Mamadou Mboup

Cédric Join
email: cedric.join@cran.uhp-nancy.fr

Michel Fliess
email: michel.fliess@polytechnique.edu

A revised look at numerical differentiation with an application to nonlinear feedback control

We are presenting new and efficient methods for numerical differentiation, i.e., for estimating derivatives of a noisy time signal. They are illustrated, via convincing numerical simulations, by the analysis of an academic signal and by the feedback control of a nonlinear system.

I. INTRODUCTION

A. Numerical differentiation

Numerical differentiation, i.e., the derivatives estimation of noisy time signals, is an important but difficult ill-posed theoretical problem. It has attracted a lot of attention in many fields of engineering and applied mathematics (see, e.g., in the recent control literature [START_REF] Chitour | Time-varying high-gain observers for numerical differentiation[END_REF], [START_REF] Dabroom | Discrete-time implementation of high-gain observers for numerical differentiation[END_REF], [START_REF] Duncan | Numerical differentiation and parameter estimation in higher-order linear stochastic systems[END_REF], [START_REF] Ibrir | Online exact differentiation and notion of asymptotic algebraic observers[END_REF], [START_REF] Ibrir | Linear time-derivatives trackers[END_REF], [START_REF] Ibrir | A numerical procedure for filtering and efficient high-order signal differentiation[END_REF], [START_REF] Kelly | Global regulation of flexible joint robots using approximate differentiation[END_REF], [START_REF] Levant | Robust exact differentiation via sliding mode technique[END_REF], [START_REF] Levant | Higher-order sliding modes, differentiation and output-feedback control[END_REF], [START_REF] Su | A simple improved velocity estimation for low-speed regions based on position measurements only[END_REF], and the references therein). Our purpose here is to improve a new approach which started in [START_REF] Fliess | Compression différentielle de transitoires bruités[END_REF], [START_REF] Fliess | Control via state estimations of some nonlinear systems[END_REF], and in [START_REF] Fliess | Estimation des dérivées d'un signal multidimensionnel avec applications aux images et aux vidéos[END_REF], [START_REF] Fliess | Analyse et représentation de signaux transitoires : application à la compression, au débruitage et à la détection de ruptures[END_REF], [START_REF] Fliess | Closed-loop faulttolerant control for uncertain nonlinear systems[END_REF], for solving various questions in control and in signal and image processing. Let us briefly describe our differentiators which are obtained via iterated time integrals of the noisy signal.

B. Short summary of our approach

Start with the first degree polynomial time function p 1 (t) = a 0 + a 1 t, t ≥ 0, a 0 , a 1 ∈ R. Rewrite thanks to classic operational calculus (cf. [START_REF] Yosida | Operational Calculus: A Theory of Hyperfunctions[END_REF]) p 1 as P 1 = a0 s + a1 s 2 . Multiply both sides by s 2 :

s 2 P 1 = a 0 s + a 1 (1)
Take the derivative of both sides with respect to s, which corresponds in the time domain to the multiplication by -t (cf. [START_REF] Yosida | Operational Calculus: A Theory of Hyperfunctions[END_REF]):

s 2 dP 1 ds + 2sP 1 = a 0 (2)
The coefficients a 0 , a 1 are obtained via the triangular system of equations ( 1)- [START_REF] Chitour | Time-varying high-gain observers for numerical differentiation[END_REF]. We get rid of the time derivatives, i.e., of sP 1 , s 2 P 1 , and s 2 dP1 ds , by multiplying both sides of equations ( 1)-( 2) by s -n , n ≥ 2. The corresponding iterated time integrals are low pass filters which attenuate the corrupting noises, which are viewed as highly fluctuating phenomena (see [START_REF] Fliess | Analyse non standard du bruit[END_REF] for more details). A quite short time window is sufficient for obtaining accurate values of a 0 , a 1 .

The extension to polynomial functions of higher degree is straightforward. For derivatives estimates up to some finite order of a given smooth function f : [0, +∞) → R, take a suitable truncated Taylor expansion around a given time instant t 0 , and apply the previous computations. Utilizing sliding time windows permit to estimate derivatives of various orders at any sampled time instant.

C. Difficulties and improvements

The above method becomes more and more ill-conditioned for higher order truncation of the Taylor expansion. This is a major impediment for obtaining good estimates for higher order derivatives in a noisy setting. Using elimination techniques and Jacobi orthogonal polynomials (see, e.g., [START_REF] Abramowitz | Handbook of Mathematical Functions[END_REF], [START_REF] Szegö | Orthogonal Polynomials[END_REF]), we propose here individual and independent derivatives estimators for each given order. A judicious choice of the point at which the derivatives are estimated in each sliding time window permits to take advantage of the extramodeling capability afforded by a higher order truncation.

D. Organization of our paper

Section II discusses the mathematical foundations of our differentiators. The first illustration in Section III applies the above techniques for estimating the derivative of a noisy academic signal. The second illustration in Section IV, which is borrowed from [START_REF] Fan | Observer design for systems with multivariable monotone nonlinearities[END_REF], deals with the nonlinear feedback control of a DC motor joined to an inverted pendulum through a torsional spring. We provide in both cases convincing computer simulations 1 .

II. DERIVATIVES ESTIMATION

Let y(t) = x(t) + n(t) be a noisy observation on a finite time interval of a real-valued signal x(t), the derivatives of which we want to estimate. Assume that x(t) is analytic on this time interval. Consider without any loss of generality the convergent Taylor expansion x(t) = i 0 c i

t i i! at t = 0. The truncated Taylor expansion x N (t) = N i=0 c i t i i! satisfies the differential equation d N +1
dt N +1 x N (t) = 0. It reads in the operational domain as

s N +1 xN (s) = s N x N (0) + s N -1 ẋN (0) . . . + x (N ) N (0) (3)
where xN (s) is the operational analog of x N (t). To ease the notation, we subsequently ignore the argument s.

A. Simultaneous estimation

Replace x N (t) by the noisy observed signal y(t). Then the estimates x(i) N (0) of the derivatives at the origin x

(i) N (0) △ = d i dt i x(t) t=0
are directly obtained from the linear triangular system of equations (see [START_REF] Fliess | Compression différentielle de transitoires bruités[END_REF], [START_REF] Fliess | Control via state estimations of some nonlinear systems[END_REF])

s -ν d m ds m Ò xN (0)s N + . . . + x(N-1) N (0)s + x(N) N (0) Ó = s -ν d m ds m Ò s N+1 ŷÓ (4)
m = 0, . . . , N , where ν N + 2 ensures strict properness.

To obtain the numerical estimates, it suffices to express (4) back in the time domain, using the classical rules of operational calculus ( [START_REF] Yosida | Operational Calculus: A Theory of Hyperfunctions[END_REF]). Denote by T the estimation time.

We end up with the following closed form expression

P ν (T )      xN (0) xN (0) . . . x(N) N (0)      = T 0 Q ν (τ )y(τ )dτ (5)
where the nonzero entries of the triangular matrix P ν (T ) are given, for i = 0, . . . , N , j = 0, . . . , Ni, by

{P ν (T )} ij = (N -j)! (N -i -j)! T ν-N +i+j-1 (ν -N + i + j -1)! and {Q ν (τ )} i = i ℓ=0 q i,ℓ (T -τ ) ν-N -2-ℓ τ i-ℓ with q i,ℓ = i ℓ (N + 1)! (N + 1 -ℓ)! (-1) i-ℓ (ν -N -2 -ℓ)!
Finally, for each estimation time interval of length T , I T t+ = [t, t + T ], we obtain the derivatives estimates at time t by replacing y(t) in ( 5) by y(t+τ ). These estimates are however not causal. To obtain causal estimates, i.e., the estimates at time t, based on the signal observation in

I T t-= [t -T, t], it suffices to replace y(t + τ ) by -y(t -τ ), τ ∈ [0, T ].

B. Individual estimation

The matrix P ν (T ) in ( 5) is in general ill-conditioned, and yields therefore poor estimates especially in a noisy setting. A solution to this problem is to obtain an independent estimator for each order of derivation2 . Reconsider (3). Examine for 0 n N the n th order derivative. Annihilate the remaining coefficients x (j) N (0), j = n by multiplying by linear differential operators of the form

Π N,n κ = d n+κ ds n+κ • 1 s • d N -n ds N -n κ 0
It yields the following estimator for x (n) (0)

x(n) N (0) s ν+n+κ+1 = (-1) n+κ (n + κ)!(N -n)! 1 s ν Π N,n κ s N +1 x (6)
which is strictly proper whenever ν is of the form ν = N + 1 + µ, µ 0. We obtain a family of strictly proper estimators which is parametrized by κ, µ and N . Write

x(n) 0 (κ, µ; N ) the corresponding estimator. If N = n the differential operator Π N,n κ reduces to Π n κ = d n+κ ds n+κ • 1 s : we will then use the simplified notation x(n) 0 (κ, µ).
The following result is straightforward:

Lemma 1: For any N n and µ 0, x(n) 0 (κ, µ; N ) in (6) belongs to the set

F = span Q {x (n) 0 (κ ℓ , µ ℓ ), ℓ = 0, . . . , min (n + κ, N -n)} (7)
where κ ℓ = κ + Nnℓ and µ ℓ = µ + ℓ.

Proof: Set q = Nn and p = n + κ. The proof follows by direct inspection, upon writing Π N,n κ (s N +1 x) in the form

Π N,n κ (s N+1 x) = q i=0 q i (q + 1)! (q + 1 -i)! d p ds p Ò s q-i (s n x) (q-i) Ó = q i=0 min (p,q-i) j=i ai,js q-j Π n κ+q-j {s n+1 x} (8)
where

a i,j = q i p j -i (q + 1)! (q + 1 -i)(q -j)!
This lemma shows that an n th -order truncated Taylor expansion is appropriate for estimating the n th -order derivative. 3A common way for estimating the derivatives of a signal is to resort to a least squares polynomial fitting on an interval and then take the derivatives of the resulting polynomial function. The estimators derived here rely however on a different approach: the derivatives are estimated pointwise. This depature is furthermore apparent with the developments of the preceding subsection. Nonetheless, a least squares interpretation may be attached to our approach, as shown below.

C. Least squares interpretation

Start with the estimation of the first order derivative.

a) N = 1: With ν = µ + 2, equation (6) becomes 1 s µ+κ+4 x0(κ, µ) = (-1) κ+1 (κ + 1)! x(κ+1) s µ+1 + (κ + 1) x(κ) s µ+2 (9)
It reads in the time domain:

x0 (κ, µ) = µ + 2 T µ + κ + 3 κ + 1 1 0 p(τ )τ κ (1 -τ ) µ y(T τ )dτ (10
) where p(τ ) = (µ+ κ+ 2)τ -(κ+ 1), and T is the estimation time (the estimation interval is I T 0+ ). We replace of course the signal x by its noisy observation y.

Consider now the Jacobi orthogonal polynomials (cf. [START_REF] Szegö | Orthogonal Polynomials[END_REF], [START_REF] Abramowitz | Handbook of Mathematical Functions[END_REF]) {P κ,µ i (t)} i 0 , associated to the weight function

w κ,µ (t) = t κ+1 (1 -t) µ+1 on the interval [0, 1].
Lemma 2: The first order derivative estimate, given in equation [START_REF] Fliess | Closed-loop faulttolerant control for uncertain nonlinear systems[END_REF], reads as

x0 (κ, µ) = 1 P κ,µ 0 (t) 2 1 0 P κ,µ 0 (τ )w κ,µ (τ ) ẏ(T τ )dτ △ = P κ,µ 0 (τ ), ẏ(T τ ) κ,µ P κ,µ 0 (τ ) 2 (11)
Proof: Observe that P κ,µ 0 (t) = 1 and p(τ

)τ κ (1 -τ ) µ = -d dτ {τ κ+1 (1 -τ ) µ+1 }.
Integration by parts shows that the integral in (10) reduces to T P κ,µ 0 (τ ), ẏ(T τ ) κ,µ . The equality P κ,µ 0 2 = (µ+1)!(κ+1)! (κ+µ+3)! completes the proof. This estimate of the first order derivative appears as the orthogonal projection of the unobserved signal derivatives, on P κ,µ 0 (t). Expanding ẋ(T τ ), τ ∈ [0, 1] in the basis of the Jacobi polynomials ẋ(T τ ) = a 0 P κ,µ 0 (τ ) + a 1 P κ,µ 1 (τ ) + a 2 P κ,µ 2 (τ ) + • • • (12) shows that x0 (κ, µ), which coincides with a 0 , actually turns out to correspond to an estimate of ẋ(T τ 0 ) for some τ 0 > 0. Using a first order approximation of ẋ(T τ ) ẋ(T τ 0 ) ≈ a 0 P κ,µ 0 (τ 0 ) + a 1 P κ,µ 1 (τ 0 ) = a 0 allows one to identify τ 0 as the solution of P κ,µ 1 (τ 0 ) = 0. This value of τ 0 , given by

τ 0 = κ + 2 µ + κ + 4 (13)
is "experimentally" confirmed by the numerical simulations below. The resulting derivative estimation is thus subject to a time delay. b) N = 2: It allows to avoid such a delay, which may not be tolerable in real-time processing. Equation (8) yields x0 (κ, µ; 2)

s µ+κ+5 = (-1) κ+1 (κ + 1)! x(κ+2) s µ+1 + (κ + 2) x(κ+1) s µ+2 + (κ + 3)
x(κ+1)

s µ+2 + (κ + 1) x(κ) s µ+3 (14)
It reads in the time domain after some algebraic manipulations:

x0 (κ, µ; 2) = P κ,µ 0 (τ ), ẏ(T τ ) κ,µ P κ,µ 0 2
+ P κ,µ 1 (0)

P κ,µ 1 (τ ), ẏ(T τ ) κ,µ P κ,µ 1 2 (15)
where P κ,µ 1 (0) = -τ 0 = -κ+2 µ+κ+4 . We therefore deduce that x0 (κ, µ; 2) corresponds to an estimate of the first order derivative ẋ(t) at t = 0. The estimation of ẋ(t) from a second order truncation of the Taylor expansion is therefore delay free. We now show how this interpretation can be exploited to obtain better estimates. According to Lemma 1 and using [START_REF] Ibrir | Online exact differentiation and notion of asymptotic algebraic observers[END_REF], it is easy to verify that the relation

x0 (κ, µ; 2) = λ 0 x0 (κ, µ + 1) + λ 1 x0 (κ + 1, µ) (16)
holds for λ 0 = κ + 3 and λ 1 = -(κ + 2). Let us now extend the set F in [START_REF] Fliess | Estimation des dérivées d'un signal multidimensionnel avec applications aux images et aux vidéos[END_REF], by allowing the coefficients of the linear combinations therein to be real, rather than rational. In particular, given a point τ 1 ∈ [0, 1], one may always choose the λ i 's such that (16) becomes

xτ1 (κ, µ; 2) = P κ,µ 0 (τ ), ẏ(T τ ) κ,µ P κ,µ 0 2 + P κ,µ 1 (τ 1 ) P κ,µ 1 (τ ), ẏ(T τ ) κ,µ P κ,µ 1 2 (17)
In this case, xτ1 (κ, µ; 2) will represent the estimate of ẋ(T τ 1 ), obtained from the truncated Taylor expansion

x 2 (t) = x(τ ) + ẋ(τ )(t -τ ) + ẍ(τ ) 2 (t -τ ) 2 = x 2 (0) + [ ẋ(τ ) - τ ẍ(τ )]t + ẍ(τ )
2 t 2 at τ = T τ 1 . A direct verification will show that the values for λ 0 and λ 1 , associated to τ 1 are given by λ 0 = (κ + 3) -(µ + κ + 5)τ 1 and λ 1 = 1λ 0 .

Let us now turn to the question pertaining to the selection of a value for τ 1 . Expansion [START_REF] Fliess | Flatness and defect of non-linear systems: introductory theory and examples[END_REF] shows that a good choice for τ 1 is given by the smallest (resp. largest) root of the polynomial P κ,µ 2 when the estimation interval is I T t+ (resp. I T t-). Indeed, choosing τ 1 as a zero of P κ,µ 2 annihilates the contribution of the orthogonal projection of the signal on P κ,µ 2 in the estimation error. On the other hand, recall that τ 1 (resp. 1τ 1 for I T t-) represents a delay in the estimation. Choosing the smallest (resp. largest) root thus translates to a lower delay. Note also that the delay τ 1 is smaller than τ 0 , since the zeros of P κ,µ 2 and P κ,µ 1 interlace. The above analysis may be easily generalized to the n th order derivative estimation, n ≥ 2.

III. FIRST ILLUSTRATION: DERIVATIVE OF A NOISY

SIGNAL

Let y(t) = x(t)+n(t), 0 ≤ t ≤ 5, be a noisy measurement of the signal

x(t) = tanh(t -1) + e -t/1.2 sin(6t + π)
The noise level, measured by the signal to noise ratio in dB, i.e., SN R = 10 log 10 1). In all the subsequent numerical simulations, the integrals are computed via the classical trapezoidal rule.

È |y(ti)| 2 È |n(ti)| 2 , corresponds to SN R = 25 dB (see Figure
Begin with the first order derivative and N = 1. The estimates x0 (κ, µ), obtained from [START_REF] Fliess | Closed-loop faulttolerant control for uncertain nonlinear systems[END_REF] derivative of the noise-free signal is also displayed (dashed line) in order to gauge the estimation accuracy. For this, we have introduced a shift corresponding to the delay τ 0 in [START_REF] Fliess | Control via state estimations of some nonlinear systems[END_REF]. Observe how this predicted value of τ 0 fits with the experiment. Of course with N = 1, the truncated Taylor series model is linear, resulting in poor estimates on the intervals where the signal's dynamic is strong. For high signal-to-noise ratio, the estimates may be improved by reducing the estimation time T . Alternatively, one may consider a richer signal model, e.g., with N = 2. This is the case of the next experiment. We now consider the sliding windows I T t-, with T = 110T s . The estimates x0 (κ, µ; 2), based on ( 14), [START_REF] Ibrir | Linear time-derivatives trackers[END_REF], are plotted (solid line) in Figure 3 below, for κ = µ = 0.

There is no estimation delay, as expected. However, the performance significantly degrades as compared to the preceding results although the signal model is more precise. If we now relax this delay-free constraint, it becomes possible to take advantage of the more flexible second order model for the signal. This is illustrated in the following simulation (see Figure 4), where we keep the same settings for T , κ and µ. The solid line curve in Figure 4 from ( 16)-( 17) where τ 1 is the largest root of P κ,µ 2 . Using the same idea with N = 3, we obtain the estimates x0 (κ, µ; 3), plotted with dashed line. The associated estimator reads as

x0 (κ, µ; 3) = 2 i=0 λ i x0 (κ + 2 -i, µ + i)
where the real coefficients λ i , i = 0, 1, 2 are chosen so that x0 (κ, µ; 3) corresponds to ẋ(τ 2 T ), with τ 2 being a root of P κ,µ 3 . If κ = µ, which is the case here, then τ 2 = 0.5 is a common root of all Jacobi polynomials of odd degree. The dashed line curve was obtained with this choice of τ 2 .

IV. SECOND ILLUSTRATION: NONLINEAR FEEDBACK CONTROL

A. System description

Consider with [START_REF] Fan | Observer design for systems with multivariable monotone nonlinearities[END_REF] the mechanical system, which consists of a DC-motor joined to an inverted pendulum through a torsional spring:

Jm θm = κ θ l -θm ¡ -B θm + Kτ u J l θl = -κ θ l -θm ¡ -mgh sin(θ l ) y = θ l
θ m and θ l represent respectively the angular deviation of the motor shaft and the angular position of the inverted pendulum. J m , J l , h, m, κ, B, K τ and g are physical parameters which are assumed to be constant and known. This system, which is linearizable by static state feedback, is flat (cf. [START_REF] Fliess | Flatness and defect of non-linear systems: introductory theory and examples[END_REF], [START_REF] Sira-Ramírez | Differentially Flat Systems[END_REF]); y = θ l is a flat output, which is measured.

B. Control design

Asymptotic tracking of a given smooth reference trajectory y * (t) = θ * l (t) is achieved by the feedback controller

u = 1 Kτ Jm κ J l v + κÿe + mgh(ÿe cos ye -( ẏe) 2 sin ye) ¡ +J l ÿe + mgh sin ye B κ J l y (3)
e + κ ẏe + mgh ẏe cos ye

¡ where v = [y * (t)] (4) -γ4(y (3)
e -[y

* (t)] (3) ) -γ3(ÿe -ÿ * (t)) -γ2( ẏe -ẏ * (t)) -γ1(ye -y * (t))
The subscript "e" denotes the estimated value. The design parameters γ 1 , ..., γ 4 are chosen so that the resulting characteristic polynomial is Hurwitz. The state θ m is estimated via [θ m ] e = 1 κ J l ÿe + mgh sin y e + y e

C. Numerical simulations

The physical parameters have the same numerical values as in [START_REF] Fan | Observer design for systems with multivariable monotone nonlinearities[END_REF]:

J m = 3.7 × 10 -3 kgm 2 , J l = 9.3 × 10 -3 kgm 2 , h = 1.5 × 10 -1 m, m = 0.21 kg, B = 4.6 × 10 -2 m, K τ = 8 × 10 -2 NmV -1 .
The numerical simulations (see also [START_REF] Fliess | Non-linear estimation is easy[END_REF]), which are much better than in [START_REF] Fliess | Control via state estimations of some nonlinear systems[END_REF], where less efficient differentiators were employed, are presented in Figures 5. Robustness has been tested with an additive white Gaussian noise N (0; 0.01) on the output y (this noise level is quite relevant in such applications). Note that the off-line estimations of ÿ and θ m , where a "small" delay is allowed, are more accurate than the on-line estimation of ÿ.

V. CONCLUSION

The basic elements of our differentiators are essentially integrators, the advantage of which is twofold: easy implementation and good robustness with respect to noise corruption. Our estimators may be given a least squares interpretation (see [START_REF] Mboup | Parameter estimation via differential algebra and operational calculus[END_REF] for more details), although our approach is based on quite different mathematical ingredients, which are mainly of algebraic flavor. This interpretation is a key point leading to improved numerics.

5 Fig. 1 .

 51 Fig. 1. Noisy observation signal, SN R = 25dB.

Fig. 2 .

 2 Fig. 2. Estimation of the signal derivative: N = 1.

Fig. 3 .

 3 Fig. 3. Estimation of the signal derivative: N = 2, no delay

Fig. 4 .

 4 Fig. 4. Estimation of the signal derivative: N = 2, with delay τ 1 .

Fig. 5 .

 5 Fig. 5. Simulation results for DC-motor

Interested readers may obtain the corresponding computer programs from one of the authors (cedric.join@cran.uhp-nancy.fr). Reference[START_REF] Fliess | Non-linear estimation is easy[END_REF], from which the second example is taken, contains many more applications to various topics in nonlinear control. Most useful discussions and comparisons may be found in[START_REF] Nöthen | Beiträge zur Rekonstruktion nicht direkt gemessener Größen bei der Silizium-Einkristallzüchtung nach dem Czochralski-Verfahren[END_REF] where an interesting concrete casestudy is analyzed.

The system[START_REF] Fan | Observer design for systems with multivariable monotone nonlinearities[END_REF] being triangular, a closed-form expression for the estimator of x (i) (t) may be derived from it. The corresponding solutions would however exhibit the same sensitivity to noise perturbations.

The authors would like to thank A. Sedoglavic for bringing this question to their attention.


Pierre Debs
email: pierre.debs@iecn.u-nancy.fr

Mihai Gradinaru
email: mihai.gradinaru@iecn.u-nancy.fr

Penalization for birth and death processes

Keywords: Birth and death Markov processes, penalization, sojourn time, Dynkin's formula, random walk, Brownian motion with drift, Bessel chain and process, change of probability. 2000 Mathematics Subject Classification: 60J80, 60J10, 60J27, 60J75, 60B10, 60G42, 60G44

In this paper we study a transient birth and death Markov process penalized by its sojourn time in 0. Under the new probability measure the original process behaves as a recurrent birth and death Markov process. We also show, in a particular case, that an initially recurrent birth and death process, behaves as an transient birth and death process after penalization with the event that it can reach zero in infinite time. We illustrate some of our results with the Bessel random walk example.

Introduction

Consider a transient Markov chain with positive states. Is it possible to force the chain to have an infinite number of visits at the state 0? Does one have a reasonable manner to do this? Conditionally to this behaviour, is the resulting chain recurrent or transient? These are some natural questions when studying penalization for Markov chains, that is to modify their laws in order to alter some of their properties. In the present paper by penalization one understands to condition by an event with probability zero, in our case, the event that the transient Markov chain returns infinitely often at 0. The penalization of a probability measure (for instance, the distribution of some Markov process) by some appropriate weights is the main tool of some classical probabilistic ideas as Doob's h-transform, Feynman-Kac transform or Girsanov transform. For instance, in the latter case, an exponential martingale occurs as weight factor and, under some natural conditions, a new probability measure is well defined. Since, dividing a probability distribution by a family of weights does not create necessarily a probability family of distributions, one needs to made some limiting procedure (often as the parameter goes to infinity). The main features of the penalization procedure have been pointed out in a series of papers by Roynette,Vallois,Yor (see [START_REF] Roynette | Pénalisations et quelques extensions du théorème de Pitman, relatives au mouvement brownien et à son maximum unilatère[END_REF] for a survey). Essentially, one considers continuous parameter Markov processes (Brownian motion or Bessel processes) and one penalizes by some function of local time, maximum, one-sided maximum etc.

For instance, one considers (Ω, {F t } 0≤t≤∞ , {R t } t≥0 , ) the canonical Bessel process of index α ∈ (-1, 0), and one denotes {L t } t≥0 its local time at level 0. Then one can be interested on the limit

lim t→∞ ¡ 0 [¢ As h(L t )] ¡ 0 [h(L t )] = Q (h) 0 (A s ) = ¡ 0 ¢ As M (h)

s

, A s ∈ F s (s fixed), h : £ + → £ + , 1 and one proves that the process {M (h) s } s≥0 is a martingale. Then one studies the canonical process {R t } t≥0 under the new probability Q (h) 0 and some interesting properties of this process are pointed out.

The discrete parameter counterpart of the same problems was performed very recently by Debs [START_REF] Debs | Pénalisation de la marche aléatoire standard par une fonction du maximum unilatère, du temps local en zéro et de la longueur des excursions[END_REF] for random walks. For instance, if (Ω, {F n } 0≤n≤∞ , {S n } n≥0 , ) is the canonical standard random walk and if one calls the local time at 0 the following sequence 0 = 0, n+1 = n + ¢ {Sn=0} , n ≥ 0, then one can be interested on the limit

lim p→∞ ¡ ¢ An (h( p )¢ {Sp>0} + g( p )¢ {Sp<0} ) ¡ h( p )¢ {Sp>0} + g( p )¢ {Sp<0} = Q (h,g) (A n ) = E[¢ An M (h,g) n ],
A n ∈ F n (n fixed), g, h : → £ + .

Again, one proves that {M (h,g) n } n≥0 is a martingale and one studies the canonical sequence {S n } n≥0 under the probability Q (h,g) . Let us notice that in almost of cases considered in the cited works, one uses as important tools precise informations on the processes : explicit laws of hitting times, explicit martingales related to considered processes and so on. This allows to perform exact computations when proving certain properties.

In the present paper we consider the birth and death Markov chains. Recall that these processes have the property that only transitions to neighbouring states are possible. This model comes from applications in biology, the developing in time of a population of particles: each particle lives for a random length of time at the end of which it splits into two particles or dies. The process is quite simple and constitutes a generalisation of the standard random walk on (in the discrete parameter setting) and have some similarities with Bessel processes (in the continuous parameter setting).

We address the previous simple questions, for birth and death Markov chains with discrete and continuous parameter. Precisely, we penalize the distribution of the transient birth and death process by the number of visits at the state 0 (which is like local time type penalization). When we force the chain to visit an infinitely often the state zero it is reasonable to think that what we get is a recurrent chain. Indeed, we prove that, under the new probability measure induced by penalization, the chain behaves as a recurrent birth and death chain. The same procedure works in continuous parameter setting. We point out that in [START_REF] Roynette | Brownian penalizations: metatheorem and correct results[END_REF] one studies the Bessel process of dimension greater than 2 (which is a transient process) penalized in some sense by its local time at 0 (more precisely, by its sojourn time in an interval, see also Remark 4.1 below).

Our approach is elementary in the discrete parameter setting, and it is based on Dynkin's formula for pure jump processes in the continuous parameter setting. We would like to stress that for both situations all we need is the transient feature of the process, but we do not need neither explicit expressions of transition probabilities (or rates), nor other explicit informations related to our processes as in previous cited works. In some sense this is one of the originalities of the present paper, besides the study of the penalization for birth and death chains.

Is it possible to penalize a recurrent birth and death chain in order to obtain a transient one? We give a particular example where we penalize a recurrent random walk by the event to reach zero in infinite time and we get a recurrent birth and death chain. Our proof for this case uses the expression of the law of the hitting time of zero for a biased random walk and we obtain a particular birth and death chain, the so-called Bessel random walk. For this reason it seems that the case of a general birth and death chain is more complicate. We illustrate some other phenomena by further study of Bessel random walks.

The paper is organised as follows : the main results concerning the penalization of transient birth and death chains, with discrete and also with continuous parameter, are stated in the next section. Section 2 also contains an example of penalization of recurrent birth and death chain. All the proofs are given in Section 3. The example of Bessel random walks, but also some other examples are discussed in the last section of the paper.

Notations and main results

Discrete parameter setting

Let {X n } n≥0 be a birth and death chain on the canonical probability space (Ω, F ∞ , ). In other words, {X n } n≥0 is a Markov chain, with state space and having the transition probabilities given by:

p := (X n+1 = + 1 | X n = ), q := (X n+1 = -1 | X n = ), r := (X n+1 = | X n = ) = 1 -p -q , ≥ 1, (2.1)
and p 0 := (X n+1 = 1 | X n = 0), q 0 := 0, r 0 := 1 -p 0 .

(2.2)

We will denote

γ 0 := 1, γ := k=1 q k p k , ≥ 1, (2.3)
and it can proved (see for instance [START_REF] Lawler | Introduction to Stochastic Processes 2nd edition[END_REF], p. 77) the following classical:

Proposition 2.1 The birth and death chain {X n } n≥0 is transient if and only if ≥0 γ < ∞, (2.4)
and in this case we will denote

S := ≥0 γ . (2.5) Set U n := n k=0 ¢ {X k =0} , 0 ≤ n ≤ ∞. (2.6)
We are able to state the first main result:

Theorem 2.1 Let {F n } n≥0 be the natural filtration associated to the transient birth and death chain {X n } n≥0 .

1. For any integer n ≥ 0 and any event A n ∈ F n ,

lim κ→∞ ¡ 0 [¢ An ¢ {U∞≥κ} ] ¡ 0 [¢ {U∞≥κ} ] = ¡ 0 [¢ An M n ], (2.7)
where

M n := 1 S 1 - p 0 S -Un+ {Xn=0}   ≥Xn γ   . (2.8) 2.
{M n } n≥0 is a positive {F n }-martingale which tends toward zero, almost surely. Henceforth {M n } n≥0 is not uniformly integrable.

3. The equality (2.7) induces a probability measure Q 0 on (Ω, F ∞ ): for any integer n ≥ 0 and any event

A n ∈ F n , Q 0 (A n ) := ¡ 0 [¢ An M n ].
(2.9)

Then, under Q 0 , {X n } n≥0 is a recurrent birth and death chain with transition probabilities given by:

p = Q 0 (X n+1 = + 1 | X n = ) := p ( P k≥ +1 γ k ) /( P k≥ γ k ), q = Q 0 (X n+1 = -1 | X n = ) := q ( P k≥ -1 γ k ) /( P k≥ γ k ), r = Q 0 (X n+1 = | X n = ) := r , ≥ 1, (2.10)
and p0 = Q 0 (X n+1 = 1 | X n = 0) := p 0 (S-1) /(S-p 0 ), q0 := 0, r0 := r 0 S /(S-p 0 ).

(2.11)

Remark 2.2 1. We suppose that the chain starts from zero only to simplify the computations, but it is possible to assume that the Markov chain {X n } n≥0 is starting from x = 0. One can obtain a similar result with minor modifications: for instance in (2.8) one needs to replace the factor

1 /S by ≥x γ -1
. 2. As a main example of birth and death chain, we will consider (see also §4.1 below) the so-called Bessel random walk with index α > -1 or of dimension δ = 2(α + 1) > 0 (see also [START_REF] Gall | Une approche élémentaire des théorémes de décomposition de Williams[END_REF], p. 448, or [START_REF] Mishchenko | A discrete Bessel process and its properties[END_REF]). The transition probabilities are given by

p 0 = 1, p = + 2α + 1 2 + 2α + 1 , q = 2 + 2α + 1 , r = 0, ≥ 1.
(2.12)

If α = -1 /2, {X n } n≥0
is the absolute value of the symmetric standard random walk. Let us note that {X n } n≥0 is transient for α > 0, as we can easily see by using Proposition 2.1. 2

Continuous parameter setting

Let {X(t)} t≥0 be a Markov process parametrized by t ∈ [0, ∞) and with the state space . For instance, we can think that X(t) is the size of a population of particles at time t. We follow [START_REF] Resnick | Adventures in Stochastic Processes, 4th printing[END_REF], p. 373, to describe the continuous time version of the birth and death process {X(t)} t≥0 . The changes of states yields as follows: when the process is in state , assume that are given two independent random variables B( ) and D( ), which are independent from {X(t)} t≥0 , and exponentially distributed with parameters b and d , respectively. A transition from to + 1 is made if B( ) ≤ D( ), which occurs with probability

[B( ) ≤ D( )] = b /(b +d );
(2.13) otherwise a transition from to -1 is made. The holding time in state is B( ) ∧ D( ), which is exponential with parameter b + d (we assume that d 0 = 0). We may think of B( ) as the time until a birth when the population is and similarly, D( ) is the time until a death when the population is ; the population increases by one if a birth occurs prior to a death; otherwise the population decreases by one. Usually, the birth and death process in continuous time is described in terms of birth and death rates : in a population of size a particle is born at rate b and dies at rate d . These refer to the infinitesimal transition probabilities: (2.14)

(X(t + h) = + k|X(t) = ) =    b h + o(h), if k = 1 d h + o(h), if k = -1 o(h), if |k| > 1, as h → 0.
We will denote

J 0 = 0 and, for n ≥ 1, J n = inf{t ≥ J n-1 : X(t) = X(J n-1 )}. (2.15)
Let us introduce the sequence of random variables {Y n } n≥0 , given by

Y n := X(J n ), n ≥ 0. (2.16)
The proof of the following proposition is classical (see, for instance [START_REF] Lawler | Introduction to Stochastic Processes 2nd edition[END_REF], p. 76-77 and [START_REF] Norris | Markov chains. 2nd printing[END_REF], p. 115):

Proposition 2.3
1. {Y n } n≥0 given by (2.16) is a birth and death discrete Markov chain with transition probabilities given by p = b /(b +d ), q = d /(b +d ), ≥ 1 and p 0 = 1. We set as previously

γ 0 = 1 and γ = k=1 q k p k = k=1 d k b k , ≥ 1.
S := ≥0 γ = 1 + ≥1 k=1 d k b k < ∞.
(2.17)

Let us denote

V (t) := t 0 ¢ {X(s)=0} ds, 0 ≤ t ≤ ∞. (2.18)
The second main result is:

Theorem 2.2 Let {F t } t≥0 be the natural filtration associated to the transient birth and death process {X(t)} t≥0 on the canonical probability space (Ω, F ∞ , ).

1. For any t ≥ 0 and any event

A t ∈ F t , lim κ→∞ ¡ 0 [¢ At ¢ {V (∞)≥κ} ] ¡ 0 [¢ {V (∞)≥κ} ] = ¡ 0 [¢ At M (t)], (2.19)
where

M (t) := 1 S exp b 0 V (t) S   ≥X(t) γ   . (2.20)
2. {M (t)} t≥0 is a positive {F t }-martingale which tends toward zero, almost surely. Henceforth {M (t)} t≥0 is not uniformly integrable.

3. The equality (2.19) induces a probability measure Q 0 on (Ω, F ∞ ): for any t ≥ 0 and any event

A t ∈ F t Q 0 (A t ) := ¡ 0 [¢ At M (t)]. (2.21)
Then, under Q 0 , {X(t)} t≥0 is a continuous recurrent birth and death process with birth and death rates given by:

b := b ( P k≥ +1 γ k ) /( P k≥ γ k ), ≥ 0, d := d ( P k≥ -1 γ k ) /( P k≥ γ k ), > 0. (2.22)

Bessel random walk as a penalized classical random walk

In this section we try to make the reverse work, more precisely, to penalize a recurrent birth and death Markov chain in order to obtain a transient birth and death Markov chain. We present a particular example of birth and death chain but the method of proof we have used in this case (see §3.3 below) seems more complicated for the general situation. Thus it remains an open question.

Let p ∈ (0, 1 /2) and we consider a recurrent random walk {Z n } n≥0 with transition probabilities

p := (Z n+1 = + 1 | Z n = ) = p, q := (Z n+1 = -1 | Z n = ) = q = 1 -p, ∀ ≥ 1 and p 0 = (Z n+1 = 1 | Z n = 0) = 1. (2.23) It is a recurrent Markov chain by Proposition 2.1, since ≥1 γ = ≥1 ( q /p) = ∞.
We shall penalize this chain to reach the state zero in an infinite time. As usual, we denote by T 0 the hitting time of state zero by the Markov chain {Z n } n≥0 and it is not very difficult to get an equivalent for x (T 0 ≥ κ), as κ → ∞ (see Lemma 3.5 below). With this tool in hand, we can prove: Proposition 2.4 Let {F n } n≥0 be the natural filtration associated to the recurrent birth and death chain {Z n } n≥0 with the transition probabilities given by (2.23).

For any integers

x ≥ 1, n ≥ 0 and any event A n ∈ F n , lim κ→∞ ¡ x [¢ An ¢ {T 0 ≥κ} ] ¡ x [¢ {T 0 ≥κ} ] = ¡ x [¢ An M n ], (2.24)
where

M n := ¢ {T 0 >n} Z n x q p (Zn -x) /2 (4pq) -n /2 (2.25)
is a positive {F n }-martingale which tends toward zero, almost surely.

2. The equality (2.24) induces a probability measure Q x on (Ω, F ∞ ): for any integer n ≥ 0 and any event

A n ∈ F n , Q x (A n ) := ¡ x [¢ An M n ]. (2.26)
Then, under Q x , {Z n } n≥0 is a transient birth and death chain with transition probabilities given by:

Q x (Z n+1 = + 1 | Z n = ) = + 1 2 , Q x (Z n+1 = -1 | Z n = ) = -1 2 , ∀ ≥ 1,
and

Q x (Z n+1 = 0 | Z n = 1) = 0, (2.27)
that is a Bessel random walk with index 1 /2 (or dimension 3).

Now, a very natural problem is to make a similar study for a recurrent birth and death process in continuous time setting. Consider a continuous time random walk {Z(t)} t≥0 : when the process is in a transition in + 1 or in -1 appears with probability b /(b+d) or d /(b+d), respectively. Here b, d > 0 are the parameters of the exponential times until a birth or a death, respectively. Thus, the holding times in any state are exponential with parameter b + d. According to Proposition 2.3, the process {Z(t)} t≥0 is recurrent if and only if d > b.

It seems clearly that, if we penalize this process to reach the state zero in an infinite time, we obtain a transient Markov process, at least at a heuristic level. Moreover, in the light of Proposition 2.4, the underlying Markov chain should be a Bessel random walk with dimension 3. If T 0 denotes the hitting time of state zero by {Z(t)} t≥0 , it seems that the study of the equivalent of x (T 0 ≥ κ), as κ → ∞, is more complicate as in the discrete time setting and it remains an open problem.

Let us point out that our intuition for the continuous time setting is reinforced by the study of the drifted recurrent Brownian motion. Indeed, we get the same phenomenon as in the discrete time setting : when one penalizes the process to reach 0 at an infinite time, the new process which we obtain is a Bessel process of dimension 3. We state this result in the following: Proposition 2.5 Let {B(t)} t≥0 be a drifted Brownian motion B(t) := B(t) + µt, where µ < 0 and {B(t)} t≥0 is a standard Brownian motion starting from x. We denote, as usually, {F 1. For any t ≥ 0 and any

A t ∈ F (B,µ) t , lim κ→∞ ¡ x ¢ At ¢ {T (B,µ) 0 >κ} ¡ x ¢ {T (B,µ) 0 >κ} = ¡ x ¢ At M (B,µ) (t) , (2.28)
where

M (B,µ) (t) := ¢ {T (B,µ) 0 >t} B(t) x exp -µ B(t) -x + tµ 2 2 (2.29)
is a positive F (B,µ) -martingale which tends toward zero, almost surely.

2. The equality (2.28) induces a probability measure Q (B,µ) x on: for any t ≥ 0 and any

A t ∈ F (B,µ) t , Q (B,µ) x (A t ) := ¡ x ¢ At M (B,µ) (t)
(2.30)

and under Q (B,µ) x , the infinitesimal generator of {B(t)} t≥0 is L = 1 /2 d 2 /dx 2 + 1 /x d /dx.
0 [¢ An ¢ {U∞≥κ} ] into two terms ¡ 0 [¢ An ¢ {U∞≥κ} ] = ¡ 0 [¢ An {¢ {Un≥κ} + ¢ {U∞≥κ} ¢ {Un<κ} }] = ¡ 0 [¢ An ¢ {Un≥κ} ] + ¡ 0 [¢ An ¢ {U∞≥κ} ¢ {Un<κ} ]
and we note that ¢ {Un≥κ} vanishes if κ ≥ n /2 . Consider the second term on the right hand side of the latter equality and the Markov chain

{ Xp := X n+p -X n } p≥0 . Then, for κ ≥ n /2 , ¡ 0 [¢ An ¢ {U∞≥κ} ¢ {Un<κ} ] = ¡ 0 [¢ An ¢ {U∞≥κ} ] = ¡ 0 ¢ An ¢ {U∞•θn≥κ-Un+ {Xn=0} } = ¡ 0 ¢ An ¡ Xn ¢ { Ũ∞≥κ-Un+ {Xn=0} } .
Here we denoted Ũ∞ = p≥0 ¢ { Xp=0} and, as usually, {θ n } n≥0 the family of shift operators. At this level we need the following:

Lemma 3.1 For all integers n ≥ 0, n (U ∞ ≥ κ) = ( Rn /S)(1 -p 0/S) κ-1 , (3.1)
where

R 0 = S and R n := ≥n γ , n ≥ 1. (3.2)
Let us postpone the proof of this lemma and we finish the proof of (2.7). We deduce that, for κ ≥ n /2 :

¡ 0 [¢ An ¢ {U∞≥κ} ] = ¡ 0 ¢ An ( R Xn /S) (1 -p 0/S) κ-Un+ {Xn=0} -1
and, when we combine with (3.1) for n = 0, we obtain, for κ

≥ n /2 , ¡ 0 [¢ An ¢ {U∞≥κ} ] ¡ 0 [¢ {U∞≥κ} ] = ¡ 0 ¢ An ( R Xn/S) (1 -p 0/S) -Un+ {Xn=0} .
and the proof of the first part of Theorem 2.1 is done.

Proof of Lemma 3.1.

Let us consider f : → £ given by f (0) = 0 and, for n

≥ 1 integer, f (n) = n-1 =0 γ . Then, it is a direct computation to prove that {f (X n ) : n ≥ 0} is a F n -martingale.
We denote by T n the hitting time of n ∈ by the birth and death chain and we claim that, for all n ≥ 1,

n (T 0 < ∞) = ( P ≥n γ ) /S. (3.3)
Indeed, if m ≥ n, by the optional stopping theorem, we get

f (n) = ¡ n [f (X Tm∧T 0 )] = f (m) n (T m < T 0 ) + f (n -1)[1 -n (T 0 > T m )]
and we deduce

n (T m < T 0 ) = (f (n)-f (0)) /(f(m)-f(0)) = ( P n-1 =0 γ ) /( P m-1 =0 γ ).
Since {X n } n≥0 is a transient Markov chain, by letting m → ∞ we obtain

n (T 0 = ∞) = ( P n-1 =0 γ ) /S or n (T 0 < ∞) = ( P ≥n γ ) /S = Rn /S .
Let us introduce the following stopping times :

T (0) 0 = T 0 and T ( ) 0 := inf{n ≥ T ( -1) 0 : X n = 0}, if ≥ 1.
By using (2.6), we can write

0 (U ∞ = ) = 0 (T (1) 0
+ . . . + T ( -1) 0 < ∞, T (1) 0
+ . . . + T ( ) 0 = ∞) = 0 (T (1) 0 < ∞) 0 (T (2) 0 < ∞ | T (1) 0 < ∞) . . . 0 (T ( -1) 0 < ∞ | T (1) 0 + . . . + T ( -1) 0 < ∞) × 0 (T ( ) 0 = ∞ | T (1) 0 + . . . + T ( -1) 0 < ∞) = 0 (T (1) 0 < ∞) -1 0 (T (1)
0 = ∞) = 1 (T 0 < ∞) -1 1 (T 0 = ∞) = ( p 0/S)(1 -p 0/S) -1 .
Therefore, on the one hand

0 (U ∞ ≥ κ) = ≥κ 0 (U ∞ = ) = (1 -p 0/S) κ-1 , while, on the other hand, if n ≥ 1, n (U ∞ ≥ κ) = n (T 0 < ∞) 0 (U ∞ ≥ κ)
from which we get (3.1) and Lemma 3.1 is proved.

2 Remark 3.2
Let us point out that in fact the limit in (2.7) is an exact equality for κ > n /2 . This is a somehow different situation with respect to those in [START_REF] Debs | Pénalisation de la marche aléatoire standard par une fonction du maximum unilatère, du temps local en zéro et de la longueur des excursions[END_REF] or [START_REF] Roynette | Pénalisations et quelques extensions du théorème de Pitman, relatives au mouvement brownien et à son maximum unilatère[END_REF].

0 ≤ M n ≤ (1 -p 0/S) -Un ≤ (1 -p 0/S) -n ,
we see that M n is integrable. We can write:

¡ [M n+1 |F n ] = ¡ M n+1 ¢ {X n+1 =Xn+1} |F n + ¡ M n+1 ¢ {X n+1 =Xn-1} ¢ {Xn =0} |F n + ¡ M n+1 ¢ {X n+1 =Xn} |F n . (3.4)
By using (2.8) and (3.2), the first term in (3.4) can be written as follows

¡ ( 1 /S)(1 -p 0/S) -Un R Xn +1 ¢ {X n+1 =Xn+1} |F n = ( 1 /S)(1 -p 0/S) -Un R Xn +1 ¡ ¢ {X n+1 =Xn+1} |F n = ( 1 /S)(1 -p 0/S) -Un R Xn+1 p Xn .
For the second term in (3.4) we split the expectation into two terms

¡ M n+1 ¢ {X n+1 =Xn-1} ¢ {Xn =0,1} |F n + ¡ M n+1 ¢ {X n+1 =Xn-1} ¢ {Xn=1} |F n = ¡ ( 1 /S)(1 -p 0/S) -Un R Xn-1 ¢ {X n+1 =Xn-1} ¢ {Xn =0,1} |F n + ¡ ( 1 /S)(1 -p 0/S ) -Un-1+1 R Xn -1 ¢ {X n+1 =Xn-1} ¢ {Xn=1} |F n = ( 1 /S)(1 -p 0/S) -Un R Xn-1 ¢ {Xn =0} (¢ {Xn =1} + ¢ {Xn=1} ) ¡ ¢ X n+1 =Xn-1 |F n = ( 1 /S)(1 -p 0/S) -Un R Xn -1 ¢ {Xn =0} q Xn .
By a similar reasoning we can prove that the third term in (3.4) can be written

¡ M n+1 ¢ {X n+1 =Xn} |F n = ( 1 /S)(1 -p 0/S) -Un R Xn r Xn .
We replace in (3.4) and we can write on the one hand,

¡ M n+1 ¢ {Xn =0} |F n = ( 1 /S)(1 -p 0/S) -Un ¢ {Xn =0} p Xn R Xn+1 + q Xn R Xn-1 + r Xn R Xn = ( 1 /S)(1 -p 0/S) -Un ¢ {Xn =0} (p Xn + q Xn + r Xn )R Xn + q Xn γ Xn-1 -p Xn γ Xn = M n ¢ {Xn =0} ,
by using (2.1), (2.3) and (3.2). On the other hand

¡ M n+1 ¢ {Xn=0} |F n = ( 1 /S)(1 -p 0/S) -Un ¢ {Xn=0}    p 0 ≥1 γ + r 0 ≥0 γ    = ( 1 /S)(1 -p 0/S) -Un ¢ {Xn=0} S(1 -p 0/S) = ( 1 /S)(1 -p 0/S) -Un+1 ¢ {Xn=0} = M n ¢ {Xn=0} ,
where we used the fact that ≥1 γ = S -1 and r 0 = 1-p 0 . Hence {M n } n≥0 is a martingale.

q k R k-1 p k R k+1 = ≥1 γ k=1 R k-1 R k+1 = ≥1 γ R 0 R 1 R R +1 = SR 1 ≥1 ( 1 /R +1 -1 /R ) = SR 1 lim N →∞ ( 1 /R N +1 -1 /R 1 ) = ∞, (3.6)
since lim N →∞ R N +1 = 0, by Proposition 2.1. We prove that, under Q 0 , {X n } n≥0 is a birth and death Markov chain with transition probabilities given by (2.10). Consider, for all integers n ≥ 1 and all integers a 0 , . . . , a n-1 ≥ 0, the event A n-1 := {X 0 = a 0 , . . . , X n-1 = a n-1 }.

We can write, for ≥ 1 and all integers n ≥ 1,

p = Q 0 [X n+1 = + 1|X n = , A n-1 ] = ¡ 0 ¢ {X n+1 = +1,Xn= }∩A n-1 M n+1 ¡ 0 ¢ {Xn= }∩A n-1 M n , by (2.9)
= ¡ 0 ¢ {X n+1 = +1,Xn= }∩A n-1 (1 -p 0/S) -U n+1 R +1 ¡ 0 ¢ {Xn= }∩A n-1 (1 -p 0/S) -Un R , by (2.8) = R +1 R ¡ 0 ¡ 0 ¢ {X n+1 = +1,Xn= }∩A n-1 (1 -p 0/S) -U n+1 |F n ¡ 0 ¢ {Xn= }∩A n-1 (1 -p 0/S) -Un = R +1 R ¡ 0 ¢ {Xn= }∩A n-1 (1 -p 0/S) -Un 0 (X n+1 = X n + 1|F n ) ¡ 0 ¢ {Xn= }∩A n-1 (1 -p 0/S) -Un , since ≥ 1 = R +1 R ¡ 0 ¢ {Xn= }∩A n-1 (1 -p 0/S) -Un p Xn ¡ 0 ¢ {Xn= }∩A n-1 (1 -p 0/S) -Un = p R +1 R .
With a similar computation we get that, for ≥ 1, r = r . Assume now that = 0. Then, by using a similar reasoning, for all a 0 , . . . , a n-1 ≥ 0,

p0 = Q 0 [X n+1 = 1|X n = 0, A n-1 ] = ¡ 0 ¢ {X n+1 =1,Xn=0}∩A n-1 (1 -p 0/S) -U n+1 R 1 ¡ 0 ¢ {Xn=0}∩A n-1 (1 -p 0/S) -Un+ {Xn =0} R 0 , by (2.9) and (2.8) = R 1 S ¡ 0 ¢ {Xn=0}∩A n-1 (1 -p 0/S) -Un p Xn ¡ 0 ¢ {Xn=0}∩A n-1 (1 -p 0/S) -Un+1 = p 0 R 1 S 1 - p 0 S -1
.

Furthermore: r0 = Q 0 [X n+1 = 0|X n = 0, A n-1 ] = ¡ 0 ¢ {X n+1 =0,Xn=0}∩A n-1 (1 -p 0/S) -U n+1 + {X n+1 =0} R 0 ¡ 0 ¢ {Xn=0}∩A n-1 (1 -p 0/S) -Un+ {Xn =0} R 0 = ¡ 0 ¢ {X n+1 =0,Xn=0}∩A n-1 (1 -p 0/S) -Un-1+1 ¡ 0 ¢ {Xn=0}∩A n-1 (1 -p 0/S) -Un+1 = r 0 1 - p 0 S -1 .

Proof of Theorem 2.2

The proof contains several similar ideas as in the proof of Theorem 2.1. We shall emphasize the main differences.

a. Proof of (2.19)-(2.20). This part is quite similar to the discrete case. We split

¡ 0 [¢ At ¢ {V (∞)≥κ} ] = ¡ 0 [¢ At ¢ {V (t)≥κ} ] + ¡ 0 [¢ At ¢ {V (∞)≥κ} ¢ {V (t)<κ} ]
and we note that ¢ {V (t)≥κ} vanishes, if κ > t. Set { X(s) := X(t + s) -X(t)} s≥0 and note that V (∞) = V (t) + V (∞) • θ t . Again and, as usually, {θ t } t≥0 denotes the family of shift operators. Then, for κ > t:

¡ 0 [¢ At ¢ {V (∞)≥κ} ¢ {V (t)<κ} ] = ¡ 0 ¢ At ¢ {V (∞)•θt≥κ-V (t)} = ¡ 0 ¢ At ¡ X(t) ¢ { Ṽ (∞)≥κ-V (t) , where Ṽ (∞) := ∞ 0 ¢ { X(t)=0} ds.
As for the discrete case we can state: Lemma 3.3 For all integers n ≥ 0, and for any t ≥ 0,

n (V (∞) > t) = ( Rn /S)e -b 0 t /S , (3.7)
where R n is given by (3.2).

Let us finish the proof of (2.19). For κ > t:

¡ 0 [¢ At ¢ {V (∞)≥κ} ] = ¡ 0 [¢ At ( R X(t) /S)e -b 0 (t-V (t)) /S ] t J 1 J 4 J 5 J 6 J 7
e 2 e 3 e 1

X(t)

Figure 1: X(t) conditionnally to the event U ∞ = 3 and using the preceding lemma again, we obtain for κ > t:

¡ 0 [¢ At ¢ {V (∞)≥κ} ] ¡ 0 [¢ {V (∞)≥κ} ] = ¡ 0 [¢ At ( R X(t) /S)e b 0 V (t) /S ], that is (2.19)-(2.

20).

Proof of Lemma 3.3. Recall that {Y n } n≥0 is given by (2.16) and we shall denote U (Y )

∞ := n≥0 ¢ {Yn=0} . It is no difficult to see that, on the event {U (Y ) ∞ = k}, V (∞)
is the sum of k independent random variables e 1 , . . . , e k , the holding times at 0, having exponential distribution of same parameter λ 0 = b 0 , and being independent from U (Y ) ∞ (see Figure 1). Therefore, by using (3.1), for all integers n ≥ 0,

n (V (∞) > t) = k≥0 n U (Y ) ∞ = k, e 1 + ... + e k > t = k≥0 n U (Y ) ∞ = k (e 1 + . . . + e k > t) = k≥0 R n S 1 - 1 S k-1 1 S e -b 0 t k-1 i=0 (b 0 t) i i! = R n S 1 S e -b 0 t i≥0 (b 0 t) i i! k≥i+1 1 - 1 S k-1 = R n S 1 S e -b 0 t i≥0 (b 0 t) i i! 1 - 1 S i S = R n S e -b 0 t S .
The proof of Lemma 3.3 is done.

Remark 3.4

We point out again that in fact the limit in (2.19) is an exact equality for κ > t, as was noted in the discrete parameter setting. 2

b. Proof of the second part of Theorem 2.2. As in the discrete case we point out a useful martingale. At this level we need to recall Dynkin's formula for pure jump processes (see, for instance [START_REF] Klebaner | Introduction to Stochastic Calculus with Applications 2nd edition[END_REF], p. 262): if f : → £ is a bounded function such that {f (X(t))} t≥0 is a pure jump process, then

M f (t) := f (X(t)) -f (X(0)) - t 0 Lf (X(s))ds (3.8)
is a local martingale, where we denoted

Lf (x) := b x [f (x + 1) -f (x)] + d x [f (x -1) -f (x)] . (3.9)
If we take in (3.9) f (x) = ≥x γ = R x , we obtain:

Lf (x) = b x (R x+1 -R x ) + d x (R x-1 -R x ) = -b x γ x + d x γ x-1 = -b x x =1 d b + d x x-1 =1 d b = -b 0 γ 0 ¢ {x=0} = -b 0 ¢ {x=0} .
Hence {M f (t)} t≥0 given by (3.8) is a local martingale. Therefore, we can apply the integration by parts formula (see, for instance [START_REF] Klebaner | Introduction to Stochastic Calculus with Applications 2nd edition[END_REF], p. 220) to the semimartingale

R X(t) = M f (t) + S - t 0 b 0 ¢ {X(s)=0} ds
and to the finite variation process A(t) := exp( b 0 V (t) /S). We note that

dR X(t) = dM f (t) -b 0 ¢ {X(t)=0} dt, dA(t) = ( b 0/S)e b 0 V (t) /S ¢ {X(t)=0} dt
and R X(•) , A = 0. We obtain, by using (2.20),

S M (t) = R X(t) e b 0 V (t) /S = S + t 0 R X(s) dA(s) + t 0 A(s) dR X(s) + R X(•) , A t = S + t 0 R X(s) b 0 S e b 0 V (s) /S ¢ {X(s)=0} ds + t 0 e b 0 V (s) /S dM f (s) -b 0 ¢ {X(s)=0} ds = S + t 0 e b 0 V (s) /S dM f (s)
and consequently, {M (t)} t≥0 is a local martingale. In fact, it is a (true) martingale since, for all T ≥ 0,

¡ ( sup 0≤t≤T |M (t)|) ≤ exp( b 0 T /S).
b. Proof of the third part of Theorem 2.2. First, we prove that, under Q 0 , {X(t)} t≥0 is a Markov process. For any n ≥ 1 and arbitrary sequences 0 ≤ t 0 < t 1 < . . . < t n-1 and a 0 , ..., a n-1 ∈ , we consider the event

A n-1 = {X(t 0 ) = a 0 , ..., X(t n-1 ) = a n-1 }.
Then, for all integers k, ≥ 0,

Q 0 (X(t n+1 ) = | X(t n ) = k, A n-1 ) = ¡ 0 ¢ {X(t n+1 )= ,X(tn)=k}∩A n-1 M (t n+1 ) ¡ 0 ¢ {X(tn)=k}∩A n-1 M (t n ) = ¡ 0 ¢ {X(t n+1 )= ,X(tn)=k}∩A n-1 R e b 0 V (t n+1 ) /S ¡ 0 ¢ {X(tn)=k}∩A n-1 R k e b 0 V (tn) /S = R R k ¡ 0 ¢ {X(t n+1 )= ,X(tn)=k}∩A n-1 exp ( b 0 (V (tn)+V (t n+1 -tn)•θt n ) /S) ¡ 0 ¢ {X(tn)=k}∩A n-1 e b 0 V (tn) /S = R R k ¡ 0 ¢ {X(tn)=k}∩A n-1 e b 0 V (tn) /S ¡ ¢ {X(t n+1 )= } exp ( b 0 (V (t n+1 -tn)•θt n ) /S) | F tn ¡ 0 ¢ {X(tn)=k}∩A n-1 e b 0 V (tn) /S = R R k ¡ 0 ¢ {X(tn)=k}∩A n-1 e b 0 V (tn ) /S ¡ k ¢ { X(t n+1 -tn)= } e b 0 Ṽ (t n+1 -tn) /S ¡ 0 ¢ {X(tn)=k}∩A n-1 e b 0 V (tn) /S = R R k ¡ k ¢ { X(t n+1 -tn)= } e b 0 Ṽ (t n+1 -tn ) /S ,
which does not depend on A n-1 . This proves that, under Q 0 , {X(t)} t≥0 is a Markov process.

Let us study the holding times, under Q 0 , of the Markov process {X(t)} t≥0 , for instance we shall compute the distribution of J 1 . For any integer ≥ 1 and for any α ≥ 0:

¡ Q 0 [e -αJ 1 ] = ¡ Q 0 [e -αJ 1 ¢ {X(0)= } ] ¡ Q 0 [X(0) = ] = ¡ [¢ {X(0)= } e -αJ 1 M (J 1 )] ¡ [¢ {X(0)= } M (0)] = S R ¡ [e -αJ 1 M (J 1 )] = S R ¡ ¢ {B( )≤D( )} e -αB( ) M (B( )) + ¡ ¢ {D( )≤B( )} e -αD( ) M (D( )) = 1 R ¡ ¢ {B( )≤D( )} e -αB( ) R +1 + ¡ ¢ {D( )≤B( )} e -αD( ) R -1 ,
since ¢ {B( )≤D( )} V (J 1 ) = 0, -a.s. and by using (2.20) and the stopping theorem. Hence

¡ Q 0 [e -αJ 1 ] = R +1 R ¡ ¢ {B( )≤D( )} e -αB( ) + R -1 R ¡ ¢ {D( )≤B( )} e -αD( ) .
By using (2.13) we can compute :

¡ [¢ {B( )≤D( )} e -αB( ) ] = ∞ 0 dx e -αx e -b x b ∞ x dy e -d y d = b b + d + α .
Therefore, we obtain:

¡ Q 0 [e -αJ 1 ] = b R +1 + d R -1 R (b + d + α) = b + d b + d + α .
We conclude that the holding time at ≥ 1 is an exponential random variable with parameter λ = b + d . We stress that for ≥ 1 the distribution of the holding time is the same under the probabilities and Q 0 .

The reasoning is similar for = 0, :

¡ Q 0 0 [e -αJ 1 ] = ¡ [¢ {X(0)=0} e -αJ 1 M (J 1 )] ¡ [¢ {X(0)=0} M (0)] = ¡ 0 [e -αB(0) R 1 S e b 0 V (B(0)) /S ] = R 1 S ¡ 0 [e -αB(0) e b 0 B(0) S ] = S -1 S ¡ 0 [e B(0)(-α+ b 0/S) ] = b 0 -b 0/S b 0 -b 0/S + α ,
and we deduce that the holding time at 0 is an exponential random variable of parameter b 0 (S-1) /S. Now, we study the underlying Markov chain {Y n } n≥0 under Q 0 . For any positive Borel function g : → £ and ≥ 1, as previously:

¡ Q 0 [g(Y 1 )] = ¡ Q 0 [g(X(J 1 ))] = ¡ Q 0 [¢ {X(0)= } g(X(J 1 ))] ¡ Q 0 [¢ {X(0)= } ] = ¡ [¢ {X(0)= } ¢ {B( )≤D( )} g( + 1)R +1 ] ¡ Q 0 [¢ {X(0)= } R ] + ¡ [¢ {X(0)= } ¢ {D( )≤B( )} g( -1)R -1 ] ¡ Q 0 [¢ {X(0)= } R ] = g( + 1) R +1 R ¡ [¢ {B( )≤D( )} ] + g( -1) R -1 R ¡ [¢ {D( )≤B( )} ] = g( + 1) R +1 R b b + d + g( -1) R -1 R d b + d .
For = 0, the reasoning is the same. We conclude that under Q 0 , {Y n } n≥0 is a birth and death Markov chain with discrete parameter and the transition probabilities:

p = R +1 R b b + d , q = R -1 R d b + d
, for ≥ 1, and p0 = 1.

Then if we set γ0 = 1 and γ = i=1 qi pi , for ≥ 1, we have:

≥0 γ = 1 + ≥1 i=1 d i R i-1 b i R i+1 = 1 + ≥1 i=1 q i R i-1 p i R i+1 .
According to (3.6) and Proposition 2.1, {Y n } n≥0 is recurrent. Furthermore, according to the second point of the Proposition (2.3), {X(t)} t≥0 is recurrent under Q 0 . If we denote b and d the birth and death rates of (X(t), t ≥ 0) under Q 0 ,we have:

b + d = b + d and p = b b + d = R +1 R b b + d . Consequently, b = b R +1 R and d = d R -1 R ,
and (2.22) is proved.

Proofs of Propositions 2.4 and 2.5

Proof of Proposition 2.4. To begin with, we need to find an equivalent, as κ → ∞, for

x (T 0 ≥ κ), when x ≥ 1 is an integer. For the sake of completeness we state this in the following:

Lemma 3.5 Let x ≥ 1 an integer. As κ → ∞, x (T 0 ≥ κ) ∼ 4 2 π 1 /2 x q p x /2 κ -3 /2 (4pq) κ /2 . (3.10)
As previously, we postpone the proof of this lemma and we finish the proof of (2.24)-(2.25).

As κ → ∞, we obtain the following equivalent

¡ x ¢ {An, T 0 ≥κ} ¡ x ¢ {T 0 ≥κ} = ¡ x ¢ {Λn, T 0 ≥n} ¡ Zn ¢ e T 0 ≥κ-n ¡ x ¢ {T 0 ≥κ} ∼ κ→∞ ¡ x ¢ {An, T 0 ≥n} 4 2 π 1 /2 Z n q p Zn /2 (κ -n) -3 /2 (4pq) (κ-n) /2 4 2 π 1 /2 x q p x /2 κ -3 /2 (4pq) κ /2
and we deduce the expression of M n , (2.25). As usually, we verify the fact that it is a martingale. Firstly,

¡ x M n+1 ¢ {Zn≥2} | F n = ¢ {T 0 ≥n, Zn≥2} Z n x q p (Zn-x) /2 (4pq)
-(n+1) /2 ¡ q p ξ n+1/2 | F n + ¢ {T 0 ≥n, Zn≥2} 1 x q p (Zn -x) /2 (4pq)
-(n+1) /2 ¡ ξ n+1 q p ξ n+1/2 | F n = ¢ {T 0 ≥n, Zn≥2} Z n x q p (Zn-x) /2 (4pq) -(n+1) /2 p q p 1 /2 + q q p -1 /2 + ¢ {T 0 ≥n, Zn≥2} 1 x q p (Zn-x) /2 (4pq) -(n+1) /2 p q p 1 /2 -q q p -1 /2 = ¢ {Zn≥2} M n ,
and, secondly,

¡ x M n+1 ¢ {Zn=1} | F n = ¡ x ¢ {T 0 ≥n Z n+1 =2, Zn=1} 2 x q p (2-x) /2 (4pq)
-(n+1) /2 = ¢ {T 0 ≥n,Zn=1} 2 x q p (1-x) /2 (4pq) -n /2 q p 1 /2 (4pq) -1 /2 p = ¢ {Zn=1} M n .
Finally, to verify (2.27), we denote again A n-1 := {Z 0 = a 0 , . . . , Z n-1 = a n-1 }. If ≥ 1:

Q x (Z n+1 = + 1 | Z n = , A n-1 ) = ¡ x ¢ {Z n+1 = +1,Zn= }∩A n-1 M n+1 ¡ x ¢ {Zn= }∩A n-1 M n = ¡ x ¢ {T 0 ≥n+1,Z n+1 = +1,Zn= ,}∩A n-1 +1 x q p ( +1-x) /2 (4pq) -(n+1) /2 ¡ x ¢ {T 0 ≥n,Zn= ,}∩A n-1 x q p ( -x) /2 (4pq) -n /2 = + 1 q p 1 /2 (4pq) -1 /2 ¡ x ¢ { T 0 ≥n,Z n+1 = +1,Zn= }∩A n-1 ¡ x ¢ {T 0 ≥n,Zn= }∩A n-1 = + 1 2 p × x (T 0 ≥ n, Z n+1 = + 1 | Z n = , A n-1 ) = + 1 2 p (Z n+1 = + 1 | Z n = ) = + 1 2 ,
where we used the Markovian feature of {Z n } under x . Similarly, we show that

Q x (Z n+1 = -1 | Z n = ) = -1 2 .
Also, it is clear that Q x (Z n+1 = 0 | Z n = 1) = 0 and (2.27) is proved.

Proof of Lemma 3.5. To begin with, recall that (see for instance [START_REF] Feller | An introduction to probability theory and its applications[END_REF], p. 351), for all s ∈ [0, 1],

¡ x s T 0 = 1 -1 -4pqs 2 2ps x = r≥x x (T 0 = r)s r
It is a direct computation to get:

1 -1 -4pqs 2 2ps x = 1 (2p) x ≥x (-4pq) ! s 2 -x x k=0 (-1) k x k -1 i=0 k 2 -i .
Therefore, if x and r are of same evenness we find

P x (T 0 = r) = (-4pq) (n+r) /2 (2p) x ( (x+r) /2)! x k=0 (-1) k x k (x+r) /2-1 i=0 k 2 -i = - (-4pq) (x+r) /2 (2p) x (x+r) 2 ! x-1 2 k =0 x 2k + 1 (x+r) 2 -1 i=0 k -i + 1 2 .
After some algebraic computations we obtain

x (T 0 = r) = (4pq) (x+r) /2 π(2p) x ( (x+r) /2)! x-1 2 k =0 (-1) k x 2k + 1 Γ k + 3 2 Γ x + r 2 -k - 1 2 .
As r → ∞, the only significant term in the latter sum corresponds to k = 0, hence

x (T 0 = r) ∼ r→∞ xΓ( 3 /2) π x+r 2 ! q p x /2 (4pq) r 2 Γ x + r 2 - 1 2 ∼ 2x √ π q p x /2 r 2 -3 /2 (4pq) r /2 .
Finally, since x (T 0 ≥ κ) = r≥κ x (T 0 = r), it is no difficult to prove that, as κ → ∞,

x (T 0 ≥ κ) ∼ x (T 0 = κ), the first term again. The proof of (3.10) is done.

Proof of Propositions 2.5. It is a classical result, using the Girsanov theorem (see, for instance [START_REF] Karatzas | Brownian motion and stochastic calculus[END_REF], p. 197), that, for x > 0:

x (T

(B,µ) 0 > κ) = 0 (T (B,µ) -x > κ) = ∞ κ x √ 2πs 3 exp - (-x -µs) 2 2s ds,
and we easily deduce that:

x (T

(B,µ) 0 > κ) ∼ κ→∞ x √ 2πκ 3 e -µx-κµ 2 /2 .
Then, as in the proof of Proposition 2.4:

¡ x ¢ At ¢ {T (B,µ) 0 >κ} ¡ x ¢ {T (B,µ) 0 >κ} ∼ κ→∞ ¡ x ¢ At ¢ {T (B,µ) 0 >κ} ¡ B(t) ¢ { e T (B,µ) 0 >κ-t} ¡ x ¢ {T (B,µ) 0 >κ} ∼ κ→∞ ¡ x ¢ At ¢ {T (B,µ) 0 >κ} ( B(t) / √ 2π(κ-t) 3 ) exp(-µB(t) -(κ-t)µ 2 /2) ( x / √ 2πκ 3 ) exp(-µx -κµ 2 /2)
and we deduce the expression of M (B,µ) . We prove the fact that it is a F (B,µ) -martingale by using the classical Itô's formula. We write

M (B,µ) (t) = e µx x B(t)Y (t),
where Y (t) = e -µB(t)-µ 2 t /2 . (3.11)

Clearly,

Y (t) = e -µx -µ t 0 Y (s)dB(s)
and then, by the integration by parts formula,

M (B,µ) (t) = e µx x B(t)Y (t) = 1 - e µx x t 0 Y (s)(1 -µB(s))dB(s). (3.12)
Let us now study the process {B(t)} t≥0 under Q (B,µ) x

. We consider an arbitrary function f ∈ C 2 (£ ; £ ). Then, using Itô's formula and (3.12) we can write

f (B(t))M (B,µ) (t) = f (x) + t 0 f (B(s))dM (B,µ) (s) + M (B,µ) (s)df (B(s)) + f (B), M (B,µ) (t) = f (x) + t 0 e µx x f (B(s))Y (s)(1 -µB(s)) + f (B(s))M (B,µ) (s) dB(s) + t 0 µf (B(s))M (B,µ) (s) + 1 2 f (B(s))M (B,µ) (s) + e µx x Y (s)(1 -µB(s))f (B(s)) ds.
Taking the expectation and using (3.11) we get

¡ (B,µ) x f B(t) = ¡ x f B(t) M (B,µ) (t) = f (x) + ¡ x t 0 1 2 f (B(s))M (B,µ) (s) + e µx x Y (s)f (B(s)) ds = f (x) + t 0 ¡ x Lf (B(s))M (B,µ) (s) ds = ¡ (B,µ) x f (x) + t 0 Lf (B(s))ds , where L = 1 /2 d 2 /dx 2 + 1 /x d /dx. We deduce that under Q (B,µ) x
, the infinitesimal generator of {B(t)} t≥0 is the infinitesimal generator of a Bessel process of dimension 3.

Bessel chains and other examples

Bessel chains

We shall illustrate our results with two examples. In the discrete situation we consider the so-called Bessel random walk with index α > 0, while in the continuous parameter setting we adapt this example.

According to Theorem 2.1 one can obtain:

Corollary 4.1 Let {X n } n≥0 the Bessel random walk with index α > 0.

1. For any integer n ≥ 0 and any event A n ∈ F n , (2.7) is verified with the positive no uniformly integrable martingale

M n := Γ(2α + 1)X n ! Γ(X n + 2α + 1)(2α + 1) -Un+ {Xn =0} . (4.1)
2. Let Q 0 be the probability measure on (Ω, F ∞ ) given by (2.9). Then, under Q 0 , {X n } n≥0 is a recurrent birth and death chain with transition probabilities given by: p0 = 1 and p := + 1 2 + 2α + 1

, q := + 2α 2 + 2α + 1

, for ≥ 1. (4.2)

Proof. We compute R n , ∀n ≥ 0. First, by (2.3):

γ = i=1 q i p i = i=1 i i + 2α + 1 = Γ (2α + 2) Γ( + 1) Γ( + 2α + 2) = (2α + 1)β(2α + 1, + 1).
Then, by (3.2), for n ≥ 1:

R n = (2α + 1) ≥n β(2α + 1, + 1) = (2α + 1) ≥0 β(2α + 1, + n + 1) = (2α + 1)β(2α, n + 1),
where we used the following equality k≥0 β(x, y + k) = β(x -1, y) (see for instance, [START_REF] Gradstein | Table of integrals, Series and Products[END_REF], p. 950). Hence,

R n = (2α + 1)Γ(2α)n! Γ(n + 2α + 1) , for n ≥ 1, and R 0 = S = 1 + R 1 = 2α + 1 2α . ( 4.3)
The expression (4.1) is then a consequence of (4.3) and (2.8). Furthermore, according to (2.10), pk = p k R k+1/R k , and using again (4.3) we obtain (4.2).

Remark 4.1

In [START_REF] Roynette | Brownian penalizations: metatheorem and correct results[END_REF] one considers the penalization of the transient continuous Bessel process {R t } t≥0 with dimension d > 2 (or of index n = d /2 -1 > 0) by its sojourn time in an interval. Set, for r > 0, U

t = t 0 ¢ [0,r] (R s )ds (0 ≤ t ≤ ∞). One studies the limit lim t→∞ ¡ x ¢ As ¢ {U (r) ∞ >t} ¡ x ¢ {U (r) ∞ >t} = ¡ x ¢ As M (r) s =: Q (r) x (A s ), A s ∈ F s (s fixed). (r)
One proves in [START_REF] Roynette | Brownian penalizations: metatheorem and correct results[END_REF] that :

• the positive martingale M (r)
s is given by M (r)

t := h (r) (R t ) exp( z 1 U (r) t /2r 2 ), where h (r) (z) := Γ(n + 1) [( 2r /z 1 z) n J n ( z 1 z /r)¢ 0≤z≤r + ( 2r 2 /z 1 z 2 ) n J n (z 1 )¢ z≥r ],
and z 1 is the first positive zero of the Bessel function J n-1 .

• for x ≥ r, under Q (r)

x , the process {R t } t≥0 has the infinitesimal generator L := 1 /2 d 2 /dx 2 + (1-2n) /x d /dx. This process is recurrent and behaves, in a certain sense, as r → 0, as a Bessel process of dimension 4d (or of index -n).

We return to the case of Bessel random walk with index α > -1 and transition probabilities given by (2.12) and we illustrate now a similar phenomenon as for the continuous Bessel process. Let us remak that the Bessel random walk could also be characterized by the following transition probabilities:

(X n+1 = -1 | X n = ) = 1 2 1 - α + 1 /2 + α + 1 /2 , (X n+1 = + 1 | X n = ) = 1 2 1 + α + 1 /2 + α + 1 /2 , for , n ≥ 1 integers. Clearly, if α ∈ (0, 1) \ { 1 /2} (or of dimension δ ∈ (2, 4) \ {3}), Q(X n+1 = -1 | X n = ) = 1 2 1 - -α + 1 2 -α + 1 2 + H( ) 2 , with H( ) = α /2 -α 2 ( -α + 1 /2)( + α + 1 /2) .
As → ∞, we can write

Q(X n+1 = -1 | X n = ) = 1 2 1 - -α + 1 2 -α + 1 2 + o( 1 / ).
In other words, if α = 1 /2, the behaviour of the chain {X n } n≥0 under Q, is that of a Bessel random walk with index α = -α (or with dimension δ = 4 -δ ∈ (0, 2) \ {1}), at least for large states . If α = 1 /2, the chain {X n } n≥0 under Q is a standard symmetric random walk.

2

Let us turn now to the continuous parameter example which is built using the previous example. Let us call it Bessel jump process with index α > 0. It is a pure jump process {X(t)} t≥0 with birth and death rates: 2. Let Q be the probability measure on (Ω, F ∞ ) given by (2.21). Then, under Q, {X t } t≥0 is a recurrent birth and death chain with transition probabilities given by: b := + 1, ≥ 0, d := + 2α, ≥

b = + 2α + 1, d = ,
The proof of this corollary is based on the following fact : the underlying jump chain associated to {X(t)} t≥0 is a Bessel random walk with index α. The remaining details are left to the reader.

Other examples

Let us state other natural questions: assuming the hypothesis of Theorem 2.2, the process {X(t)} t≥0 , under Q, is it recurrent positive? Does there exist an invariant probability distribution? The answer at these questions is no. Here are two simple examples.

Set, for ≥ 0, b := a , d := a 2 , where 0 < a < 1. Therefore, the associated birth and death process is transient since

γ n = n =1 a = a n(n+1) /2 , S = n≥0 a n(n+1) /2 < ∞.
Again, the process is recurrent under Q, by Theorem 2.2. Using (2.22) we get

b = b R +1 R = a R +1 R , d = d R -1 R = a 2 R -1 R .
We compute ≥0 a ( +2j+1) /2 = 1, then lim j→∞ U j+1/U j = 0. By D'Alembert criterion, the series ( 1 /S 2 ) j≥1 U j converges and therefore the necessary condition for the existence of an invariant distribution (see, for instance [START_REF] Lawler | Introduction to Stochastic Processes 2nd edition[END_REF], p. 78 or [START_REF] Resnick | Adventures in Stochastic Processes, 4th printing[END_REF], p. 412) is satisfied. Hence, in this case there exists an invariant distribution for the process {X(t)} t≥0 under Q.

j≥1 j =1 b -1 d = j≥1 R j S 2 j =1 1 a +1 = 1 S 2 j≥1
Set, for ≥ 0, b := a , d := a +1 , where 0 < a < 1. Again, the associated birth and death process is transient since On the one hand, by Theorem 2.2 one knows that the process is recurrent under Q. On the other hand:

γ n = n =1 a = a n , S = n≥0 γ n = 1 1 -a < ∞.
j≥1 j =1 b -1 d = j≥1 1 = ∞.
Hence in this case there is not invariant distribution for {X(t)} t≥0 under Q.

Let us note that in previous examples there is no explosion. A natural question is then the following : if the process explodes is it possible to prove the existence of an invariant distribution? Again the answer is negative as we can see below.

Let us recall, according to [START_REF] Norris | Markov chains. 2nd printing[END_REF] p. As previously, we have: b -1/ d behaves as the series j≥1 (j +1) 2-2a , that is convergent if a > 3 /2 and divergent if a ≤ 3 /2.

Finally, the explosion is not a criterion for the existence of an invariant distribution.

  It can be seen that these assumptions lead to a model of jump Markov process with the holding time parameter at λ := b + d and the jump from ξ = +1 with probability b /(b +d ) -1 with probability d /(b +d ) .

  (B,µ) t } t≥0 its natural filtration and T (B,µ) 0 the first hitting time of 0 by B.

Using ( 2

 2

  90 , that {X(t)} t≥0 can explode if sup ≥0 b + d = ∞. Set, for ≥ 0, b = (1 + ) a and d = a , with a > 1. Then, obviously sup ≥0 b + d = ∞, and ) a < ∞.

  2. The continuous time birth and death process {X(t)} t≥0 is transient if and only if the discrete Markov chain {Y n } n≥0 is transient, that is, if and only if

  2b. Proof of the second part of Theorem 2.1. By using again the fact that {X n } n≥0 is a transient Markov chain, lim n→∞ X n = ∞ almost surely, and by using (2.1) the remainder of the series (2.6) tends to zero. Furthermore U n is almost surely finite, hence by (2.8) we obtain that lim n→∞ M n = 0, almost surely. Then, necessarily, by assuming that {M n } n≥0 is a martingale, it can not be uniformly integrable, since¡ (M 0 ) = 1.Hence the only thing we need to prove is that {M n } n≥0 is a martingale. Since, by (2.8) and (2.6), for each n ≥ 0,

  c. Proof of the third part of Theorem 2.1. Assuming that, under Q 0 , the sequence {X n } n≥0 is a birth and death Markov chain, with transition probabilities given by (2.10), it is no difficult to prove that, under Q 0 , the chain is recurrent. Indeed, it suffices to use the result of Proposition 2.1. Set

		γ0 := 1, γ :=	k=1	qk pk	, ≥ 1.	(3.5)
	Therefore, by using (2.3), (2.10) and (3.2), we can write
	γ =
	≥1	≥1 k=1

  for ≥ 0 integer. We can state a consequence of Theorem 2.2: Corollary 4.2 1. For any t ≥ 0 and any event A t ∈ F t , (2.19) is verified with the positive no uniformly integrable martingale

	M (t) :=	2α 2α + 1	e 2αV (t) Γ(2α + 1) X(t)! Γ(2α + X(t) + 1)	.	(4.4)

Acknowledgement: We thank Bernard Roynette for many valuable discussions on the subject and especially about Proposition 2.5 and Remark 4.1.