d explodes in high dimension. More precisely, after computations, one can show that, when d → +∞,

C d ∼ d(d -1) 4λ π 2 d .
Contrary to the previous example, the probability measure

P µ [X t ∈ •|τ ∂ > t]
does not converge to α OU for any initial distribution µ. The curious reader can read the paper [START_REF] Lladser | Domain of attraction of the quasi-stationary distributions for the Ornstein-Uhlenbeck process[END_REF], where it is shown, in the one-dimensional case, that there exists an infinity of quasi-stationary distributions for Ornstein-Uhlenbeck processes absorbed by D c , each associated with their own domain of attraction. It is then expected that the property χ 2 (η OU • µ|η OU • α OU ) < +∞ cannot be satisfied for every initial distributions. However, α OU has the property that, for any x ∈ D,

P x [X t ∈ •|τ ∂ > t] -→
t→∞ α OU (this can be seen for example in [START_REF] Champagnat | General criteria for the study of quasi-stationarity[END_REF]). Such a quasi-stationary distribution is called a Yaglom limit (see for example [START_REF] Méléard | Quasi-stationary distributions and population processes[END_REF] for the definition). Our aim is then to show that there exists t 0 > 0 such that

χ 2 (η OU • φ t 0 (δ x )|η OU • α OU ) < +∞, ∀x ∈ D. ( 29
)
To do so, we will use Corollary 2, stating that this holds when η OU (x) > 0 for all x ∈ D, which is satisfied, and when

D dφ t 0 (δ x ) dα OU η OU dφ t 0 (δ x ) < +∞.
For sake of simplicity, let us only deal with the case d = 1 (the result in general dimension could be deduced by tensorization). In this case, for any x ∈ (0, +∞), one claims that the positive measure

δ x P 1 = P x [X 1 ∈ •, τ ∂ > 1]
admits a density function with respect to the measure 2λ π 1 x∈D γ(dx), denoted by f x , which admits the following representation

f x = n odd e -λn F n (x)F n , in L 2 (γ) , (30)
with

F n (x) = H n ( √ λx) √ n! , ∀n ∈ Z + , ∀x ∈ (0, +∞),
where (H n ) n∈Z + are Hermite polynomials. See the Appendix for the definition of Hermite polynomials and the proof of this claim. Now, let us prove (29

) for t 0 = 1. Since P x [τ ∂ > 1] > 0 for all x ∈ D, (29) is equivalent to ∞ 0 δ x P 1 dα OU η OU d(δ x P 1 ) < +∞.
Thus, using that δ x P 1 (dy) = 2λ π f x (y)γ(dy) and α OU (dy) = λη OU (y)γ(dy), ( 29) is equivalent to

∞ 0 f 2 x (y) 2λ π γ(dy) < +∞.
Then, using the representation [START_REF] Röckner | Weak Poincaré inequalities and L 2 -convergence rates of Markov semigroups[END_REF] and the fact that

(F n ) n∈N is orthonormal in L 2 2λ π γ (this is proved in the Appendix), ∞ 0 f 2 x (y) 2λ π γ(dy) = n odd e -2λn F 2 n (x) ≤ n∈Z + e -2λn F 2 n (x).
However, the representation n∈Z + e -λn F n (x)F n is no less than the density function, with respect to λ 2π γ, of the marginal law at time 1 of an Ornstein-Uhlenbeck process of parameter λ, starting at x (this fact is also proved in the Appendix, see (47)). In other terms, for any y ∈ R,

n∈Z + e -λn F n (x)F n (y) ∝ e - λ(y-xe -λ ) 2 1-e -2λ e -λ y 2 2 , in L 2 (γ).
This entails that, for any x ∈ (0, +∞),

n∈Z + e -2λn F 2 n (x) ∝ R e - 2λ(y-xe -λ ) 2 1-e -2λ
e -λ y 2 2 dy < +∞.

Tu sum up, (29) holds for t 0 = 1 and for any x ∈ (0, +∞), which entails that

χ 2 (η OU • φ 1 (δ x )|η OU • α OU ) < +∞, ∀x ∈ D.

Diffusion processes on (0, ∞) d

In general, contrary to the two previous examples, the eigenfunction η cannot be explicitly given, so the assumptions of Theorem 4 cannot be checked in practice. In this subsection, one will see how to bypass this problem for diffusion processes living on D = (0, +∞) d and absorbed when one of its component reaches 0.

When d = 1

Take a one-dimensional diffusion process following

dX t = √ 2dB t -V (X t )dt (31)
living on D = (0, +∞) and absorbed at ∂ = 0, where V is a C 2 -function. Then, one gets the following proposition.

Proposition 2. Assume that V is convex on (0, +∞) and

lim x→+∞ V (x) = +∞.
Then there exists an eigenfunction η such that log(η) is concave.

Proof. In [10, Corollary 4.2.], it is shown that, under the condition lim x→+∞ V (x) = +∞, there exists a unique positive eigenfunction η ∈ C 2 (D) for P t , for all t ≥ 0, such that P t η = e -λ 0 t and

η (x) -V (x)η (x) = -λ 0 η(x), ∀x ∈ D, (32)
with λ 0 > 0, and such that there exists C, θ > 0 such that, for any x ∈ D and t ≥ 0,

η(x) -e λ 0 t P x (τ ∂ > t) ≤ Ce -θt ϕ 1/p (x), (33)
where p > 1 and ϕ is a Lyapunov function such that there exists D 0 ⊂ (0, +∞), C > 0 and λ > 0 large enough such that

ϕ (x) -V (x)ϕ (x) ≤ -λϕ(x) + C 1 x∈D 0 , ∀x ∈ D.
For any x ≥ 0, h > 0 and t ≥ 0,

P x+h (τ ∂ > t) = P x+h (τ ∂ > t, τ x ≤ t) + P x+h (τ x > t) = E x+h [1 τx≤t P x (τ ∂ > t -u)| u=τx ] + P x+h (τ x > t),
where τ x is the hitting time of x by the process (X t ) t≥0 , and where the strong Markov property is used for the second equality.

Considering the process (X t∧τx ) t≥0 absorbed at x, it is also a diffusion process coming down from infinity. So there exists also a positive function η x on (x, +∞) and a positive constant λ x such that, for any y > x, η x (y) = lim t→∞ e λxt P y (τ x > t).

Since τ 0 dominates stochastically τ x , λ 0 < λ x for any x > 0, so lim t→+∞ e λ 0 t P x+h (τ x > t) = 0. Now remark that for any x ≥ 0, h > 0 and t ≥ 0,

e λ 0 t E x+h [1 τx≤t P x (τ ∂ > t -u)| u=τx ] = E x+h [1 τx≤t e λ 0 τx × [e λ 0 (t-u) P x (τ ∂ > t -u)] u=τx ].
The random variable 1 τx≤t e λ 0 τx × [e λ 0 (t-u) P x (τ ∂ > t -u)] u=τx is upper bounded by e λ 0 τx [η(x) + Cϕ 1/p (x)], and, for h small enough, E x+h (e λ 0 τx ) < ∞ (see [START_REF] Méléard | Quasi-stationary distributions and population processes[END_REF]Proposition 3]). Moreover, it converges to η(x)e λ 0 τx when t goes to infinity. So, by the Lebesgue's theorem,

lim t→+∞ e λ 0 t E x+h [1 τx≤t P x (τ ∂ > t -u)| u=τx ] = E x+h [e λ 0 τx ]η(x).
In conclusion, one has

η(x + h) = lim t→∞ e λ 0 t P x+h (τ ∂ > t) = E x+h [e λ 0 τx ]η(x).
So, for any h > 0 small enough,

η(x + h) -η(x) h = η(x) E x+h [e λ 0 τx ] -1 h .
Then, since η ∈ C 2 ((0, +∞)), for any x > 0, lim h↓0

E x+h [e λ 0 τx ]-1 h exists and η (x) = η(x) lim h↓0 E x+h [e λ 0 τx ] -1 h .
In other words, one has

log(η) (x) = lim h↓0 E x+h [e λ 0 τx ] -1 h .
Now, for h fixed, since V is convex, the derivative V is non-decreasing and, by [20, Theorem 1.1, Chapter VI, p.437], the function x → E x+h (e λ 0 τx ) is non-increasing, so one has for any x ≤ x ,

lim h↓0 E x+h [e λ 0 τx ] -1 h ≥ lim h↓0 E x +h [e λ 0 τ x ] -1 h .
So the function log(η) is non-increasing, which implies that the function log(η) is concave.

The previous proposition actually tells us that, assuming V convex, the second derivative of V -2 log(η) is greater than the one of V . In particular, Proposition 2 entails the following corollary: Corollary 4. Let (X t ) t≥0 satisfying (31) and assume that there exists κ > 0 such that

V (x) ≥ κ, ∀x ∈ (0, +∞).
Then there exists a quasi-stationary distribution α, which is absolutely continuous with respect to γ, and a constant C > 0 such that, for any µ ∈ M 1 (D) and for t large enough,

P µ [X t ∈ •|τ ∂ > t] -α T V ≤ Cχ 2 (η • µ|η • α)e -κt ,
and

W 1 (P µ [X t ∈ •|τ ∂ > t], α) ≤ Cχ 2 (η • µ|η • α)e -κt ,
where η := dα dγ .

Proof. Integrating twice the Bakry-Émery condition, there exists two constants a 1 , a 2 ∈ R such that, for any x > 0,

V (x) ≥ a 1 + a 2 x + κ 2 x 2 . (34)
Hence, one has lim x→+∞ V (x) = +∞ and there exists an eigenfunction η satisfying [START_REF] Seneta | On quasi-stationary distributions in discrete-time Markov chains with a denumerable infinity of states[END_REF] and [START_REF] Steinsaltz | Quasistationary distributions for one-dimensional diffusions with killing[END_REF]. Moreover, since V is convex, log(η) is concave by Proposition 2, so for any x > 0,

(V -2 log(η)) (x) ≥ V (x) ≥ κ,
which implies also that γ(η 2 ) < +∞. Hence the conditions (BE1)-(BE2) hold. Finally, by [START_REF] Van Doorn | Quasi-stationary distributions and convergence to quasi-stationarity of birth-death processes[END_REF],

∞ 0 (1 + x) 2 e -V (x) dx < +∞,
which entails the exponential decay in total variation and 1-Wasserstein distance by Theorem 4, setting α := η • γ.

One-dimensional processes coming down from infinity

Let (X t ) t≥0 be a solution of (31) coming down from infinity, which means that there exists a constant ρ > 0 such that sup x≥0 E x (e ρτ ∂ ) < +∞ (see [START_REF] Bansaye | Diffusions from infinity[END_REF] for alternative definitions). Quasi-stationarity for such processes have been already studied in [START_REF] Champagnat | Uniform convergence of conditional distributions for absorbed one-dimensional diffusions[END_REF], in particular (X t ) t≥0 absorbed at 0 admits a unique quasi-stationary distribution α absolutely continuous with respect to γ and an eigenfunction η, unique up to a multiplicative constant, satisfying the following relation (see [START_REF] Champagnat | Uniform convergence of conditional distributions for absorbed one-dimensional diffusions[END_REF]Theorem 4.1.]):

η(x) = 4λ 0 ∞ 0 (x ∧ y)η(y)γ(dy), (35)
where -λ 0 < 0 is the eigenvalue associated to α and η. Moreover, [START_REF] Champagnat | Uniform convergence of conditional distributions for absorbed one-dimensional diffusions[END_REF]Proposition 4.2.] states that η is proportional to the function

x → ∞ 0 (x ∧ y)α(dy).
In particular, log(η) is concave, whatever the convexity of the potential V . For these processes, one can state the following result :

Theorem 5. Let (X t ) t≥0 following (31) coming down from infinity such that

κ := inf x>0 V (x) + 8λ 0 e -V (x) > 0.
Then there exists a constant C > 0 such that, for any µ ∈ M 1 (D) and for t large enough,

P µ [X t ∈ •|τ ∂ > t] -α T V ≤ Cχ 2 (η • µ|η • α)e -κt , (36)
and

W 1 (P µ [X t ∈ •|τ ∂ > t], α) ≤ Cχ 2 (η • µ|η • α)e -κt , (37)
If moreover V (x) > 0 for any x > 0, then the previous statement holds for

κ := inf x>0    V (x) + 8λ 0 e -V (x) + 2λ 2 0 1 -4e -V (x) V (x) 2    .
Remark 10. In other words, this theorem states that the rate of convergence κ coming from the Bakry-Émery condition V ≥ κ can actually be improved replacing it by κ. Moreover, this entails that the exponential convergences ( 36) and ( 37) holds even if V is concave in a neighborhood of 0, as soon as the function x → V (x) + 8λ 0 e -V (x) is lower-bounded by a positive constant.

Proof of Theorem 5. The idea is simply to apply Theorem 4 and to compute the best κ satisfying [V -2 log(η)] (x) ≥ κ, ∀x > 0, knowing [START_REF] Van Doorn | Quasi-stationary distributions for discrete-state models[END_REF]. First of all, for any x > 0,

[V -2 log(η)] (x) = V (x) -2 η (x) η(x) - η (x) η(x) 2 .
By the equality [START_REF] Van Doorn | Quasi-stationary distributions for discrete-state models[END_REF],

η(x) = 4λ 0 x 0 yη(y)γ(dy) + 4λ 0 x ∞ x η(y)γ(dy).
Then, for any x > 0,

η (x) = 4λ 0 ∞ x η(y)γ(dy), η (x) = -4λ 0 η(x)e -V (x) .
Hence, for any x > 0,

[V -2 log(η)] (x) = V (x) + 8λ 0 e -V (x) + 2 η (x) η(x) 2 .
As a result, assuming κ := inf x>0 V (x) + 8λ 0 e -V (x) > 0, one has [V -2 log(η)] (x) ≥ κ, ∀x ∈ (0, +∞). Now, assuming moreover V (x) > 0 for any x > 0, and using that η (x) -V (x)η (x) = -λ 0 η(x) for any x > 0, one has

η (x) = η (x) + λ 0 η(x) V (x) = λ 0 η(x) 1 -4e -V (x) V (x) , which entails that [V -2 log(η)] (x) = V (x) + 8λ 0 e -V (x) + 2λ 2 0 1 -4e -V (x) V (x) 2 ,
which concludes the proof.

Example 1. Considering V : x → (x + 1) δ , δ > 2,
the underlying process (X t ) t≥0 satisfying (31) comes down from infinity, so Theorem 5 applies and the inequalities ( 36) and ( 37) hold for κ :

= inf x>0 V (x) + 8λ 0 e -V (x) + 2λ 2 0 1-4e -V (x) V (x)

2

. For this example, the eigenvalue -λ 0 is not explicitly known, but it is possible to compare it with the eigenvalue -λ OU associated to the one-dimensional absorbed Ornstein-Uhlenbeck process satisfying

dX OU t = √ 2dB t -(X OU t + 1)dt,
and such that, for any x > 0,

λ OU = -lim t→∞ log P OU t [1 (0,+∞) ](x) t ,
where (P OU t ) t≥0 is the sub-Markovian semi-group associated to (X OU t ) t≥0 . This eigenvalue is explicitly known :

λ OU = 1.

Likewise one has, for any x > 0, lim t→∞

- log P x [τ ∂ > t] t = λ 0 .
Hence, since V (x) ≥ x + 1 for any x > 0, one deduces from the theorem of comparison [20, Theorem 1.1, Chapter VI] that, for any x > 0, λ 0 ≥ λ OU = 1.

As a result, one has a lower-bound for λ 0 and one can choose κ as

κ := inf x>0    V (x) + 8e -V (x) + 2 1 -4e -V (x) V (x) 2    .

Multi-dimensional diffusion processes

Now consider one d-dimensional diffusion process (X t ) t≥0 := (X 1 t , . . . , X d t ) t≥0 satisfying

dX t = √ 2dB t -∇V (X t )dt
where, for any x 1 , . . . , x d ∈ (0, +∞) d ,

V (x 1 , . . . , x d ) = d i=1 V i (x i ), (38)
where, for any 1

≤ i ≤ d, V i : [0, +∞) → R is a convex C 2 ([0, +∞))-function such that lim x→∞ V i (x) = +∞.
We consider this process as absorbed by the boundary of [0, +∞) d . In particular, D = (0, +∞) d and ∂D = {(x 1 , . . . , x d ) ∈ [0, +∞) d : x i = 0 for some i}.

Denote η a common nonnegative eigenfunction of (P t ) t≥0 . Then, for any x 1 , . . . , x d , η can be expressed as follows

η(x 1 , . . . , x d ) := d i=1 η i (x i ),
where (η i ) i=1,...,d are functions such that, for any i, there exists λ 0,i > 0 such that, for any t ≥ 0 and x ∈ (0, +∞), P i t η i (x) = e -λ 0,i t η i (x), where

P i t f (x) = E x [f (X i t )1 τ i ∂ >t ], ∀f ∈ B((0, +∞)), ∀t ≥ 0, where τ i ∂ := inf{t ≥ 0 : X i t = 0}
. η is therefore associated to λ 0 := d i=1 λ 0,i , and one has for any

x 1 , . . . , x d ∈ (0, +∞) d (Hess log(η)(x 1 , . . . , x d )) i,j = log(η i (x i )) if i = j 0 otherwise.
By what it was shown previously, for any i = 1, . . . , d, log(η i ) is concave. As a result, one can state the following result, which is the the multi-dimensional version of Corollary 4, already stated in the Introduction.

Theorem 6. Assume that the potential can be written as (38) and that there exists κ > 0 such that, for any i = 1, . . . , d, V i (x) ≥ κ, ∀x ∈ [0, +∞).

Then there exists a quasi-stationary distribution α := η • γ and a constant C d > 0 (depending on the dimension d) such that, for any µ ∈ M 1 (D) and t large enough,

W 1 (P µ [X t ∈ •|τ ∂ > t], α) ≤ C d χ 2 (η • µ|η • α)e -κt , and P µ [X t ∈ •|τ ∂ > t] -α T V ≤ C d χ 2 (η • µ|η • α)e -κt .
Previously, it was seen, with the two examples of Subsection 3.2, that the constant C d could explode when the dimension d goes to infinity. However, it is possible to improve this result when the initial measure µ is the tensorial product of d probability measures on (0, +∞). In this case, since (38) is assumed, the one-dimensional processes (X i ) i=1,...,d are mutually independent. Moreover, since {X t = 0} = i=1,...,d {X i t = 0}, then for any t ≥ 0 and µ 1 , . . . , µ d ∈ M 1 ((0, +∞)),

P µ 1 ⊗•••⊗µ d [X t ∈ •|τ ∂ > t] = P µ 1 [X 1 t ∈ •|τ 1 ∂ > t] ⊗ • • • ⊗ P µ d [X d t ∈ •|τ d ∂ > t],
Then, one obtains the following theorem, which was also stated previously in the Introduction.

Theorem 7. Assume the assumptions of Theorem 6. Then there exists a constant C > 0, which does not depend on the dimension, such that, for any µ 1 , . . . , µ d ∈ M 1 ((0, +∞)), and for t large enough,

P µ 1 ⊗•••⊗µ d [X t ∈ •|τ ∂ > t] -α T V ≤ C d i=1 χ 2 (η i • µ i |η i • α i ) e -κt ,
and

W 1 (P µ 1 ⊗•••⊗µ d [X t ∈ •|τ ∂ > t], α) ≤ C d i=1 χ 2 (η i • µ i |η i • α i ) e -κt ,
where α i (dx) := η i (x)e -V i (x) dx.

Proof. The first result comes from the inequalities

µ 1 ⊗ • • • ⊗ µ d -ν 1 ⊗ • • • ⊗ ν d T V ≤ d i=1 µ i -ν i T V ,
which can be shown using the equality

1 2 µ -ν T V = inf (X,Y )∈Π(µ,ν) P(X = Y ), ∀µ, ν ∈ M 1 (D),
and the result is deduced from the one obtained for d = 1. In the same way, by the definition of W 1 and recalling that W 1 is defined through the L 1 -distance defined in ( 16), one has

W 1 (µ 1 ⊗ • • • ⊗ µ d , ν 1 ⊗ • • • ⊗ ν d ) = d i=1 W 1 (µ i , ν i ),
which implies the second inequality in the statement of Theorem 7.

Obvioulsy, one can also state a result similar to Theorem 5 for multi-dimensional diffusion processes coming down from infinity: Theorem 8. Assume that, for any i = 1, . . . , d, X i comes down from infinity and V i (x) > 0 for any x > 0. Then the statements of Theorem 6 and 7 hold replacing κ by

κ := min i=1,...,d inf x>0    V i (x) + 8λ 0,i e -V i (x) + 2λ 2 0,i 1 -4e -V i (x) V i (x) 2    .
These functions are therefore eigenfunctions of the Laplacian ∆: for all k 1 , . . . , k d ∈ N, for any (

x 1 , . . . , x d ) ∈ C N , Lη k 1 ,...,k d (x 1 , . . . , x d ) = ∆η k 1 ,...,k d (x 1 , . . . , x d ) = 1 √ N d d i=1 ∂ x i ,x i d i=1 sin k i π 2N (x i + N ) = - 1 √ N d d i=1 k i π 2N 2 d i=1 sin k i π 2N (x i + N ) = -λ k 1 ,...,k d η k 1 ,...,k d (x 1 , . . . , x d ), (39)
where

λ k 1 ,...,k d := π 2 4N 2 d i=1 k 2 i .
In particular, by the definition of the function η Bm in [START_REF] Littin | Uniqueness of quasistationary distributions and discrete spectra when ∞ is an entrance boundary and 0 is singular[END_REF],

η Bm = √ N d η 1,...,1 .
Hence, by (39), ∆η Bm = -λ 0 η Bm , where ) k 1 ,...,k d is a total orthonormal basis of L 2 (β Bm ) and one obtains by (26) that 1/C P = λ 1 -λ 0 , where λ 1 is the smallest λ k 1 ,...,k d different from λ 0 . In other words,

λ 0 := λ 1,...,1 = dπ 2 4N 2 .
λ 1 = λ 1,...,1,2 = (d -1)π 2 4N 2 + π 2 N 2 ,
so that 1

C P = λ 1 -λ 0 = 3π 2 4N 2 > κ,
which is exactly the claim stated in Subsection 3.2.

Spectral analysis of the Ornstein-Uhlenbeck process living in (0, +∞) d

Let us consider the d-dimensional process (X t ) t≥0 defined by Then two interesting properties can be deduced from this definition:

• For any n, m ∈ Z + , R H n (x)H m (x) 1 √ 2π e -x 2 2 dx = n!δ n,m , (40)
where δ n,m is the Kronecker delta.

• For any n ∈ Z + and x ∈ R, H n (x) -xH n (x) = -nH n (x).

In particular, by the first property, denoting γ 0 (dx) = dy, denoted by f x , whose a representation is

f x = n odd e -λn F n (x)F n . (45)
Proof. Let A ⊂ D. In this proof, let us consider the process (X t ) t≥0 as a non-absorbed process, so as an Ornstein-Uhlenbeck process living on R following dX t = √ 2dB t -λX t dt.

Since the Ornstein-Uhlenbeck process satisfies a property of reflection at 0, one has, for any x ∈ D,

P 1 1 A (x) = P x [X 1 ∈ A, τ ∂ > 1] = P x [X 1 ∈ A] -P x [X 1 ∈ A, τ ∂ ≤ 1] = P x [X 1 ∈ A] -P x [X 1 ∈ -A],
(46) where -A := {x ∈ R : -x ∈ A}. Another way to write (46) is

P 1 1 A (x) = E x [(1 A -1 -A )(X 1 )].
Since (F n ) n∈Z + is a total orthonormal basis of L 2 (γ) which are eigenfunctions for the operator S 1 , respectively associated to the eigenvalues e -λn , then for any f ∈ L 2 (γ),

E x [f (X 1 )] = S 1 f (x) = n∈Z + e -λn < F n , f > γ F n (x), (47)
where < F n , f > γ = R F n (y)f (y) λ √ 2π e -λy 2 2 dy. Noting that the function 1 A -1 -A is odd and that F n is odd (respectively even) when n is odd (respectively even), one has

< F n , 1 A -1 -A > γ = 0 if n is even.
A F n (y) 2λ π e -λy 2 2 dy otherwise.

In conclusion, using (46) and (47), The representation (45) is naturally deduced from the previous equality.

Definition 1 .

 1 dX t = √ 2dB t -λX t dt,with λ > 0, living on (0, ∞) d and absorbed at its boundary. First, we prove that the function η OU defined byη OU (x 1 , . . . , x d ) = d i=1x i is a right eigenfunction for the operator L = ∆ -λx • ∇, positive on (0, +∞) d and vanishing at its boundary. As a matter of fact,Lη OU (x) = ∆η OU (x) -λx • ∇η OU (x) = -λ d i=1 x i ∂ x i η OU (x 1 , . . . , x d ) = -λd d j=1 x j = -λdη OU (x 1 , . . . , x d ),which proves one of the claim of Subsection 3.2. It remains us to prove the formula of representation[START_REF] Röckner | Weak Poincaré inequalities and L 2 -convergence rates of Markov semigroups[END_REF] for the density function f x . Before proving this equality, let us recall some facts on Hermite polynomials: Hermite polynomials (H n ) n∈Z + are defined as follows: for any n ∈ Z + , for any x ∈ R,H n (x) = (-1) n e

P 1 1 A

 1 (x) = n odd e -λn A F n (y)2λ π e -λy 2 2 dyF n (x).

  Hence, η Bm is indeed an eigenfunction of ∆, as claimed in Subsection 3.2. It remains us to compute the Poincaré constant C P . The family (η k 1 ,...,k d ) k 1 ,...,k d is a total orthonormal basis of L 2 (γ) (recalling that γ is Lebesgue's measure in our case) and are eigenvectors of ∆, by (39). Then (η k 1 ,...,k d ) k 1 ,...,k d are also eigenvectors of P t , for all t, respectively associated to the eigenvalues (e -λ k 1 ,...,k d t ) k 1 ,...,k d . Then, defining for any k 1 , . . . , k d ∈ N ηk 1 ,...,k d = η k 1 ,...,k d η Bm , one obtains, for any t ≥ 0, Pt ηk 1 ,...,k d = e λ 0 t η Bm P t [η k 1 ,...,k d ] = e -(λ k 1 ,...,k d -λ 0) t ηk 1 ,...,k d . Since the family (η k 1 ,...,k d ) k 1 ,...,k d is orthonormal with respect to Lebesgue's measure, the family (η k 1 ,...,k d ) k 1 ,...,k d is orthogonal with respect to the measure β Bm = η Bm • α Bm . Thus, the family ( ηk 1 ,...,k d ηk 1 ,...,k d L 2 (β Bm )

1 √

 1 2π e -x 22 , the family (H n / √ n!) n∈Z + is an orthonormal basis of L 2 (γ 0 ). Consider now the family (F n ) n∈Z + defined by Hence, by (43), the family (F n ) n∈Z + is an orthonormal basis of L 2 (γ), where Furthermore, by (44), for any n ∈ N, F n is a right eigenfunction for the Ornstein-Uhlenbeck semigroup (S t ) t≥0 , defined byS t f (x) = E f xe -λt + 1 -e -2λt λ Z , ∀f ∈ B(R), ∀x ∈ R,with Z be a standard Gaussian variable. More precisely, for any n ∈ Z + , for any t ≥ 0,S t F n (x) = e -λnt F n (x), ∀x ∈ R.Now, in order to prove the claim in Subsection 3.2, let us prove the following proposition: Proposition 3. Let d = 1. For any A ⊂ D and x ∈ D, P 1 1 A (x) =where (F n ) n∈Z + is defined as in (42). In particular, for any x ∈ D, δ x P 1 admits a density function with respect to the measure 2λ π e

				γ(dx) :=	λ 2π	e -λx 2 2 dx =	λ 2π	γ(dx).
				n odd	e -λn			A	F n (y)	2λ π	e -λy 2 2 dy F n (x),
				-λy 2
				2
			F n (x) =	H n ( √ √ n! λx)	, ∀n ∈ N, ∀x ∈ (0, +∞).	(42)
	Then, by (40),
	R	F n (x)F m (x)	λ 2π	e -λ x 2 2 dx =	R	H n ( √ √ n! λx)	H m ( √ √ m! λx)	1 √ 2π	e -( √	λx) 2 2	√	λdx = δ n,m .	(43)
	Now, by (41),
		F λH n ( √	λx) -λ × √ n! √	λxH n (	√	λx)
							= -λnF n (x).	(44)

n (x) -λxF n (x) =

Institut de Mathématiques, Université de Neuchâtel, Rue Emile-Argand, Neuchâtel, Suisse-2000 E-mail: william.ocafrain@hotmail.fr

The choice of the value 0.9 is totally arbitrary, any value smaller than 1 is suitable for the proof.

Acknowledgement. I am very grateful to the anonymous referee for his/her comments and questions, which allow me to better the paper. This research was supported by the Swiss National Foundation grant 200020 196999.

Appendix Spectral analysis for the Brownian motion in a hypercube

This part is dedicated to the spectral analysis of the scaled Brownian motion (X t ) t≥0 = ( √ 2B t ) t≥0 living in the hypercube C N := (-N, N ) d .

For any k 1 , . . . , k n ∈ N, let η k 1 ,...,k d defined by


Antonios Pavlidis

Eric Faehn

Marie-Minerve Louërat

Haralampos-G Stratigopoulos

BIST-Assisted Analog Fault Diagnosis

Fault diagnosis methodologies for analog circuits lag far behind those for their digital counterparts. In this paper, we show how the generic Symmetry-based Built-In Self-Test (BIST) (or SymBIST), originally proposed for defect-oriented postmanufacturing test and on-line test, can be seamlessly reused for the purpose of diagnosis. BIST can offer better insights into the circuit and, thereby, can assist diagnosis towards resolving ambiguity groups. Using SymBIST we demonstrate high diagnosis resolution and fast diagnosis cycle for an industrial Successive Approximation Register (SAR) Analog-to-Digital Converter (ADC).

I. INTRODUCTION

Fault diagnosis refers to the analysis performed to identify the root cause of failure that occurred in an Integrated Circuit (IC) either during manufacturing or in the field during normal operation following a customer return. The output is the isolation of the defect responsible for the failure and its localization at transistor-level.

Diagnosis is a crucial step in a product life-cycle. It can reveal important statistics, such as defect distribution and yield detractors. In this way, it provides valuable feedback for improving the design to prevent failure re-occurrence and expand the safety features, and also for evaluating and improving the quality of post-manufacturing tests.

In a first step, diagnosis generates a set of candidate defects based on diagnostic measurements or observed syndromes. In a second step, the IC is physically examined, for example using a thermal camera, to highlight anomalies in the operation and narrow down further the set of candidate defects. In the third and last step, the IC is submitted to Physical Failure Analysis (PFA) where de-layering and cross-sectioning of the die is performed to confirm the defect using imaging. Since PFA is destructive and irreversible, ideally the first step should pinpoint the actual defect. However, very often the first step results in an ambiguity group of candidate defects. According to industrial experience, its size should be less than 5-10 candidate defects to increase the PFA success rate.

Diagnosis metrics include: (a) resolution, i.e., the size of the ambiguity group; (b) accuracy, i.e., whether a reported candidate corresponds to the actual defect; and (c) diagnosis cycle time, i.e., the time required to complete the diagnosis since the number of diagnoses performed per week per design can be in the order of thousands [START_REF] Huang | Towards smarter diagnosis: A learning-based diagnostic outcome previewer[END_REF].

While for digital ICs there exist several in-house frameworks and commercial Electronic Design Automation (EDA) tools for diagnosis [START_REF] Huang | Towards smarter diagnosis: A learning-based diagnostic outcome previewer[END_REF], [START_REF] Maxwell | Cell-aware diagnosis: Defective inmates exposed in their cells[END_REF], for analog ICs there exists neither a commercial tool nor a standardized diagnosis approach. Analog fault diagnosis is still an ad-hoc, manual, tedious, and timeconsuming process very often resulting in large ambiguity groups and no actionable diagnosis information.

The traditional approach is a rule-based system which takes the form "IF symptom(s) THEN defect(s)" [START_REF] Erdogan | Diagnosis of assembly failures for system-inpackage RF tuners[END_REF]; however it is difficult to acquire the knowledge to build such a system.

Model-based diagnosis approaches have also been studied extensively. The idea is to first build a model linking diagnostic measurements to circuit parameters. Then, given the diagnostic measurements from the real failing device, the model is used to identify the faulty circuit parameter, as well as its deviation from the nominal value. The model can be constructed using nonlinear circuit equations [START_REF] Sen | Fault diagnosis for linear systems via multifrequency measurements[END_REF], sensitivity analysis [START_REF] Slamani | Analog circuit fault diagnosis based on sensitivity computation and functional testing[END_REF], regression [START_REF] Huang | Fault diagnosis of analog circuits based on machine learning[END_REF], or behavioral modeling [START_REF] Cota | A new adaptive analog test and diagnosis system[END_REF]. This approach can be used for diagnosis of soft faults, i.e., deviations of circuit parameters; however, it is recognized that hard faults, i.e., defects such as short-and open-circuits, are the main root cause of IC failures [START_REF] Ferguson | A CMOS fault extractor for inductive fault analysis[END_REF]. Besides, it is challenging to build a model that faithfully reproduces the analog IC behavior.

Perhaps the most common diagnosis approach is based on the use of a fault dictionary. Given a list of defects generated by Inductive Fault Analysis (IFA) [START_REF] Ferguson | A CMOS fault extractor for inductive fault analysis[END_REF], one defect is injected at a time in the netlist, and the IC is simulated to obtain the diagnostic measurement pattern. The fault dictionary contains the pairs of defects and diagnostic measurement patterns. Then, the diagnostic measurement pattern from the failed IC is mapped to one of the logged diagnostic measurement patterns of simulated defects based on some similarity metric. The mapping can be established after training a multi-class classifier using the fault dictionary dataset [START_REF] Somayajula | Analog fault diagnosis based on ramping power supply current signature clusters[END_REF], [START_REF] Huang | Diagnosis of local spot defects in analog circuits[END_REF]. The challenge with this approach is the long defect simulation time. However, the recently proposed analog defect simulators [START_REF] Sunter | Using mixed-signal defect simulation to close the loop between design and test[END_REF]- [START_REF] Gil | Assessing AMS-RF test quality by defect simulation[END_REF] can help speed up the fault dictionary generation as demonstrated in [START_REF] Melis | Automatic fault simulators for diagnosis of analog systems[END_REF].

In this paper, we investigate the use of Built-In Self-Test (BIST) in the context of analog fault diagnosis. Traditionally, BIST aims at adding on-chip test structures so to facilitate and reduce the cost of post-manufacturing test. BIST can offer better insights into the circuit and, thereby, can assist diagnosis towards resolving ambiguity groups [START_REF] Mir | Fault-based testing and diagnosis of balanced filters[END_REF]. We assume that the BIST infrastructure is already in place for post-manufacturing test and is reused for diagnosis "as is". We opt for using generic BIST virtually applicable to any analog IC class, such that the BIST-assisted diagnosis paradigm is also generic. Our starting point is the recently proposed generic Symmetry-based BIST (SymBIST) [START_REF] Pavlidis | Symmetry-based A/M-S BIST (SymBIST): Demonstration on a SAR ADC IP[END_REF], [START_REF] Pavlidis | SymBIST: Symmetry-based analog and mixed-signal built-in self-test for functional safety[END_REF], which has been demonstrated for defectoriented post-manufacturing test and on-line test. Herein, we demonstrate how SymBIST can be utilized for fault diagnosis too. Our case study is a 65nm 10-bit Successive Approximation Register (SAR) Analog-to-Digital Converter (ADC) IP by ST Microelectronics (STM) that comprises several hundreds of transistors.

The rest of the paper is structured as follows. In Section II, we provide a concise overview of the SymBIST operation. In Section III, we present the use of SymBIST for diagnosis. In Section IV, we present the case study. In Section V, we discuss defect modelling and simulation. In Section VI, we present the diagnosis results. Section VII concludes the paper.

II. SymBIST OVERVIEW

The central idea in SymBIST [START_REF] Pavlidis | Symmetry-based A/M-S BIST (SymBIST): Demonstration on a SAR ADC IP[END_REF], [START_REF] Pavlidis | SymBIST: Symmetry-based analog and mixed-signal built-in self-test for functional safety[END_REF] is to build invariant signals (or invariances) by monitoring internal nodes. An invariance is defined as a signal with a constant value irrespective of the input. It takes the form f k (V) = 0, where V is a vector containing the voltages of the monitored nodes and k denotes the invariance number. Then, one checker is used per invariance. The checker constructs and monitors the invariance and provides a pass or fail output, corresponding to invariance compliance or violation, respectively. Invariance violation flags an anomaly in operation pointing to defect detection. The convention used is that 0/1 corresponds to pass/fail.

In practice, a true invariance cannot exist due to noise and process, voltage, and temperature (PVT) variations. The invariance is a small peak-to-peak signal and the checker examines whether it stays within a tolerance window, e.g. |f k (V)| < δ, δ > 0. The tolerance window [-δ, δ] is set by simulating the complete nominal design, including the SymBIST infrastructure, while considering noise, Monte Carlo (MC) analysis, and power supply and temperature fluctuations. The tolerance window defines a trade-off between yield loss and test escapes. By enlarging it we reduce yield loss (or false positives) possibly at the expense of test escapes (or false negatives).

The invariances can be built by exploiting existing symmetries into the design, i.e. fully-differential paths, complementary signals, replicated sub-blocks, etc. Invariance building is a circuit-specific problem, but such symmetries exist virtually in every Analog and Mixed-Signal (A/M-S) IC rendering SymBIST a generic BIST paradigm.

Since the invariances hold true for any input, we can craft a test stimulus, possibly relying on some internal reconfiguration, such that the entire design is excited in very short time, thus concluding quickly on the presence or not of a defect. This property results in a small simulation time, enabling defect coverage evaluation of large A/M-S designs.

In practice, a small number of invariances can be used to cover the entire design thanks to feedback loops and block chains. Thus, the SymBIST area overhead, which is dominated by the checkers' area, can be very small.

Finally, since the invariances hold true for a running input, SymBIST can be reused for on-line concurrent error detection during normal operation [START_REF] Pavlidis | SymBIST: Symmetry-based analog and mixed-signal built-in self-test for functional safety[END_REF].

III. DIAGNOSIS USING SymBIST

Let C k (t) denote the time-varying checker output for the k-th invariance, k = 1, • • • , K, where K is the number of invariances. For a checker clocked at a frequency f clk , C k (t) is a bitstring with period 1/f clk . We consider the vector

C k [t i ] of n = T • f clk
bits, where T is the duration of the test stimulus and t i are sampling times at half-period, i.e.

t i = i•(1/2f clk ), i = 1, • • • , n.
If the invariance is permanently satisfied (violated), then C k [t i ] will be a vector of zeros (ones). It is likely, however, that a defect will cause the invariance to slide outside the tolerance window at specific clock cycles, in which case C k [t i ] will be a vector containing both zeros and ones. We use the vector C k [t i ] as an n-bit digital diagnostic measurement pattern.

This diagnostic measurement pattern is a function of the parameters of the SymBIST setup. In general, this set of parameters includes the primary test stimulus to the IC denoted by I pri , internally generated test stimuli denoted by I int , and the tolerance window defined by δ. We can consider unequal tolerance window limits δ -and δ + , i.e. the tolerance window becomes [δ -, δ + ]. A setup may involve also some internal reconfiguration denoted by R. Let

S = {I pri , I int , R , δ -, δ + } (1)
denote the set of parameters of the -th SymBIST setup, = 1, • • • , L, where L is the number of SymBIST setups.

In post-manufacturing test, the tolerance window sets a desired trade-off between test escapes and yield loss. In contrast, in diagnosis, the placement of the tolerance window can target uniquely exposing defects. For example, consider a defect that results in a time-varying invariant signal that is permanently above the upper limit of the tolerance window. By placing a higher upper limit at the average value of the invariant signal, we can obtain a vector C k [t i ] that toggles between 1 and 0, potentially offering a good diagnostic measurement pattern for this defect.

We follow a fault dictionary approach assuming a list of N f defects denoted by F j , j = 1, • • • , N f . The diagnostic measurement pattern for defect F j obtained by the k-th checker using the -th SymBIST setup is denoted by

C Fj , k [t i ].
Combining the diagnostic measurement patterns of K checkers monitoring the K invariances and considering L SymBIST setups, we obtain a diagnostic measurement pattern for defect

F j in the form of a K × L × n bitstring DM (F j ) = [C Fj ,1 1 , • • • , C Fj ,1 K , • • • , C Fj ,L 1 , • • • , C Fj ,L K ]. (2)
The diagnosis objective is that the Hamming Distance (HD) of the diagnostic measurement patterns of any two defects differs at least by 1, which can be expressed as

HD (DM (F a ), DM (F b )) > 1, a = b. (3)
An ambiguity group is a set of defects for which the HD of the diagnostic measurements of any pair of defects is 0.

In the diagnosis phase of faulty device X, the diagnostic measurement pattern DM X is first obtained and then matched with a row in the fault dictionary matrix    DM (F 1 ) . . .

DM (F N f )    , (4)
which has the diagnostic measurement pattern of defect j in row j. The row number points to the diagnosed defect. SymBIST setups are sequentially added by searching in the space of test stimuli, tolerance window limits, and reconfigurations, so as to enrich the digital diagnostic measurement pattern towards resolving the ambiguity groups.

IV. CASE STUDY

We use the same case study, i.e., a 10-bit SAR ADC IP by STM, employed in [START_REF] Pavlidis | Symmetry-based A/M-S BIST (SymBIST): Demonstration on a SAR ADC IP[END_REF], [START_REF] Pavlidis | SymBIST: Symmetry-based analog and mixed-signal built-in self-test for functional safety[END_REF]. Herein, we reintroduce the case study in Sections IV-A-IV-F for the purpose of completeness. The SymBIST setups for diagnosis are discussed in Section IV-G.

Previous BIST approaches for SAR ADCs are functional aiming at measuring Differential Non-Linearity (DNL) and Integral Non-Linearity (INL) [START_REF] Azais | Optimizing sinusoidal histogram test for low cost ADC BIST[END_REF], [START_REF] Feitoza | Static linearity BIST for Vcm-based switching SAR ADCs using a reduced-code measurement technique[END_REF]. They require long simulation times and, thus, large-scale defect simulation to generate the fault dictionary becomes cumbersome. In contrast, SymBIST offers a negligible simulation time and can be used in this context.

A. SAR ADC IP architecture

B. Invariances

The following invariances are identified covering the entire A/M-S part of the SAR ADC IP:

C. Test stimulus

The test stimulus has two parts, namely a static and a dynamic, and both parts can be robustly generated on-chip. The static part is simply a DC input to the ADC, denoted by ∆IN DC . The dynamic part is a set of digital test patterns that are applied sequentially to the inputs of the two SUBDACs. More specifically, we cycle through all 2 5 bit combinations at the inputs of each SUBDAC. The SUBDAC input sequence generator is shown in Fig. 3. A 5-bit digital counter is used to generate the incremental counting W< 0 : 4 > and a shuffling block is used to shuffle its outputs so as to generate the digital test stimuli Q< 0 : 4 > and Q< 5 : 9 > that cycle nonincrementally through all 2 5 bit combinations at the inputs of each SUBDAC. The shuffling block can be programmed to implement different cycles. Then, 10 2:1 multiplexers are used to switch during test mode the inputs of the SUBDACs, denoted now by Bnew< 0 : 4 > and Bnew< 5 : 9 >, from the SAR Logic outputs B< 0 : 4 > and B< 5 : 9 > to Q< 0 : 4 > and Q< 5 : 9 >, respectively.

D. Checker design

Invariances in Eqs. ( 5)-( 9) are of type V 1 + V 2 = c and are constructed and monitored using the checker in Fig. 4. It can be shown that

V o = A + G • (V 1 + V 2 )
, where A is a function of the DC components of V 1 and V 2 and G is a gain quantity. In error-free operation, V o is a DC signal with value A + G • c. A defect will shift the DC component of V o and/or will add an AC component, causing V o to slide outside the tolerance window [δ -, δ + ], and raising the output from 0 to 1. The switches are used to disconnect the checker when SymBIST is disabled or to set the checker into self-test mode so as to verify its operation before it is used for diagnosis [START_REF] Pavlidis | SymBIST: Symmetry-based analog and mixed-signal built-in self-test for functional safety[END_REF].

Fig. 5 shows the checker design for the invariance in Eq. [START_REF] Huang | Diagnosis of local spot defects in analog circuits[END_REF]. Note that this checker does not implement a tolerance window since the comparison is built from digital signals.

E. SymBIST test time

In total, 6 checkers are used, and the same test stimulus is used to check all invariances in parallel. The duration of the test stimulus is 2 5 clock cycles, which corresponds to a test time T = 2 5 • (1/f clk ) = 0.205µs per SymBIST setup.

F. Overheads

The test stimulus is composed of a DC static part and a digital dynamic part, while the checkers' output is an 1-bit response. Thus, SymBIST can be interfaced with a standard digital test access and control mechanism based on two external pins which is the minimum [START_REF] Sunter | Streaming access to ADCs and DACs for mixed-signal ATPG[END_REF].

The SymBIST infrastructure is totally non-intrusive to the SAR ADC operation. The checkers in Figs. 4 and5 have buffers as their first stage and the test stimulus generator in Fig. 3 is in a non-critical signal path. The zero performance penalty is confirmed by transistor-level DNL and INL simulations using a ramp histogram test [START_REF] Pavlidis | SymBIST: Symmetry-based analog and mixed-signal built-in self-test for functional safety[END_REF].

The SymBIST infrastructure area overhead is roughly 5%.

G. SymBIST for diagnosis

A SymBIST setup is composed of the triplet:

S = {∆IN DC , Q < 0 : 9 >, δ }, (11)
where denotes the setup index and δ is a 2 × 5 vector containing the lower and upper limits of the tolerance window of the checkers monitoring the invariances in Eqs. ( 5)- [START_REF] Somayajula | Analog fault diagnosis based on ramping power supply current signature clusters[END_REF].

Each checker provides a 32-bit diagnostic measurement pattern. Their concatenation results in a diagnostic measurement pattern of length 6 × 32 = 192 bits per SymBIST setup.

The first SymBIST setup is the same one used for postmanufacturing defect-oriented test [START_REF] Pavlidis | Symmetry-based A/M-S BIST (SymBIST): Demonstration on a SAR ADC IP[END_REF], [START_REF] Pavlidis | SymBIST: Symmetry-based analog and mixed-signal built-in self-test for functional safety[END_REF]. This setup uses ∆IN DC = 0.1V, the tolerance windows for the invariances in Eqs. ( 5)-( 9) are placed at ±5 • σ so as to have negligible yield loss, and the implemented shuffling in Fig. 3

is Q[j+5]=W[4- j] for SUBDAC1 and Q[j]=W[4-j] for SUBDAC2, j = 0, • • • , 4. For example, for SUBDAC1, the input sequence is {2 4 , 2 3 , 2 4 + 2 3 , 2 2 , 2 2 + 2 4 , • • • }.
Then, additional SymBIST setups are used to resolve ambiguity groups. These SymBIST setups have a different combination of DC fully-differential input, DAC input sequence, and tolerance window widths for each checker. Generating the fault dictionary matrix in Eq. ( 4) for one SymBIST setup takes roughly one day. Thus, using an optimization algorithm to search in the space of SymBIST setups is computationally expensive, even if this is an offline analysis. To this end, the next SymBIST setup is crafted based on the unresolved ambiguity groups so far with the aim to split them into smaller size groups, preferably uniquely detecting the defects composing them. For example, we can try a different input DAC sequence to split an ambiguity group with defects into the SUBDACs, we can increase (decrease) the upper (lower) tolerance window limit for an ambiguity group where the defects cause the invariance to exceed the nominal upper (lower) tolerance window limit, etc. We stop adding SymBIST setups when diagnosis resolution stops improving.

V. DEFECT MODELLING AND SIMULATION

We consider only the analog blocks and analog sections of the mixed-signal blocks of the SAR ADC IP. Digital blocks and digital sections of mixed-signal blocks can be tackled by digital diagnosis tools. Referring to Figs. 1 and2, purely digital blocks excluded from the analysis are the SAR Control, Phase Generator, SAR Logic, and RS Latch. Moreover, the digital sections of the two SUBDACs are excluded.

To automate defect simulation we rely on the mixedsignal defect simulator tool Tessent®DefectSim by Mentor®, A Siemens Business [START_REF] Sunter | Using mixed-signal defect simulation to close the loop between design and test[END_REF].

We make a single defect assumption and we adopt the default defect model in the tool [START_REF] Sunter | Using mixed-signal defect simulation to close the loop between design and test[END_REF]. In particular, for MOS transistors we use only gate open and drain-to-source short defects. We use a default drain-to-source short resistance of 10Ω [START_REF] Sunter | Using mixed-signal defect simulation to close the loop between design and test[END_REF]. Regarding gate opens, we use the modeling approach proposed in [START_REF] Esen | Effective DC fault models and testing approach for open defects in analog circuits[END_REF]. In particular, V GS is controlled by V DS with a gain coefficient that is set to a default value of 0.5 [START_REF] Sunter | Using mixed-signal defect simulation to close the loop between design and test[END_REF]. For resistors and capacitors, we consider ±50% variations. The total defect count is N f = 648.

VI. RESULTS

Fig. 6 shows a transistor-level transient simulation for two short defects within the SUBDAC1. The middle subplot shows the invariant signal in Eq. ( 6) for the defect-free and the two defective cases. For each defect we show the invariance for two different SymBIST setups 10 and 11, which differ only in the dynamic part of the test stimulus, i.e., the input DAC sequence. The top and bottom subplots show the diagnostic measurement pattern of each defect for these two setups. In SymBIST setup 10, for both defects the invariance violates the lower limit of the tolerance window, thus their diagnostic measurement pattern is the same, i.e., a vector of ones, and the two defects are inseparable. With SymBIST setup 11, the invariant signal for defect 2 now toggles across the lower limit, whereas the invariant signal for defect 1 is still permanently violated. Thus, the two diagnostic measurement patterns are now different and the defects are successfully distinguished. Fig. 7 shows the diagnosis results achieved with SymBIST following the strategy described in Section IV-G. For a given SymBIST setup, we show 3 bars corresponding to the percentage of uniquely diagnosed defects, i.e., ambiguity groups of size 1, the percentage of defects in ambiguity groups of size ≤ 5, and the percentage of defects in ambiguity groups of size ≤ 10. The inclusion of additional SymBIST setups stops when these three metrics saturate. As it can be seen, saturation occurs after using 17 different SymBIST setups. Using only the first SymBIST setup, 21% of the defects are uniquely diagnosed, 44% are in an ambiguity group of size ≤ 5, and 49% are in an ambiguity group of size ≤ 10. After using 17 different SymBIST setups, these metrics increase to 62%, 77%, and 80%, respectively. Fig. 8(a) sheds more light into the resultant ambiguity groups and their sizes. For example, 263 defects are uniquely diagnosed, there are 38/2 = 19 ambiguity groups of size 2, 1 ambiguity group of size 3, etc.

It should be noted that the percentages in Fig. 7 are given in terms of detectable defects using nominal tolerance windows placed at 5 • σ. The first SymBIST setup provides the best absolute defect coverage of 57%, while the cumulative defect coverage increases to 65% considering all 17 SymBIST setups. Note that in practice we are interested in the likelihoodweighted (L-W) defect coverage that considers the likelihood of defect occurrence [START_REF] Sunter | Using mixed-signal defect simulation to close the loop between design and test[END_REF], which is shown to be over 86% for the complete A/M-S part of the SAR ADC IP using the first SymBIST setup [START_REF] Pavlidis | Symmetry-based A/M-S BIST (SymBIST): Demonstration on a SAR ADC IP[END_REF], [START_REF] Pavlidis | SymBIST: Symmetry-based analog and mixed-signal built-in self-test for functional safety[END_REF]. Thus, we are diagnosing over 86% of occurring defective cases considering defect likelihoods.

Next, we aimed at further improving the diagnosis result in Fig. 7 achieved with 17 SymBIST setups. Our first idea was to extract two diagnostic measurement patterns per defect per invariance by considering separately the two tolerance window limits. If an invariance violates permanently either the upper or the lower limit, the diagnostic measurement pattern is in both cases a vector of ones. If we consider separately the two tolerance window limits, then we can distinguish two defects where one violates the upper and the other the lower limit. This effectively doubles the size of the diagnostic measurement pattern per checker. This new SymBIST setup uses the same test stimulus and comparison window as SymBIST setup 1 and is numbered 18 in Figs. 7 and8(b). As it can be seen, it improves the diagnosis resolution. For example, the number of uniquely diagnosed defects increases to 277 and the largest ambiguity group has now size 21.

Our second idea was to use the existing VREFP test pin, shown in the top-level architecture of Fig. 1. This pin outputs the VREF[32] reference voltage of the Reference Buffer used by the DAC during the conversion. We use the DC VREFP measurement as an analog diagnostic measurement. The nominal VREFP is 0.5V and we consider a nominal tolerance window set at ±5 • σ. Let VREFP(F j ) denote the diagnostic measurement value for defect F j . Considering a range of 20mV centered at each VREFP(F j ), a defect is detectable if its corresponding range does not overlap with the nominal tolerance window, and two defects are distinguishable if their corresponding ranges do not overlap. This test is numbered 19 in Figs. 7 and8(c). It improves further the diagnosis resolution since the number of uniquely diagnosed defects increases to 291 and the largest ambiguity group has now size 17.

Our third idea was to use internal re-configuration. We used the topology modification approach proposed in [START_REF] Coyette | Automatic generation of test infrastructures for analog integrated circuits by controllability and observability co-optimization[END_REF]. The underlying idea is to connect Pull-Up (PU) PMOS and Pull- Since the Bandgap provides biases for all blocks inside the SAR ADC, the effect of a defect in the Bandgap can propagate to the outputs of the checkers in a complex way. This test is named "PX re-config" and is numbered 20 in Figs.

VII. CONCLUSIONS

We presented the use of SymBIST for analog fault diagnosis. Different SymBIST setups were employed to extract rich digital diagnostic measurement patterns capable of resolving defect ambiguities. SymBIST applied for diagnosis of an industrial SAR ADC IP resulted in high diagnosis resolution, i.e. 73% correct diagnosis of defects and over 97% defects in ambiguity groups of maximum size 5, while the largest ambiguity group has size 9. SymBIST offers a fast diagnosis cycle in the order of a few µs.

Fig. 1 :

 1 Fig. 1: Top-level architecture of 10-bit SAR ADC IP by STM.

Fig. 2 :

 2 Fig. 2: Top-level architecture of SARCELL block.

Fig. 1

 1 Fig.1shows the top-level architecture. The circuit accepts a fully-differential input ∆IN = IN+ -IN-. CLK is the master clock with frequency f clk = 156 MHz. The 10-bit digital code during the conversion is denoted by B< 0 : 9 >, and at the end of the conversion it is latched at the output D< 0 : 9 >. The SARCELL, whose top-level architecture is shown Fig.2, is the main block which implements the SAR algorithm.

Fig. 3 :

 3 Fig. 3: On-chip generation of dynamic test stimulus. M+ + M-= VREF[32] (5) L+ + L-= VREF[32] (6) DAC+ + DAC-= 2V cm (7) LIN+ + LIN-= 2V cm2 (8) Q+ + Q-= VDD (9) sgn (Q+ -Q-)sgn (LIN+ -LIN-) = 0 (10) where VREF[32] is the largest reference voltage created by the Reference Buffer, M+ and M-are complimentary comparison levels created by SUBDAC1 using the reference voltages VREF< 0 : 32 > while converting the 5 Most Significant Bits (MSBs) B< 5 : 9 >, L+ and L-are complimentary comparison levels created by SUBDAC2 using the reference voltages VREF< 0 : 32 > while converting the 5 Least Significant Bits (LSBs) B< 0 : 4 >, DAC+ and DAC-are the fully-differential outputs of the DAC, LIN+ and LIN-are the fully-differential outputs of the pre-amplifier, Q+ and Qare the complimentary digital outputs of the RS Latch, V cm is the common mode used inside the DAC, and V cm2 is the common mode at the outputs of the pre-amplifier. The signals composing the invariances are annotated in Figs. 1 and 2.

Fig. 4 :

 4 Fig. 4: Checker design for the invariances in Eqs. (5)-(9).

Fig. 5 :

 5 Fig. 5: Checker design for the invariance in Eq. (10).

Fig. 6 :

 6 Fig. 6: Transient simulation of SymBIST invariance in Eq. (6) for two defects considering two different SymBIST setups.

Fig. 7 :

 7 Fig. 7: Percentage of defects uniquely diagnosed and in ambiguity groups of sizes ≤ 5 and ≤ 10.

( a )

 a SymBIST setups 1-17. (b) SymBIST setups 1-18. (c) SymBIST setups 1-18 and VREFP. (d) SymBIST setups 1-18, VREFP, and PX re-config.

Fig. 8 :

 8 Fig. 8: Gradual diagnosis resolution improvement.Down (PD) NMOS transistors that bring an internal node to VDD or ground, respectively. This changes the topology of the circuit and potentially can expose defects differently. First, we observed that the two largest ambiguity groups of sizes 16 and 17 in Fig.8(c) concern defects inside the Bandgap. Thus, we apply topology modifications only in the Bandgap. More specifically, we inserted 13 PU and PD transistors enabling 13 re-configurations, where in each re-configuration only one PU or PD transistor is activated. For each re-configuration, we read out the diagnostic measurements using SymBIST setup 1. Since the Bandgap provides biases for all blocks inside the SAR ADC, the effect of a defect in the Bandgap can propagate to the outputs of the checkers in a complex way. This test is named "PX re-config" and is numbered 20 in Figs.7 and 8(d). By adding this SymBIST variant, we were able to uniquely diagnose 73% of the defects, while the largest ambiguity group has size 9 and all but 8 + 9 = 17 defects, i.e., 97.4% of the defects, are in ambiguity groups of size ≤ 5, which overall is an excellent diagnosis resolution.Finally, in total we used L = 18 + 13 = 31 SymBIST setups, thus maximum diagnosis time is 0.205 • 31 = 6.35 µs, plus the time of a DC measurement from the VREFP pin.

  7 and 8(d). By adding this SymBIST variant, we were able to uniquely diagnose 73% of the defects, while the largest ambiguity group has size 9 and all but 8 + 9 = 17 defects, i.e., 97.4% of the defects, are in ambiguity groups of size ≤ 5, which overall is an excellent diagnosis resolution.Finally, in total we used L = 18 + 13 = 31 SymBIST setups, thus maximum diagnosis time is 0.205 • 31 = 6.35 µs, plus the time of a DC measurement from the VREFP pin.

ACKNOWLEDGMENTS

The authors would like to acknowledge the financial support in the framework of the ANR EDITSoC project with N o ANR-17-CE24-0014-02.