P Del Moral

A Kurtzmann

J Tugaut

On the stability and the uniform propagation of chaos of Extended Ensemble Kalman-Bucy filters

Keywords: Extended Kalman-Bucy filter, Ensemble Kalman filters, Monte Carlo methods, mean field particle systems, stochastic Riccati matrix equation, propagation of chaos properties, uniform estimates Mathematics Subject Classification : 60J60, 60J22, 35Q84, 93E11, 60M20, 60G25

This article is concerned with the exponential stability and the uniform propagation of chaos properties of a class of Extended Ensemble Kalman-Bucy filters with respect to the time horizon. This class of nonlinear filters can be interpreted as the conditional expectations of nonlinear McKean Vlasov type diffusions with respect to the observation process. In contrast with more conventional Langevin nonlinear drift type processes, the mean field interaction is encapsulated in the covariance matrix of the diffusion. The main results discussed in the article are quantitative estimates of the exponential stability properties of these nonlinear diffusions. These stability properties are used to derive uniform and non asymptotic estimates of the propagation of chaos properties of Extended Ensemble Kalman filters, including exponential concentration inequalities. To our knowledge these results seem to be the first results of this type for this class of nonlinear ensemble type Kalman-Bucy filters.

Introduction

From the probabilistic viewpoint, the Ensemble Kalman filter (abbreviated EnKF) proposed by G. Evensen in the beginning of the 1990s [START_REF] Evensen | Sequential data assimilation with a non-linear quasi-geostrophic model using Monte Carlo methods to forecast error statistics[END_REF] is a mean field particle interpretation of extended Kalman type filters. More precisely, Kalman type filters (including the conventional Kalman filter and extended Kalman filters) can be interpreted as the conditional expectations of a McKean-Vlasov type nonlinear diffusion. The key idea is to approximate the Riccati equation by a sequence of sample covariance matrices associated with a series of interacting Kalman type filters.

In the linear Gaussian case these particle type filters converge to the optimal Kalman filter as the number of samples (a.k.a. particles) tends to 8. Little is known for nonlinear and/or non Gaussian filtering problems, apart that they don't converge to the desired optimal filter. This important problem is rather well known in signal processing community. For instance, we refer the reader to [START_REF] Kelly | Well-posedness and accuracy of the Ensemble Kalman filter in discrete and continuous time[END_REF][START_REF] Le Gland | Large sample asymptotics for the ensemble Kalman filter The Oxford Handbook of Nonlinear Filtering[END_REF] for a more detailed discussion on these questions in discrete time settings. In this connection, we mention that these ensemble Kalman type filters differ from interacting jump type particle filters and related sequential Monte 1 Carlo methodologies. These mean field particle methods are designed to approximate the conditional distributions of the signal given the observations. It is clearly not the scope of this article to give a comparison between these two different particle methods. For a more thorough discussion we refer the reader to the book [START_REF] Moral | Mean field simulation for Monte Carlo integration[END_REF] and the references therein. We also mention that the EnKF models discussed in this article slightly differ from more conventional EnKF used to approximate nonlinear filtering problems. To be more precise we design a new class of EnKF that converge to the celebrated extended Kalman filter as the number of particles goes to 8.

These powerful Monte Carlo methodologies are used with success in a variety of scientific disciplines, and more particularly in data assimilation method for filtering high dimensional problems arising in fluid mechanics and geophysical sciences [START_REF] Kalnay | Atmospheric modeling, data assimilation, and predictability[END_REF][START_REF] Lisaeter | Assimilation of ice concentration in a coupled ice-ocean model, using the Ensemble Kalman Filter[END_REF][START_REF] Majda | Filtering complex turbulent systems[END_REF][START_REF] Mandel | On the convergence of the ensemble Kalman filter Applications of Mathematics[END_REF][START_REF] Naevdal | Reservoir monitoring and continuous model updating using ensemble Kalman filter[END_REF][START_REF] Ott | A local ensemble Kalman filter for atmospheric data assimilation[END_REF][START_REF] Seiler | Using the EnKF for history matching and uncertainty quantification of complex reservoir models[END_REF][START_REF] Skjervheim | Incorporating 4D seismic data in reservoir simulation models using ensemble Kalman filter[END_REF][START_REF] Wen | Real-Time Reservoir Model Updating Using Ensemble Kalman Filter SPE-92991-MS[END_REF]. A more thorough discussion on the origins and the application domains of EnKF is provided in the series of articles [START_REF] Burgers | On the analysis scheme in the ensemble Kalman filter[END_REF][START_REF] Del Moral | On the stability and the uniform propagation of chaos properties of Ensemble Kalman-Bucy filters[END_REF][START_REF] Evensen | The Ensemble Kalman Filter: theoretical formulation and practical implementation[END_REF][START_REF] Evensen | Using the EnKF for assisted history matching of a North Sea Reservoir Model SPE[END_REF] and in the seminal research monograph by G. Evensen [START_REF] Evensen | Data assimilation : The ensemble Kalman filter[END_REF].

The mathematical foundations and the convergence of the EnKF has started in 2011 with the independent pioneering works of F. Le Gland, V. Monbet and V.D. Tran [START_REF] Le Gland | Large sample asymptotics for the ensemble Kalman filter The Oxford Handbook of Nonlinear Filtering[END_REF], and the one by J. Mandel, L. Cobb, J. D. Beezley [START_REF] Mandel | On the convergence of the ensemble Kalman filter Applications of Mathematics[END_REF]. These articles provide L δ -mean error estimates for discrete time EnKF and show that they converge towards the Kalman filter as the number of samples tends to infinity. We also quote the recent article by D.T. B. Kelly, K.J. Law, A. M. Stuart [START_REF] Kelly | Well-posedness and accuracy of the Ensemble Kalman filter in discrete and continuous time[END_REF] showing the consistency of Ensemble Kalman filters in continuous and discrete time settings. In the latter the authors show that the Ensemble Kalman filter is well posed and the mean error variance doesn't blow up faster than exponentially. The authors also apply a judicious variance inflation technique to strengthen the contraction properties of the Ensemble Kalman filter. We refer to the pioneering article by J.L. Anderson [START_REF] Anderson | An ensemble adjustment Kalman filter for data assimilation[END_REF][START_REF] Anderson | An adaptive covariance inflation error correction algorithm for ensemble filters[END_REF][START_REF] Anderson | Spatially and temporally varing adaptive covariance inflation for ensemble filters[END_REF] on adaptive covariance inflation techniques, and to the discussion given in the end of section 1.2 in the present article.

In a more recent study by X. T. Tong, A. J. Majda and D. Kelly [START_REF] Tong | Nonlinear stability and ergodicity of ensemble based Kalman filters[END_REF] the authors analyze the long-time behaviour and the ergodicity of discrete generation EnKF using Foster-Lyapunov techniques ensuring that the filter is asymptotically stable w.r.t. any erroneous initial condition. These important properties ensure that the EnKF has a single invariant measure and initialization errors of the EnKF will not dissipate w.r.t. the time parameter. Beside the importance of these properties, the only ergodicity of the particle process does not give any information on the convergence and the accuracy of the particle filters towards the optimal filter nor towards any type of extended Kalman filter, as the number of samples tends to infinity.

Besides these recent theoretical advances, the rigorous mathematical analysis of long time behavior of these particle methods is still at its infancy. As underlined by the authors in [START_REF] Kelly | Well-posedness and accuracy of the Ensemble Kalman filter in discrete and continuous time[END_REF], many of the algorithmic innovations associated with the filter, which are required to make a useable algorithm in practice, are derived in an ad hoc fashion. The divergence of ensemble Kalman filters has been observed numerically in some situations [START_REF] Gottwald | A mechanism for catastrophic filter divergence in data assimilation for sparse observation networks[END_REF][START_REF] Kelly | Concrete ensemble kalman filters with rigorous catastrophic filter divergence[END_REF][START_REF] Majda | Catastrophic filter divergence in filtering nonlinear dissipative systems[END_REF], even for stable signals. This critical phenomenon, often referred as the catastrophic filter divergence in data assimilation literature, is poorly understood from the mathematical perspective. Our objective is to better understand the long time behavior of ensemble Kalman type filters from a mathematical perspective. Our stochastic methodology combines spectral analysis of random matrices with recent developments in concentration inequalities, coupling theory and contraction inequalities w.r.t. Wasserstein metrics.

These developments have been started in two recent articles [START_REF] Del Moral | On the stability and the uniform propagation of chaos properties of Ensemble Kalman-Bucy filters[END_REF][START_REF] Del Moral | On the stability and the concentration of extended Kalman-Bucy filters[END_REF]. The first one pro-vides uniform propagation of chaos properties of ensemble Kalman filters in the context of linear-Gaussian filtering problems. The second article is only concerned with extended Kalman-Bucy filters. It discusses the stability properties of these filters in terms of exponential concentration inequalities. These concentration inequalities allow to design confidence intervals around the true signal and extended Kalman-Bucy filters.

The first contribution of the article is to extend these results as the level of the McKean-Vlasov type nonlinear diffusion associated with the ensemble Kalman-Bucy filter. Under some natural regularity conditions we show that these nonlinear diffusions are exponentially stable, in the sense that they forget exponentially fast any erroneous initial condition. These stability properties are analyzed using coupling techniques and expressed in terms of δ-Wasserstein metrics.

The main objective of the article is to analyze the long time behavior of the mean field particle interpretation of these nonlinear diffusions. We present new uniform estimates w.r.t. the time horizon for the bias and the propagation of chaos properties of the mean field systems. We also quantify the fluctuations of the sample mean and covariance particle approximations.

The rest of the article is organized as follows: Section 1.1 presents the nonlinear filtering problem discussed in the article, the Extended Kalman-Bucy filter, the associated nonlinear McKean Vlasov diffusion and its mean field particle interpretation. The two main theorems of the article are described in section 1.2. In a preliminary short section, section 2, we show that the conditional expectations and the conditional covariance matrices of the nonlinear McKean Vlasov diffusion coincide with the EKF. We also provide a pivotal fluctuation theorem on the time evolution of these conditional statistics. Section 3 is mainly concerned with the stability properties of the nonlinear diffusion associated with the EKF. Section 4 is dedicated to the propagation of chaos properties of the extended ensemble Kalman-Bucy filter.

Description of the models

Consider a time homogeneous nonlinear filtering problem of the following form

# dX t " ApX t q dt `R1{2 1 dW t dY t " BX t dt `R1{2 2 dV t
and we set G t " σ pY s , s ď tq.

In the above display, pW t , V t q is an pr 1 `r2 q-dimensional Brownian motion, X 0 is a r 1 -valued random vector with mean and covariance matrix pEpX 0 q, P 0 q (independent of pW t , V t q), the symmetric matrices R 1{2 1 and R 1{2 2 are invertible, B is an pr 2 ˆr1 q-matrix, and Y 0 " 0. The drift of the signal is differentiable vector valued function A :

x P R r 1 Þ Ñ Apxq P R r 1 with a Jacobian denoted by BA : x P R r 1 Þ Ñ Apxq P R pr 1 ˆr1 q .
The Extended Kalman-Bucy filter (abbreviated EKF) and the associated stochastic Riccati equation are defined by the evolution equations

# d p X t " Ap p X t q dt `Pt B 1 R ´1 2 " dY t ´B p X t dt ı with p X 0 " EpX 0 q B t P t " BAp p X t qP t `Pt BAp p X t q 1 `R ´Pt SP t with pR, Sq :" pR 1 , B 1 R ´1 2 Bq
In the above display, B 1 stands for the transpose of the matrix B.

We associate with these filtering models the conditional nonlinear McKean-Vlasov type diffusion process

dX t " A `Xt , ErX t | G t s ˘dt `R1{2 1 dW t `Pηt B 1 R ´1 2 " dY t ´´BX t dt `R1{2 2 dV t ¯ı (1)
with the nonlinear drift function Apx, mq :" A rms `BA rms px ´mq

In the above display pW t , V t , X 0 q stands for independent copies of pW t , V t , X 0 q (thus independent of the signal and the observation path), and P ηt stands for the covariance matrix P ηt " η t " pe ´ηt peqqpe ´ηt peqq 1 ‰ with η t :" LawpX t | G t q and epxq :" x.

We call the stochastic process defined in (1) the Extended Kalman-Bucy diffusion or simply the EKF-diffusion.

In section 2 (see proposition 2.1) we will see that the G t -conditional expectation of the states X t and their G t -conditional covariance matrices coincide with the EKF filter and the Riccati equation presented in [START_REF] Anderson | An ensemble adjustment Kalman filter for data assimilation[END_REF].

The Extended Kalman-Bucy filter (abbreviated En-EKF) coincides with the mean field particle interpretation of the nonlinear diffusion process [START_REF] Anderson | An ensemble adjustment Kalman filter for data assimilation[END_REF].

To be more precise, let pW i t , V i t , ξ i 0 q 1ďiďN be N independent copies of pW t , V t , X 0 q. In this notation, the En-EKF is given by the Mckean-Vlasov type interacting diffusion process

dξ i t " Apξ i t , m t q dt `R1{2 1 dW i t `pt B 1 R ´1 2 " dY t ´´Bξ i t dt `R1{2 2 dV i t ¯ı (2)
for any 1 ď i ď N , with the sample mean and the rescaled particle covariance matrices defined by

m t :" 1 N ÿ 1ďiďN ξ i t and p t :" ˆ1 ´1 N ˙Pη N t " 1 N ´1 ÿ 1ďiďN `ξi t ´mt ˘`ξ i t ´mt ˘1 (3)
with the empirical measures η N t :"

1 N ř 1ďiďN δ ξ i t .
We also consider the N -particle model ζ t " `ζi t ˘1ďiďN defined as ξ t " `ξi t ˘1ďiďN by replacing the sample variance p t by the true variance P t (in particular we have ξ 0 " ζ 0 ).

As mentioned in the introduction the En-EKF (2) differs from the more conventional one defined as above by replacing Apξ i t , m t q by the signal drift Apξ i t q. In this context the resulting sample mean will not converge to the EKF but to the filter defined as in (1) by replacing Ap p X t q by the conditional expectations E pApX t q | G t q. The convergence analysis of this particle model is much more involved than the one discussed in this article. The main difficulty comes from the dependency on the whole conditional distribution of the signal given the observations. We plan to analyze this class of particle filters in a future study.

Statement of the main results

To describe with some precision our results we need to introduce some terminology. We denote by λ min pSq and λ max pSq the minimal and the maximal eigenvalue of a given symmetric matrix S. We let ρpP q " λ max ppP `P 1 qq{2 the logarithmic norm of a given square matrix P . Given pr 1 ˆr2 q matrices P, Q we define the Frobenius inner product xP, Qy " trpP 1 Qq and the associated norm }P } 2 F " trpP 1 P q where trpCq stands for the trace of a given matrix. We also equip the product space R r 1 Rr 1 ˆr1 with the inner product xpx 1 , P 1 q, px 2 , P 2 qy :" xx 1 , x 2 y `xP 1 , P 2 y and the norm }px, P q} 2 :" xpx, P q, px, P qy

In the further development of the article we assume that the Jacobian matrix BA of the signal drift function A satisfies the following regularity conditions:

$ & % λ BA :" ´2 inf xPR r 1 ρpBApxqq ą 0 }BApxq ´BApyq} ď κ BA }x ´y} for some κ BA ă 8. (4)
where ρpP q :" λ max pP q stands for the logarithmic norm of a symmetric matrix P . In the above display }BApxq´BApyq} stands for the L 2 -norm of the matrix operator pBApxq´BApyqq, and }x ´y} the Euclidian distance between x and y. A first order Taylor expansion shows that p4q ùñ xx ´y, Apxq ´Apyqy ď ´λA }x ´y} 2 with λ A ě λ BA {2 ą 0.

(

) 5
Given some δ ě 1, the δ-Wasserstein distance W δ between two probability measures ν 1 and ν 2 on some normed space pE, }.}q is defined by

W δ δ pν 1 , ν 2 q " inf E ´}Z 1 ´Z2 } δ ¯.
The infimum in the above displayed formula is taken of all pair of random variable pZ 1 , Z 2 q such that LawpZ i q " ν i , with i " 1, 2.

In the further development of the article, to avoid unnecessary repetitions we also use the letter "c" to denote some finite constant whose values may vary from line to line, but they don't depend on the time parameter.

Our first main result concerns the stability properties of the EKF-diffusion [START_REF] Anderson | An ensemble adjustment Kalman filter for data assimilation[END_REF]. It is no surprise that these properties strongly depend on logarithmic norm of the drift function A as well as on the size of covariance matrices of the signal-observation diffusion. For instance, we have the uniform moment estimates

λ BA ą 0 ñ @δ ě 1 sup tě0 ! Er}X t } δ s _ trpP t q _ Er}X t ´p X t } δ s ) ď c (6)
A detailed proof of these stochastic stability properties including exponential concentration inequalities can be found in [START_REF] Del Moral | On the stability and the concentration of extended Kalman-Bucy filters[END_REF]. Observe that trpP t q is random so that the above inequality provides an almost sure estimate. To be more precise we use [START_REF] Anderson | An ensemble adjustment Kalman filter for data assimilation[END_REF] to check that B t tr pP t q ď ´λBA tr pP t q `trpRq ùñ trpP t q ď e ´λBA t tr pP 0 q `trpRq{λ BA [START_REF] Del Moral | On the stability and the concentration of extended Kalman-Bucy filters[END_REF] To get one step further in our discussion, we consider the following ratio

λ S :" λ BA ρpSq λ R :" λ BA trpRq and λ K :" λ BA κ BA
The quantity ρpSq is connected to the sensor matrix B and to the inverse of the covariance matrix of the observation perturbations. We also have the rather crude estimate

ρpSq ď trpSq " }BR ´1{2 2 B} 2 F ď trpR ´1 2 q }B} 2 F
Roughly speaking, these three quantities presented above measure the relative stability index of the signal drift with respect to the perturbation degree of the sensor, the one of the signal, and the modulus of continuity of the Jacobian entering into the Riccati equation. For instance, λ S is high for sensors with large perturbations, inversely λ R is large for signals with small perturbations.

Most of our analysis relies on the behavior of the following quantities:

λ R,S :" p8eq ´1 λ R a λ S " 1 `2 λ R λ S  ´1 p λ BA {λ BA :" ˆ1 2 ´2 λ K λ R ˙`ˆ1 2 ´1 ? λ S ˙"1 ´3 4 1 ? λ S 
To better connect these quantities with the stochastic stability of the EKF diffusion we discuss some exponential concentration inequalities that can be easily derived from our analysis. These concentration inequalities are of course more accurate than any type of mean square error estimate. Let p X t pm, pq be the solution of the EKF equation (1) starting at p p X 0 , P 0 q " pm, pq, and let X t pxq be the state of the signal starting at X 0 pxq " x. Let pδq be the function δ P r0, 8rÞ Ñ pδq :" e2 ? 2

" 1 2 `´δ `?δ

¯

For any time horizon t P r0, 8r, and any δ ě 0 the probabilities of the following events

}X t pxq ´p X t pm, pq} 2 ď 1 2e pδq a λ S {λ R,S
`2 e ´λBA t }x ´m} are greater than 1 ´e´δ . The proof of the first assertion is a consequence of theorem 1.1 in [START_REF] Del Moral | On the stability and the concentration of extended Kalman-Bucy filters[END_REF], the proof of the second one is a consequence of the L δ -mean error estimates [START_REF] Majda | Catastrophic filter divergence in filtering nonlinear dissipative systems[END_REF]. These concentration inequalities show that the quantity

a λ S {λ R,S " 8e λ S 1 λ R λ S " 1 `2 λ R λ S 
can be interpreted as the size of a confidence interval around the values of the true signal, as soon as the time horizon is large. It is also notable that the same quantity controls the fluctuations of the EKF diffusion around the values of the EKF. These confidence intervals are small for stable signals with small perturbations. The product term

1 λ R λ S " λ ´2 BA trpR 1 q trpBR ´1
We further assume that

pλ K λ R {4q ^λR,S ^pλ S {4q ą 1 (8)
This condition ensures that 0 ă p λ BA ď λ BA . Condition ( 8) is clearly met as soon as λ R and λ S are sufficiently large. As we shall see the quantity p λ BA represents the Lyapunov stability exponent of the EKF. This exponent is decomposed into two parts. The first one represents the relative contribution of the signal perturbations, the second one is related to the sensor perturbations.

In contrast with the linear-Gaussian case discussed in [START_REF] Del Moral | On the stability and the uniform propagation of chaos properties of Ensemble Kalman-Bucy filters[END_REF], the stochastic Riccati equation (1) depends on the states of the EKF. As shown in [START_REF] Del Moral | On the stability and the concentration of extended Kalman-Bucy filters[END_REF] the stability of the EKF rely on a stochastic Lyapunov exponent that depends on the random trajectories of the filter as well as on the signal-observation processes. The technical condition (8) allows to control uniformly the fluctuations of these stochastic exponents with respect to the time horizon. A more detailed discussion on the regularity condition [START_REF] Evensen | Sequential data assimilation with a non-linear quasi-geostrophic model using Monte Carlo methods to forecast error statistics[END_REF], including a series of sufficient conditions are provided in the appendix, section 5.1.

Let pX t , Xt q be a couple of EKF Diffusions (1) starting from two random states with mean p p X 0 , q X 0 q and covariances matrices pP 0 , q P 0 q. One key feature of these nonlinear diffusions is that the G t -conditional expectations p p X t , q X t q and the G t -conditional covariance matrices pP t , q P t q satisfy the EKF and the stochastic Ricatti equations discussed in (1). Whenever condition ( 8) is satisfied we recall from [START_REF] Del Moral | On the stability and the concentration of extended Kalman-Bucy filters[END_REF] that for any Ps0, 1s there exists some time horizon s such that for any t ě s we have the almost sure contraction estimate

E ´}p p X t , P t q ´p q X t , q P t q} δ S | G s ¯2{δ S ď Z s exp " ´p1 ´ q p λ BA pt ´sq ı }p p X s , P s q ´p q X s , q P s q} 2
with δ S :" 2 ´1 ? λ S , and some random process Z t satisfying the uniform moment condition

sup tě0 E pZ α t q ă 8 with α " 2λ R,S δ S . (9)
These conditional contraction estimates can be used to quantify the stability properties of the EKF. More precisely, if we set P t " Lawp p X t , P t q and q P t " Lawp q X t , q P t q then the above contraction inequality combined with the uniform estimates (6) readily implies that

@t ě t 0 W 2 δ S pP t , q P t q ď c exp " ´t p1 ´ q p λ BA ı
for any P r0, 1r, with some time horizon t 0 . This stability property ensures that the EKF forgets exponentially fast any erroneous initial condition. Of course these forgetting properties of the EKF don't give any information at the level of the process. One of the main objective of the article is to complement these conditional expectation stability properties at the level of the Mc-Kean Vlasov type nonlinear EKF-diffusion (1). Our first main result can basically be stated as follows.

Theorem 1.1. Let pη t , ηt q be the probability distributions of a couple pX t , Xt q of EKF Diffusions (1) starting from two possibly different random states. Assume condition ( 8) is met with δ 1 S :" δ S {4 ě 2. In this situation, for any P r0, 1r there exists some time horizon t 0 such that for any t ě t 0 we have

W 2 δ 1 S pη t , ηt q ď c exp r´t p1 ´ q λs with λ ě p λ BA ^pλ BA {4q (10)
Our next objective is to analyze the long time behavior of the mean field type En-EKF model discussed in [START_REF] Anderson | An adaptive covariance inflation error correction algorithm for ensemble filters[END_REF]. From the practical estimation point of view, only the sample mean and the sample covariance matrices (3) are of interest since these quantities converge to the EKF and the Riccati equations, as N tends to 8. Another important problem is to quantify the bias of the mean field particle approximation scheme. These properties are related to the propagation of chaos properties of the mean field particle model. They are expressed in terms of the collection of probability distributions

P N t " Lawpm t , p t q Q N t " Lawpξ 1 t q and Q t " Lawpζ 1 t q
Theorem 1.2. Assume that λ min pSq ą 0 and condition ( 8) is met with δ R,S :" peλ R,S q δS ě 2 In this situation, there exists some N 0 ě 1 and some β Ps0, 1{2s such that for any N ě N 0 , we have the uniform non asymptotic estimates

trpP 0 q 2 ď λ S λ R " 1 2 `1 λ R λ S  ùñ sup tě0 W δ R,S `PN t , P t ˘ď cN ´β (11)
In addition, when δ R,S ě 4 we have the uniform propagation of chaos estimate

sup tě0 W 2 `QN t , Q t ˘ď cN ´β (12)
We end this section with some comments on our regularity conditions. The condition λ min pSq ą 0 is needed to control the fluctuations of the trace of the sample covariance matrices of the En-EKF, even if the trace expectation is uniformly stable.

Despite our efforts, our regularity conditions are stronger than the ones discussed in [START_REF] Del Moral | On the stability and the uniform propagation of chaos properties of Ensemble Kalman-Bucy filters[END_REF] in the context of linear-Gaussian filtering problems. The main difference here is that the signal stability is required to compensate the possible instabilities created by highly informative sensors when we initialize the filter with wrong conditions.

Next we comment the trace condition in [START_REF] Evensen | Using the EnKF for assisted history matching of a North Sea Reservoir Model SPE[END_REF]. As we mentioned earlier, the stability properties of the limiting EKF-diffusion (1) are expressed in terms of a stochastic Lyapunov exponent that depends on the trajectories of the signal process. The propagation of chaos properties of the mean field particle approximation (2) depend on the long time behavior of these stochastic Lyapunov exponents. Our analysis is based on a refined analysis of Laplace transformations associated with quadratic type stochastic exponents. The existence of these χ-square type Laplace transforms require some regularity on the signal process. For instance at the origin we have

ptrpP 0 q ďq r 1 ρpP 0 q ď 1{p4δq ùñ E ´exp " δ}X 0 ´p X 0 } 2 ı¯ď e (13)
The proof of ( 13) and more refined estimates can be found in [START_REF] Del Moral | On the stability and the concentration of extended Kalman-Bucy filters[END_REF].

From the numerical viewpoint the trace condition in [START_REF] Evensen | Using the EnKF for assisted history matching of a North Sea Reservoir Model SPE[END_REF] is related to the initial location of the particles and the signal-observation perturbations. Signals with a large diffusion part are more likely to correct an erroneous initialization. In the same vein, the estimation problems associated with sensors corrupted by large perturbations are less sensitive to the initialization of the filter. In the reverse angle, when the signal is almost deterministic and the sensor is highly informative the particles need to be initialized close to the true value of the signal.

To better connect our work with existing literature we end our discussion with some connection with the the variance inflation technique introduced by J.L. Anderson in [START_REF] Anderson | An ensemble adjustment Kalman filter for data assimilation[END_REF][START_REF] Anderson | An adaptive covariance inflation error correction algorithm for ensemble filters[END_REF][START_REF] Anderson | Spatially and temporally varing adaptive covariance inflation for ensemble filters[END_REF] and further developed by D.T. B. Kelly, K.J. Law, A. M. Stuart [START_REF] Kelly | Well-posedness and accuracy of the Ensemble Kalman filter in discrete and continuous time[END_REF] and by X. T. Tong, A. J. Majda and D. Kelly [START_REF] Tong | Nonlinear stability and ergodicity of ensemble based Kalman filters[END_REF]. In discrete time settings this technique amounts of adding an extra positive matrix in the Riccati updating step. This strategy allows to control the fluctuations of the sample covariance matrices. In continuous time settings, this technique amounts of changing the covariance matrix P ηt in the EKF diffusion (1) by P ηt `θ Id for some tuning parameter θ ą 0. The resulting EKF-diffusion (1) is given by the equation

dX t " `A `Xt , ErX t | G t s ˘´θ S X t ˘dt `Pηt B 1 R ´1 2 " dY t ´´BX t dt `R1{2 2 dV t ¯ı `"R 1{2 1 dW t ´θ B 1 R ´1{2 2 dV t ı `θ B 1 R ´1 2 dY t
The stabilizing effects of the variance inflation technique are clear. The last term in the r.h.s. of the above displayed formula has no effect (by simple coupling) on the stability properties of the diffusion. The form of the drift also indicates that we increase the Lyapunov exponent by an additional factor θ (as soon as ρpSq ą 0). In addition we increase the noise of the diffusion by a factor θ 2 , in the sense that the covariance matrix of the perturbation term R 1{2 1

dW t ´θ B 1 R ´1{2 2
dV t is given by R 1 `θ2 S. We believe that the stability analysis of these regularized models is simplified by these additional regularity properties. This class of regularized nonlinear diffusions can probably be studied quite easily using the stochastic analysis developed in this article. We plan to develop this analysis in a forthcoming study.

Some preliminary results

This short section presents a couple of pivotal results. The first one ensures that the Extended Kalman-Bucy filter coincides with the G t -conditional expectations of the nonlinear diffusion X t . The second result shows that the stochastic processes pm t , p t q satisfy the same equation as ´p X t , P t ¯, up to some local fluctuation orthogonal martingales with angle brackets that only depends on the sample covariance matrix p t . Proposition 2.1. We have the equivalence EpX 0 q " p X 0 and P η 0 " P 0 ðñ @t ě 0 EpX t | G t q " p X t and P ηt " P t

Proof. Taking the G t -conditional expectations in (1) we find the diffusion equation

d EpX t | G t q " ApEpX t | G t q dt `Pηt B 1 R ´1 2 " dY t ´B EpX t | G t qdt ‰
Let us compute the evolution of P ηt . We set r X t " X t ´EpX t | G t q. In this notation we have

d r X t " BApEpX t | G t qq r X t dt `R1{2 1 dW t ´Pηt B 1 R ´1 2 " B r X t dt `R1{2 2 dV t ı " " BApEpX t | G t qq ´Pηt S ‰ r X t dt `R1{2 1 dW t ´Pηt B 1 R ´1{2 2 dV t This implies that dp r X t r X 1 t q " " " BAp p X t q ´Pt S ı r X t r X 1 t dt `r X t r X 1 t " BAp p X t q ´Pt S ı 1 `pR `Pηt SP ηt q * dt `"R 1{2 1 dW t ´Pηt B 1 R ´1{2 2 dV t ı r X 1 t `r X t " R 1{2 1 dW t ´Pηt B 1 R ´1{2 2 dV t ı 1
Taking the G t -conditional expectations we conclude that

B t P ηt " " BAp p X t q ´Pηt S ı P ηt dt `Pηt " Hp p X t q ´Pηt S ı 1 `pR `Pηt SP ηt q " BAp p X t qP ηt `Pηt BAp p X t q 1 `R ´Pηt SP ηt
This ends the proof of the proposition.

Theorem 2.2 (Fluctuation theorem [START_REF] Del Moral | On the stability and the uniform propagation of chaos properties of Ensemble Kalman-Bucy filters[END_REF]). The stochastic processes pm t , p t q defined in (3) satisfy the diffusion equations

dm t " A rm t s dt `pt B 1 R ´1 2 pdY t ´Bm t dtq `1 ? N dM t ( 14
)
with the vector-valued martingale M t " `M t pkq ˘1ďkďr 1 with the angle-brackets

B t xM t pkq, M t pk 1 qy t " Rpk, k 1 q `pp t Sp t q pk, k 1 q (15)
We also have the matrix-valued diffusion

dp t " `BA rm t s p t `pt BA rm t s 1 ´pt Sp t `R˘d t `1 ? N ´1 dM t ( 16
)
with a symmetric matrix-valued martingale M t " pM t pk, lqq 1ďk,lďr 1 and the angle brackets

B t @
M pk, lq, M pk 1 , l 1 q D t " pR `pt Sp t q pk, k 1 q p t pl, l 1 q `pR `pt Sp t q pl, l 1 q p t pk, k 1 q `pR `pt Sp t q pl 1 , kq p t pk 1 , lq `pR `pt Sp t q pl, k 1 q p t pk, l 1 q (17) In addition we have the orthogonality properties

@ M pk, lq, M pl 1 q D t " @ M pk, lq, V pk 1 q D t " @ M pl 1 q, V pk 1 q D t " 0
for any 1 ď k, l, l 1 ď r 1 and any 1 ď k 1 ď r 2 .

Proof. We have dpξ i t ´mt q " rBA pm t q ´pt B 1 Ss pξ i t ´mt qdt `dM i t with the martingale

dM i t :" R 1{2 1 ˜dW i t ´1 N ÿ 1ďjďN dW j t ¸´p t B 1 R ´1{2 2 ˜dV i t ´1 N ÿ 1ďjďN dV j t
Notice that B t xM i pkq, M i pk 1 qy t " ˆ1 ´1 N ˙pR `pt Sp t q pk, k 1 q and for i " j B t xM i pkq, M j pk 1 qy t " ´1 N pR `pt Sp t q pk, k 1 q

The end of the proof follows the proof of theorem 1 in [START_REF] Del Moral | On the stability and the uniform propagation of chaos properties of Ensemble Kalman-Bucy filters[END_REF], thus it is skipped. This ends the proof of the theorem.

Stability properties

This section is dedicated to the long time behavior of the EKF-diffusion (1), mainly with the proof of theorem 1.1. We use the stochastic differential inequality calculus developed in [START_REF] Del Moral | On the stability and the uniform propagation of chaos properties of Ensemble Kalman-Bucy filters[END_REF][START_REF] Del Moral | On the stability and the concentration of extended Kalman-Bucy filters[END_REF]. Let Y t be some non negative process defined on some probability space pΩ, F, Pq equipped with a filtration F " pF t q tě0 of σ-fields. Also let pZ t , Z t q be some processes and M t be some continuous F t -martingale. We use the notation

dY t ď Z t dt `dM t ðñ `dY t " Z t dt `dM t with Z t ď Z t ˘(18)
We recall some useful algebraic properties of the above stochastic inequalities. Let pY t , Z t , Z t , M t q be another collection of processes satisfying the above inequalities, and pα, αq a couple of non negative parameters. In this case it is readily checked that

dpα Y t `α Y t q ď pα Z t `α Z t q dt `dpα M t `α M t q and dpY t Y t q ď " Z t Y t `Zt Y t `Bt xM, My t ı dt `Yt dM t `Yt dM t
We consider a couple of diffusions pX t , Xt q coupled with the same Brownian motions pV t , W t q and the same observation process Y t , and we set F t :" G t _ σ ´pX s , Xs q, s ď t Next proposition provides uniform estimates of the L δ -centered moments of the EKFdiffusion with respect to the time horizon. Proposition 3.1. Assume that λ BA ą 0. In this situation, for any δ ě 1 and any time horizon s ě 0 we have the uniform almost sure estimates

E ´}X t ´p X t } δ | F s ¯2{δ ď e ´λBA pt´sq }X s ´p X s } 2 `p2δ ´1q " λ ´1 R p1 `2 pλ R λ S q ´1q `2e ´λBA pt`sq trpP 0 q 2 λ ´1 S ı (19)
Proof. We have

dpX t ´p X t q " " BAp p X t q ´Pt S ı pX t ´p X t q dt `R1{2 1 dW t ´Pt B 1 R ´1{2 2 dV t
This implies that

d}X t ´p X t } 2 " " 2xX t ´p X t , " BAp p X t q ´Pt S ı pX t ´p X t qy `trpR 1 q `trpP 2 t Sq ı dt `dM t ď " ´λBA }X t ´p X t } 2 `Ut ı dt `dM t
with the process U t :" trpRq `trpP 2 t Sq ď trpRq `ρpSqtrpP t q 2 ď trpRq `ρpSq ´e´λ BA t trpP 0 q `1{λ R

¯2

and the martingale

dM t :" 2xX t ´p X t , R 1{2 1 dW t ´Pt B 1 R ´1{2 2 dV t y
Observe that the angle bracket of this martingale satisfy the property

B t xM y t " 4xX t ´p X t , pR `Pt SP t q pX t ´p X t qy ď 4}X t ´p X t } 2 }R `Pt SP t }
By corollary 2.2 in [START_REF] Del Moral | On the stability and the concentration of extended Kalman-Bucy filters[END_REF] for any δ ě 1 we have

E ´}X t ´p X t } δ | F s ¯2{δ ď exp p´λ BA pt ´sqq }X s ´p X s } 2 `p2δ ´1q ż t s exp p´λ BA pt ´uqq `trpRq `ρpSqtrpP u q 2 ˘du
Observe that by ( 7)

ρpSq ż t s exp p´λ BA pt ´uqq trpP u q 2 du ď 2ρpSq ż t s exp p´λ BA pt ´uqq " e ´2λ BA u trpP 0 q 2 `1{λ 2 R ı du ď 2pλ 2 R λ S q ´1 `2 exp p´λ BA pt `sqqtrpP 0 q 2 λ ´1 S
This ends the proof of the proposition.

Theorem 3.2. When the initial random states X 0 and X0 have the same first and second order statistics p p X 0 , P 0 q " p q X 0 , q P 0 q we have the almost sure contraction estimates:

}X t ´X t } 2 ď exp r´λ BA ts }X 0 ´X 0 } 2
More generally, when condition ( 8) is met with λ S ě 4 4 , for any P r0, 1r there exists some s such that for any t ě s and any 1 ď δ ď 4 ´4 ? λ S we have

E ´}X t ´X t } 2δ | F s ¯1{δ ď exp " ´p1 ´ qλ BA pt ´sq ‰ " }X s ´X s } 2 `Zs ı ( 20
)
with some exponent λ BA ě p λ BA ^pλ BA {2q, and some process Z t satisfying the uniform moment condition

sup tě0 E ´Zα{4 t ¯ă 8 for any α ď λ R,S a λ S . (21)
Before getting into the details of the proof of this theorem we mention that ( 10) is a direct consequence of (20) combined with the uniform estimates [START_REF] Majda | Catastrophic filter divergence in filtering nonlinear dissipative systems[END_REF]. Indeed, applying [START_REF] Mandel | On the convergence of the ensemble Kalman filter Applications of Mathematics[END_REF], for any δ ě 2 we have

E ´}X t ´X t } δ ¯1{δ ď exp " ´p1 ´ qλ BA pt ´sq{2 ‰ ˆE " }X s ´X s } δ ı 1{δ `E " Z δ{2 s ı 1{δ
Using [START_REF] Majda | Catastrophic filter divergence in filtering nonlinear dissipative systems[END_REF] and the fact that

1 ď δ{2 ď 16 ´1 a λ S ď 4 ´1 λ R,S a λ S
we conclude that W δ pη t , ηt q ď c exp

" ´t p1 ´ qp1 ´s{tqλ BA {2 ‰ ď c exp " ´t p1 ´2 qλ BA {2 ‰
as soon as s{t ď . The end of the proof of ( 10) is now clear. Now we come to the proof of the theorem.

Proof of theorem 3.2:

We have

dX t " ApX t , p X t q dt `R1{2 1 dW t `Pt B 1 R ´1 2 " dY t ´´BX t dt `R1{2 2 dV t ¯ı
Using the decomposition q P t S Xt ´Pt SX t " ´Pt SpX t ´X t q `p q P t ´Pt qS Xt we readily check that When the initial random states X 0 and X0 are possibly different but they have the same first and second order statistics we have p X 0 " q X 0 and P 0 " q P 0 ùñ @t ě 0 p X t " q X t and P t " q P t

d ´Xt ´X t " !" ApX t , p X t q ´Ap Xt , q X t q ı ´Pt SpX t ´X t q ) dt
In this particular situation we have ApX t , p X t q ´Ap Xt , q X t q " BAp q X t q pX t ´X t q and B t ´Xt

´X t ¯" " BAp q X t q ´Pt S ı pX t ´X t q
This implies that

B t }X t ´X t } 2 " 2xpX t ´X t q, " BAp q X t q ´Pt S ı pX t ´X t qy ď ´λBA }X t ´X t } 2
This ends the proof of the first assertion.

More generally, we have ApX t , p X t q ´Ap Xt , q X t q " BAp q X t q pX t ´X t q `"Ap p X t q ´Ap q X t q ı ´BAp q X t qp p X t ´q X t q `"BAp p X t q ´BAp q X t q ı pX t ´p X t q

This yields the estimate We conclude that λ S and any t ě s we have

xX t ´X t , ´ApX t , p X t q ´Ap Xt , q X t q ¯´P t SpX t ´X t qy ď ´λBA 2 }X t ´X t } 2 `xX t ´X t , " BAp p X t q ´BAp q X t q ı pX t ´p X t qy `xX t ´X t , " Ap p X t q ´Ap q X t q ı ´BAp q X t qp p X t ´q X t qy ď ´λBA 2 }X t ´X t } 2 `} p X t ´q X t } }X t ´X t }
d}X t ´X t } 2 ď " ´λBA 2 }X t ´X t } 2 `Ut  dt `2a V t }X t ´X t }
E ´αδ{4 t } p X t ´q X t } δ{2 | F s ¯4{δ ď E ´} p X t ´q X t } δ | F s ¯2{δ E ´αδ{2 t | F s ¯2{δ ď Z s exp ´´p λ BA p1
´ qpt ´sq for some process Z s satisfying the uniform moment condition [START_REF] Naevdal | Reservoir monitoring and continuous model updating using ensemble Kalman filter[END_REF]. In the same vein we check that

E ´Uδ{4 t | F s ¯4{δ _ E ´Vδ{4 t | F s ¯4{δ ď Z s exp ´´p λ BA p1
´ qpt ´sq for any s ě t 0 . By corollary 2.2 in [START_REF] Del Moral | On the stability and the concentration of extended Kalman-Bucy filters[END_REF] we have

E ´}X t ´X t } δ{4 | F s ¯8{δ ď exp ˆ´" λ BA 2 pt ´sq ˙} X s ´X s } 2 `n Z s ż t s exp ˆ´" λ BA 2 pt ´uq `p λ BA p1 ´ qpu ´sq ˙d u ď e ´λBA 2 pt´sq }X s ´X s } 2 `n Z t 0 | p λ BA p1 ´ q ´λBA {2| |e ´λBA 2 pt´sq ´e´p λ BA p1´ qpt´sq |
The end of the proof of the theorem is now easily completed.

4 Quantitative propagation of chaos estimates

Laplace exponential moment estimates

The analysis of EKF filters and their particle interpretation is mainly based on the estimation of the stochastic exponential function

E Γ ptq :" exp "ż t 0 Γ A psq ds 
with the stochastic functional Γ A psq :" ´"λ BA ´´2κ BA trpP t q `ρpSq }X t ´p X t } ¯ı Assume condition ( 8) is satisfied and set

Λ BA r , δs {λ BA :" 1 ´2 λ K λ R `1 λ S ˆ3 4 ´δ˙´1 δ λ A 2λ BA
Observe that for any δ ą 0 we have

" 1 2
λ BA λ A ùñ Λ BA " , a λ S {2 ı " p λ BA ě Λ BA r , δs
The next technical lemma provides some key δ-exponential moments estimates. Its proof is quite technical, thus it is housed in the appendix, section 5.2. Lemma 4.1.

• For any δ ą 0 and any 0 ď s ď t we have the almost sure estimate

E ´pE Γ ptq{E Γ psqq ´δ | F s ¯1{δ ď exp `ΛΓ pt ´sq ˘with Λ Γ " λ BA " 1 ´2 λ K λ R  (22)
• For any P r0, 1s, any 0 ă δ ď e λ R,S and any initial covariance matrix P 0 such that trpP 0 q 2 ď σ 2 p , δq :"

λ S λ R " 1 2 `1 λ R λ S  pe λ R,S {δ ´1q
for any time horizon t ě 0 we have the exponential δ-moment estimate

E " E Γ ptq δ ı 1{δ ď c δ pP 0 q exp " Λ Γ p , δq t ‰ ( 23
)
with the parameters Λ Γ p , δq :" 2κ BA σpδq ´ΛBA r , δs ´pδ ´1q ρpSq c δ pP 0 q :" exp `1{δ `δχpP 0 q{p2λ S q 2 • For any Ps0, 1s there exists some time horizon s such that for any t ě s and any δ ď ? λ S {2 we have the almost sure estimate

E ´EΓ ptq δ | F s ¯1{δ ď E Γ psq Z s exp ´´! p1 ´ q p λ BA `pδ ´1qρpSq ) pt ´sq ¯(24)
for some positive random process Z t s.t.

@α ď λ R,S a λ S sup tě0 E pZ α t q ă 8

A non asymptotic convergence theorem

This section is mainly concerned with the estimation of the δ-moments of the square errors Ξ t :" }pm t , p t q ´p p X t , P t q} 2 " }m t ´p X t } 2 `}p t ´Pt } 2

F

The analysis is based on a couple of technical lemmas.

The first one provides uniform moments estimates with respect to the time parameter.

Lemma 4.2. Assume that λ min pSq ą 0. In this situation there exists some ν ą 0 such that for any 1 ď n ď 1 `νN we have

sup tě0 E ptrpp t q n q ă 8 sup tě0 E `}ξ 1 t } n ˘ă 8 and sup tě0 E `}ζ 1 t } n ˘ă 8
The second technical lemma provides a differential perturbation inequality in terms of the Laplace functionals discussed in section 4.1.

Lemma 4.3. Assume that λ min pSq ą 0. We have the stochastic differential inequality

dΞ t ď Ξ t " Γ A ptq `a2ρpSq d Υ p1q t ı `"V t dt `aV t Ξ t d Υ p2q t
ı with a couple of orthogonal martingales s.t. B t xΥ piq ¨, Υ pjq ¨yt ď 1 i"j and some non negative process V t such that sup tě0 E pV n t q 1{n ď cpnq{N for any 1 ď n ď 1 `νN and some ν ą 0.

The proofs of these two lemmas are rather technical thus they are provided in the appendix, section 5.3 and section 5.4. We are now in position to state and to prove the main result of this section. λ S q ^peλ R,S q ě 2. In this situation, there exists some N 0 ě 1 and some α Ps0, 1s such that for any N 0 ď N , 1 ď δ ď p4 ´1? λ S q ^p2 ´1eλ R,S q and any initial covariance matrix P 0 of the signal we have the uniform estimates

trpP 0 q 2 ď 1 2 λ S λ R " 1 `2 λ R λ S  ùñ sup tě0 ErΞ δ t s 1{δ ď c{N α
Proof. We set Eptq :" E Γ ptqE Υ ptq " e Lt with the exponential martingale

E Υ ptq :" exp " a 2ρpSq Υ p1q t ´ρpSqt
ı and the stochastic process

L t :" ż t 0 Γ A puq du `a2ρpSq Υ p1q t ´ρpSqt
Observe that for any δ ě 0 we have

E ´δ Υ ptq " exp " ´δa 2ρpSq Υ p1q t `δρpSqt ı " exp rδp1 `2δqρpSqts E 1{2 ´2δΥ ptq
with the exponential martingale

E ´2δΥ ptq :" exp " ´2δ a 2ρpSq Υ p1q t ´4δ 2 ρpSqt ı
In the same vein we have

E δ Υ ptq " exp " δ a 2ρpSq Υ p1q t ´δρpSqt ı " exp rδp2δ ´1qρpSqts E 1{2 2δΥ ptq
with the exponential martingale

E 2δΥ ptq :" exp " 2δ a 2ρpSq Υ p1q t ´4δ 2 ρpSqt ı This yields the estimates E ´E´δ ptq ¯" exp pδp1 `2δqρpSqtq E " E Γ ptq ´δ E 1{2 ´2δΥ ptq ı ď E " E Γ ptq ´2δ ı 1{2 exp pδp1 `2δq ρpSqtq E ´Eδ ptq ¯ď E " E Γ ptq 2δ ı 1{2 exp pδp2δ ´1q ρpSqtq
Using ( 22) and ( 23) we find the estimates

E ´E´δ ptq ¯1{δ ď exp `"p1 `2δq ρpSq `ΛΓ ‰ t ˘(25) E ´Eδ ptq ¯1{δ ď c δ pP 0 q exp `"p2δ ´1q ρpSq `ΛΓ p , δq ‰ t ˘(26)
The estimate ( 26) is valid for any P r0, 1s and any δ ď e λ R,S and trpP 0 q ď σp , δq

Using the fact that

dE ´1ptq ď ´e´Lt ´ΓA ptq dt `a2ρpSq d Υ p1q t ´ρpSqdt ¯`1 2 e ´Lt 2ρpSq B t xΥ p1q y t dt ď ´E´1 ptq ´ΓA ptq dt `a2ρpSq d Υ p1q t
we find the stochastic inequality

dpΞ t E ´1ptqq ď E ´1ptq dΞ t `Ξt dE ´1ptq ´2E ´1ptq Ξ t ρpSq dt ď E ´1ptq Ξ t " Γ A ptq `a2ρpSq d Υ p1q t ı `E´1 ptq " V t dt `aV t Ξ t d Υ p2q t ı ´E´1 ptq Ξ t " Γ A ptq dt `a2ρpSq d Υ p1q t ı ´2E ´1ptq Ξ t ρpSq dt " E ´1ptq " pV t ´2 Ξ t ρpSqq dt `aV t Ξ t d Υ p2q t ı
For any δ ě 2, this implies that

dpΞ t E ´1ptqq δ ď δ Ξ δ´1 t E ´δ ptq " pV t ´2 Ξ t ρpSqq dt `aV t Ξ t d Υ p2q t ı `δ Ξ δ´1 t Eptq ´δ pδ ´1q 2 V t dt " δ Ξ δ´1 t E ´δ ptq "ˆδ `1 2 V t ´2 Ξ t ρpSq ˙dt `aV t Ξ t d Υ p2q t 
Taking the expectation we obtain

B t E " pΞ t E ´1ptqq δ ı ď δpδ `1q 2 E " ´Ξt E ´1ptq ¯δ´1 E ´1ptq V t  ´2δ ρpSq E " ´Ξt E ´1ptq ¯δ
On the other hand using lemma 4.3 and the Laplace estimate [START_REF] Tong | Nonlinear stability and ergodicity of ensemble based Kalman filters[END_REF] we have

E ˆ´Ξ t E ´1ptq ¯δ´1 E ´1ptq V t ď E ˆ´Ξ t E ´1ptq ¯δ˙1 ´1{δ E ´E´2δ ptq ¯1{p2δq E `V2δ t ˘1{p2δq ď c N exp `"p1 `4δq ρpSq `ΛΓ ‰ t ˘E ˆ´Ξ t E ´1ptq ¯δ˙1 ´1{δ
This yields

B t E ´pΞ t E ´1ptqq δ ¯1{δ ď 1 δ E ´pΞ t E ´1ptqq δ ¯1 δ ´1 B t E ´pΞ t E ´1ptqq δ ď ´2ρpSqE ´pΞ t E ´1ptqq δ ¯1{δ `pδ `1q 2 c N exp `"p1 `4δqρpSq `ΛΓ ‰ t from
which we conclude that

E ´pΞ t E ´1ptqq δ ¯1{δ ď exp t´2ρpSqtu E ´Ξδ 0 ¯1{δ `c N exp `p1 `4δqρpSq `ΛΓ ˘t( ď c N exp `p1 `4δqρpSq `ΛΓ ˘t(
By Cauchy Schwartz inequality we also have

E ´Ξδ{2 t ¯2{δ " E ˆEptq δ{2 ´Ξt E ´1ptq ¯δ{2 ˙2{δ ď E ´pΞ t E ´1ptqq δ ¯1{δ E ´Eδ ptq ¯1{δ
Using [START_REF] Wen | Real-Time Reservoir Model Updating Using Ensemble Kalman Filter SPE-92991-MS[END_REF] we conclude that for any P r0, 1s and any δ ď e λ R,S and trpP 0 q ď σp , δq

E ´Ξδ{2 t ¯2{δ ď c δ pP 0 q c N exp `6δρpSq `ΛΓ `ΛΓ p , δq ˘t( (27)
On the other hand, by theorem 2.1 in [START_REF] Del Moral | On the stability and the concentration of extended Kalman-Bucy filters[END_REF] for any δ ě 1 we also have

E ´Ξδ{2 t | F s ¯2{δ ď E " exp ˆδ ż t s tΓ A puq `pδ ´1qρpSqu du ˙| F s  1{δ ˆ"Ξ s `1 N δ `1 2 ż t s E " V δ u | F s ı 1{δ du * ( 28
)
with the rescaled process V t :" exp ˆż t s r´Γ A puq `2p1 ´δqρpSqs du ˙Vt of the process V t defined in lemma 4.3.

On the other hand using [START_REF] Skjervheim | Incorporating 4D seismic data in reservoir simulation models using ensemble Kalman filter[END_REF] for any Ps0, 1s there exists some time horizon s " sp q such that for any t ě s and any δ ď This implies that for any 1 ď δ{2 ď 1

4

? λ S we have

E ´Ξδ{2 t | F s ¯ď c Z δ{2 s exp ˆ´δ 2 p1 ´ q p λ BA pt ´sq "Ξ δ{2 s `1 N δ{2 exp " δ 2 `2p1 ´δqρpSq `ΛΓ ˘pt ´sq *
Taking the expectation and choosing ď 1{2, there exists some time horizon t 0 such that for any s ě 0 and any τ ě s `t0

E ´Ξδ{2 τ ¯2{δ
ď c exp ´´p λ BA pτ ´ps `t0 qq{2 ¯"1 `1 N exp "`2 p1 ´δqρpSq `ΛΓ ˘pτ ´ps `t0 q ‰ * for any 2 ď δ ď 1 `νN for some ν ą 0, and for some finite constant cpδq ă 8. This implies that for any time horizon t ě 0 and any

λ Γ :" Λ Γ ´2ρpSq " λ BA 2 "ˆ1 ´4 λ K λ R ˙`ˆ1 ´4 λ S ˙ ą 0
On the other hand, by (27) for any time horizon t ě 0 and any δ ď e λ R,S and any P 0 s.t. trpP 0 q ď σp1, eλ R,S {2q we have the uniform estimates sup sPr0,t 0 `ts

E ´Ξδ{2 s ¯2{δ ď c 1 N exp " λ 1 Γ t ‰ with λ 1 Γ :" 5eλ BA λ R,S {λ S `ΛΓ `ΛΓ p1, eλ R,S {2q
We conclude that for any time horizon t ě 0

sup sě0 E ´Ξδ{2 s ¯2{δ ď c # exp « ´p λ BA 2 t ff `1 N exp " pλ Γ _ λ 1 Γ qt ‰ + Choosing t " tpN q such that t " tpN q :" log N { ! p λ BA {2 `pλ Γ _ λ 1 Γ q ) We conclude that sup sě0 E ´Ξδ{2 s ¯2{δ ď c N ´α with α " p λ BA p λ BA `2 `λΓ _ λ 1 Γ ˘Ps0, 1s
This ends the proof of the theorem.

Corollary 4.5. Assume that p4 ´1?

λ S q ^p2 ´1eλ R,S q ě 2. In this situation, there exists some N 0 ě 1 and some α Ps0, 1s such that for any N 0 ď N and any initial covariance matrix P 0 of the signal

trpP 0 q 2 ď 1 2 λ S λ R " 1 `2 λ R λ S  ùñ sup tě0 E `}ξ 1 t ´ζ1 t } 2 ˘ď cpP 0 q{N α
for some finite constant cpP 0 q ă 8 whose values depends on P 0 .

Proof. Using (2) we have

dpξ 1 t ´ζ1 t q "
" pBApm t q ´pt Sq ξ 1 t `pt SX t `Apm t q ´BApm t q m t ‰ dt

´"pBAp p X t q ´Pt Sq ζ λ S q ^p2 ´1eλ R,S q ě 2 and initial covariance matrix P 0 of the signal is chosen so that trpP 0 q 2 ď 1 2

λ S λ R " 1 `2 λ R λ S  This implies that B t E `}ξ 1 t ´ζ1 t } 2 ˘ď ´λBA 2 E `}ξ 1 t ´ζ1 t } 2 ˘`cpP 0 q{N α
The end of the proof of the corollary is now a direct consequence of Gronwall lemma.

Appendix

Regularity conditions

Notice that for any α, x ě 0 we have

x 1 `1{x ą 2α ðñ x ą α ´1 `a1 `2{α ānd by (8) λ R,S ą p8eq ´1 λ R a λ S " 1 `1 λ R ? λ S  ´1
This shows that We have

p8eq ´1 λ R ? λ S " 1 `1 λ R ? λ S ı ą α ðñ λ R a λ S ą 4
´ΓA ptq " λ BA ´´2κ BA trpP t q `ρpSq }X t ´p X t } ¯ď λ BA r1 ´2{pλ K λ R qs
The end of the proof of ( 22) is now clear. Observe that

E Γ ptq δ " exp " δ ż t 0 "´2 κ BA trpP s q `ρpSq }X s ´p X s } ¯´λ BA ı ds  ď exp " δλ BA " 2 λ K ˆtrpP 0 q `1 λ R ˙´1  t  exp " δ ρpSq ż t 0 }X s ´p X s } ds 
We let φ t pxq " X t be the stochastic flows of signal starting at X 0 " x. We recall the contraction inequality }φ t pxq ´φt pyq} ď exp p´λ BA t{2q }x ´y} (29)

A proof of (29) can be found in [START_REF] Del Moral | On the stability and the concentration of extended Kalman-Bucy filters[END_REF], section 3.1. This inequality implies that

ż t 0 }X r ´p X r } dr " ż t 0 }φ r pX 0 q ´p X r } dr ď ż t 0 }φ r pX 0 q ´φr p p X 0 q} dr `ż t 0 }φ r p p X 0 q ´p X r } dr ď ˆż t 0 e ´λBA r{2 dr ˙}X 0 ´p X 0 } `ż t 0 }φ r p p X 0 q ´p X r } dr ď 2}X 0 ´p X 0 }{λ BA `ż t 0 }φ r p p X 0 q ´p X r } dr This implies that exp " δ ρpSq ż t 0 }X s ´p X s } ds  ď exp " 2δ}X 0 ´p X 0 }{λ S ı exp " δρpSq ż t 0 }φ s p p X 0 q ´p X s } ds 
Using the estimate x ´1{4 ď x 2 , which is valid for any x we have

ż t 0 pp}φ u p p X 0 q ´p X u } ´1{4q `1{4q du ď t{4 `ż t 0 }φ r p p X 0 q ´p X r } 2 dr we find that exp " δρpSq ż t 0 }X s ´p X s } ds  ď exp " 2δ}X 0 ´p X 0 }{λ S ı exp pδtρpSq{4q ˆexp " δρpSq ż t 0 }φ s p p X 0 q ´p X s } 2 ds  This yields E " exp " δ ρpSq ż t 0 }X s ´p X s } ds  | X 0  ď exp ptδρpSq{4q exp " 2δ}X 0 ´p X 0 }{λ S ı ˆE " exp " δρpSq ż t 0 }φ s p p X 0 q ´p X s } 2 ds

2

We also have the series of inequalities

1 ρpSq 1 1 `πBA p0q λ 2 A 4trpRq ě 1 ρpSq λ 2 BA 4 1 1 `πBA p0q 1 4trpRq ě λ S λ R 2 ˆ42 1 1{2 `trpP 0 q 2 pρpSq{trpRqq `ρpSqtrpRq{λ 2 BA " 1 4 2 λ S λ R ˆ2 λ R λ S trpP 0 q 2 `"1 `2 λ R λ S ˙´1 ě e 8e a λ S λ R " 1 `2 λ R λ S  ´1 ˜1 `2 λ R λ S trpP 0 q 2 " 1 `2 λ R λ S  ´1¸´1 " e λ R,S ˜1 `2 λ R λ S trpP 0 q 2 " 1 `2 λ R λ S  ´1¸´1
This shows that δρpSq ď 1 `πBA p0q

λ 2 A 4trpRq
for some P r0, 1s as soon as

trpP 0 q 2 ď 1 2 λ S λ R " 1 `2 λ R λ S  ´e δ λ R,S ´1¯f or any δ ď e λ R,S
The end of the proof of ( 23) is a direct consequence of theorem 3.2 in [START_REF] Del Moral | On the stability and the concentration of extended Kalman-Bucy filters[END_REF].

The last assertion resumes to lemma 4.1 in [START_REF] Del Moral | On the stability and the concentration of extended Kalman-Bucy filters[END_REF]. This ends the proof of the lemma.

Proof of lemma 4.2

Using (16) we have dtrpp t q " `trppBA rm t s `BA rm t s 1 qp t q ´trpSp 2 t q `trpRq ˘dt `1 ?

N ´1 dM t ď " ´λBA trpp t q ´r´1 1 λ min pSq trpp t q 2 `trpRq ‰ dt `1 ? N ´1 dM t
with a martingale M t with an angle bracket B t xMy t " 4trppR `pt Sp t qp t q ď 4trpp t q `ρpRq `ρpSq trpp t q 2 Ȗsing lemma 4.1 in [START_REF] Del Moral | On the stability and the uniform propagation of chaos properties of Ensemble Kalman-Bucy filters[END_REF] we have The end of the proof follows the same arguments as above, so it is skipped. This completes the proof of the lemma.

Proof of lemma 4.3

By ( 14) and ( 16) we have dpp t ´Pt q " Π t dt `dM t and dpm t ´p X t q " Π t dt `dM t with the drift terms Π t " ´BApm t qp t ´BAp p X t qP t ¯`´B Apm t qp t ´BAp p X t qP t

¯1

`pP t ´pt qSP t `ppP t ´pt qSP t q 1 ´pP t ´pt qSpP t ´pt q Π t " pApm t q ´Ap p X t qq ´pt Spm t ´p X t q `pp t ´Pt qSpX t ´p X t q and the martingales dM t :" This ends the proof of the lemma.

Theorem 4 . 4 .

 44 Assume that p2´1?

1 5. 2

 12 α e ´1 `a1 `1{p2αeq ¯ùñ λ R,S ą α Also observe that λ S ą 4 and λ R ą 2 α e ´1 `a1 `1{p2αeq ¯ùñ λ R,S ą α This yields the sufficient condition pλ K λ R q ^λS ą 4 and λ R a λ S ą 4e ´1 `a1 `1{p2eq ¯ùñ p8q Also observe that for any α ě 1 we have pλ K {αq ^pλ S {4q ą 1 and λ R ą 2 α e ´1 `a1 `1{p2αeq ùñ pλ K λ R {4q ^pλ R,S {αq ^pλ S {4q ą Proof of lemma 4.1

  `"P t ´q P t ı SpX t ´X t q dt `dM t

	with the martingale
	dM t :"	" P t ´q P t	ı	B 1 R 2 ´1{2	dpV t ´V t q
	ñ B t xMy t " }	" P t ´q P t	ı	B 1 R 2 ´1{2	} 2 F " tr ˆ"P t ´q P t	ı 2	S ˙ď V

t :" ρpSq }P t ´q P t } 2 F

  ´κBA }X t ´p X t } `κBA `}BA} SpX t ´X t qy ď }P t ´q P t } F }X t ´X t } }SpX t ´X t q} ´X t } 2 `2 } p X t ´q X t } }X t ´X t } ´κBA }X t ´p X t } `κBA `}BA} ¯ı dt `"2}P t ´q P t } F }X t ´X t } }SpX t ´X t q} V t }X t ´X t } dM t with V t " ρpSq }P t ´q P t } 2 F and a rescaled continuous martingale M t such that B t xMy t ď 1. On the other hand, we have 2 }X t ´X t } } p X t ´q X t } ´κBA }X t ´p X t } `κBA `}BA} ´X t } 2 `4 λ BA } p X t ´q X t } 2 ´κBA }X t ´p X t } `κBA `}BA} ¯2 and 2}P t ´q P t } F }X t ´X t } }SpX t ´X t q} ď λ BA 4 }X t ´X t } 2 `4 λ BA }P t ´q P t } 2 F }SpX t ´X t q} 2

	This implies that
	d}X t ´X t } 2
		"
	ď	´λBA }X t ı
			dt	`2? ď
		λ BA 4	}X t
			We
	also have
			"	ı
		xX t ´X t ,	P t ´q P t

  BA `}ζ 1 t } `}m t } ˘`2}BA} ˘2 }m t ´p X t } 2

	This yields
	dpξ 1 t	´ζ1 t q
		"					ı
	"	pBApm t q ´pt Sq pξ 1 t	´ζ1 t q `pp t ´Pt qSpX t	´ζ1 t q `pBApm t q ´BAp p X t qq ζ 1 t	dt
	`"´A	pm t q ´Ap p X t q	¯`´B	Ap p X t q ´BApm t q ¯mt `BAp p X t q p p X t ´mt q	ı	dt `dM t
	with
							ÿ	B t xMpkq, Mpkqy t ď 2ρpSq } p t ´Pt } 2 F
							1ďkďr 1
	This implies that
	d}ξ 1 t	´ζ1 t } 2
	ď 2xξ 1 t	´ζ1 t , dpξ 1 t	´ζ1 t qy `2ρpSq } p t ´Pt } 2 F dt
	ď ´λBA }ξ 1 t	´ζ1 t } 2 `2ρpSq } p t ´Pt } 2 F `2}ξ 1 t	´ζ1 t }
							ı)
	ˆ"}p t ´Pt } F }SpX t	´ζ1 t q} ``κ BA `}ζ 1 t } `}m t } ˘`2}BA} ˘}m t ´p X t }	dt `dM t
	with the martingale
							dM t " 2xξ 1 t	´ζ1 t , dM t y
	Notice that
	2ρpSq } p t ´Pt } 2 F
	`2}ξ 1 t		´ζ1 t }	" }p t ´Pt } F }SpX t	´ζ1 t q} ``κ BA `}ζ 1 t } `}m t } ˘`2}BA} ˘}m t ´p X t }	ı
	ď	λ BA 2	}ξ 1 t	´ζ1 t } 2 ˆ t
	with the process
	t :" 2ρpSq } p t ´Pt } 2 F
	`4 " }p t ´Pt } 2 F }SpX t	´ζ1 t q} 2 ``κ ı	{λ BA
	By theorem 4.4 we have
							sup	Ep t q ď cpP 0 q{N α
							tě0
	as soon as p4	´1?	1 t `Pt SX t `Ap p X t q ´BAp p X t q p X t	ı	dt `dM t
	with the martingale
							dM t :" pp t ´Pt qB 1 R 2 ´1{2	dpV t	´V 1 t q

  By[START_REF] Kelly | Concrete ensemble kalman filters with rigorous catastrophic filter divergence[END_REF] we have dm t " rA rm t s ´pt Sm t `pt SX t s dt `pt B 1 R Since M t is independent of V t we have d}m t } 2 " p2 xm t , rA rm t s ´pt Sm t `pt SX t sy `trpR `pt Sp t qq dt `d Ă M t " 4 xm t , pR `pt Sp t qm t y{N `4 xm t , pp t Sp t qm t y ď V t }m t } 2 with V t :" 4 rtrpR `pt Sp t q{N `trpp t Sp t qs Observe that xm t , A rm t sy " xm t ´0, A rm t s ´Ap0qy `xm t , A r0sy ď ´λA }m t } 2 `}Ap0q} }m t } ď ´pλ A {2q }m t } 2 `}Ap0q} 2 {p2λ A q This yields the estimate d}m t } 2 ď `´λ A }m t } 2 `}Ap0q} 2 {λ A `2}m t } }p t S} }X t } `trpR `pt Sp t q ˘dt `d Ă M t from which we find that d}m t } 2 ď ˆ´λ A 2 }m t } 2 `Ut ˙dt `aV t dN t with B t xN y t ď 1 andU t :" }Ap0q} 2 {λ A `}p t S} 2 }X t } 2 {λ A `trpR `pt Sp t qArguing as in the proof of theorem 3.2 we conclude that @1 ď 3n ď 1 `pN ´1q{p2r 1 q sup " pBA rm t s ´pt Sq ξ 1 t `pt SX t `A rm t s ´BA rm t s m t " pBA rm t s ´pt Sq ξ 1 t `pt SX t `A rm t s ´BA rm t s m t ‰ y `trpRq `2trpp t Sp t qs dt `dM t , dM t y ùñ B t xMy t ď V t }ξ 1 t } 2 and U t :" 2}p t SX t `A rm t s ´BA rm t s m t } 2 {λ BA `trpRq `2trpSp 2 t q V t " 4 ptrpRq `2trpp t Sp t qq

	This implies that
	d}ξ 1 t } 2 "	" 2xξ 1 t ,				´1{2 2	dV t `1 ? N	dM t
	ď	" ´pλ BA {2q }ξ 1 t } 2 `Ut	‰	dt `dM t
	with with the martingale	dM t :" 2xξ 1 t
		d Ă M t " 2 xm t , p t B 1 R	´1{2 2	dV t y	`2 1 ? N	xm t , dM t y
	and the angle bracket
	B t x Ă M y t Using (2) we have
	dξ 1 t "					‰	dt `dM t
	1 ď n ď 1 with the martingale dM t :" R `pN ´1q 2r 1 1{2 1 dW λ min pSq λ max pSq 1 t `pt B 1 R ùñ sup tě0 ´1{2 2 dpV t E ptrpp t q n q ă 8 ´V 1 t q

tě0 E `}m t } 2n ˘ă 8

  BApm t qp t ´BAp p X t qP t " BApm t qpp t ´Pt q `pBApm t q ´BAp p X t qqP t BApm t q ´1 2 pp t `Pt qS ‰ pp t ´Pt q `pp t ´Pt q " BApm t q ´1 2 pp t `Pt qS ‰ 1`pBApm t q ´BAp p X t qqP t `Pt pBApm t q ´BAp p X t qq 1This implies that xp t ´Pt , Π t y ď ´λBA }p t ´Pt } 2 F `2κ BA trpP t q }p t ´Pt } F }m t ´p X t } from which we prove that d}p t ´Pt } 2 F " 2 xp t ´Pt , dpp t ´Pt qy `2 N ´1 rtrppR `pt Sp t qp t q `trpR `pt Sp t qtrpp t qs dt ď ! ´2λ BA }p t ´Pt } 2 F `4κ BA trpP t q }p t ´Pt } F }m t ´p X t } `2 N ´1 rtrppR `pt Sp t qp t q `trpR `pt Sp t qtrpp t qs ´1 xp t ´Pt , dM t y " trppp t ´Pt qdM t qAfter some computations we find thatB t xN y t ď 4 N ´1 }p t ´Pt } 2 F trpp t pR `pt Sp t qqIn much the same vein we havexm t ´p X t , Π t y " xm t ´p X t , pApm t q ´Ap p X t qq ´pt Spm t ´p X t q `pp t ´Pt qSpX t ´p X t qy ď ´λA }m t ´p X t } 2 `ρpSq }X t ´p X t } }p t ´Pt } F }m t ´p X t }This implies thatd}m t ´p X t } 2 " 2 xpm t ´p X t q, dpm t ´p X t qy `ˆtrpSpp t ´Pt q 2 q `1 N trpR `pt Sp t q ˙dt ď ! ´2λ A }m t ´p X t } 2 `2ρpSq }X t ´p X t } }p t ´Pt } F }m t ´p X t } `ρpSq}p t ´Pt } 2 F `1 N trpR `pt Sp t q * dt `dN twith the martingale dN t " 2 xpm t ´p X t q, dM t y " 2 xpm t ´p X t q, pp t ´Pt q B 1 R N xpm t ´p X t q, dM t yIn addition we haveB t xN y t ď 4ρpSq }m t ´p X t } 2 }p t ´Pt } 2 F `4 N xpm t ´p X t q, pR `pt Sp t qpm t ´p X t qy ď 2ρpSq ´}m t ´p X t } 2 `}p t ´Pt } 2 F ¯2 `4 N }m t ´p X t } 2 trpR `pt Sp t qCombining the above estimates we find thatd Ξ t ď ! ´2λ A }m t ´p X t } 2 `2 }p t ´Pt } F }m t ´p X t } ´2κ BA trpP t q `ρpSq }X t ´p X t } ¯) dt ´p2λ BA ´ρpSqq }p t ´Pt } 2 F dtSp t qp t q `trpR `pt Sp t qtrpp t qs * dt `dN t `dN t Recalling that 2λ A ě λ BA ą 0 and 2λ BA ´ρpSq ě λ BA this yields the estimate d Ξ t ď ! ´λBA Ξ t `2 }p t ´Pt } F }m t ´p X t } ´2κ BA trpP t q `ρpSq }X t ´p X t } ¯) dt Sp t q trpp t q `trpR `pt Sp t q ˙dt `dN t `dN t On the other hand using the inequality 2ab ď a 2 `b2 we prove that d Ξ t ď ´!λ BA ´´2κ BA trpP t q `ρpSq }X t ´p X t } ¯) Ξ t dt xpm t ´p X t q, pp t ´Pt q B 1 R Observe that xΥ p1q , Υ p2q y t " 0 B t xΥ p1q y t ď 2ρpSq ´}m t ´p X t } 2 `}p t ´Pt } 2 F ¯2 ď 2ρpSq Ξ 2 t B t xΥ p2q y t ď 4 N " }m t ´p X t } 2 `2 }p t ´Pt } 2 F trpp t q

	"
	with the martingale `1 N trpR `pt Sp t q dN t " `2N ? N 2 N ˆ4N ´1 rtrppR `pt `1 N N ˆ1 ´1 trpR `pt `1 N `4 1 ´1{N trpp p1q t `Υp2q t given by	*	dt `dN t
		d Υ p1q t	:" 2 ´1{2 2	dV t y
	1 N ? ´1 dM t Using the decomposition we check that d Υ p2q t :" 2 ? N xpm ı dM ´1{2 2 ´1{2 2 dV t y `2 ? dV t `1 ? N trpR `pt Sp t q ď dM t 4 N	U t Ξ t
	Π t "	"

t :" pp t ´Pt q B 1 R t q ˙"trpRq `trpSqtrpp t q 2 ‰ dt `dN t `dN t from which we conclude that

d Ξ t ď ˆΓA ptq Ξ t `1 N U t ˙dt `d Υ t with U t :" p1 `8trpp t qq " trpRq `ρpSq trpp t q 2 ‰

and the martingale Υ t :" Υ t ´p X t q, dM t y `2 ? N ´1 xp t ´Pt , dM t y

B 1 q can be thought as a signal to noise ratio. Given a fixed signal to noise ratio, the confidence intervals are small for high informative sensors with small perturbations.


Gaku Liu
email: gakuliu@gmail.com.

Mixed volumes of hypersimplices

Keywords: Hypersimplex, mixed volume, Eulerian numbers

scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The classical Eulerian number A(n, k) counts the number of permutations in S n with k -1 descents. They arise as the coefficients of Eulerian polynomials and as volumes of hypersimplices. These numbers have been shown to satisfy many properties which appear combinatorial in nature; see [START_REF] Postnikov | Permutohedra, associahedra, and beyond[END_REF]. In this extended abstract we give a combinatorial definition of mixed Eulerian numbers, which we use to prove both previous and new results.

For integers 1 ≤ k ≤ n, the hypersimplex ∆ k,n ⊂ R n+1 is the convex hull of all points of the form

e i1 + e i2 + • • • + e i k where 1 ≤ i 1 < i 2 • • • < i k ≤ n +
1 and e i is the i-th standard basis vector. Thus, ∆ k,n is an ndimensional polytope which lies in the hyperplane x 1 + • • • + x n+1 = k. Given a polytope P ⊂ R n+1 which lies in a hyperplane x 1 + • • • + x n+1 = α for some α ∈ R, we define its (normalized) volume Vol P to be the usual n-dimensional volume of the projection of P onto the first n coordinates. It is a classical result [START_REF] De Laplace | Oeuvres complètes[END_REF] that n! Vol ∆ k,n equals the Eulerian number A(n, k).

We now define the mixed volume of a set of polytopes. Given a polytope P and a real number λ ≥ 0, let λP = {λx | x ∈ P }. Given polytopes P 1 , . . . , P m ⊂ R n , let their Minkowski sum be

P 1 + • • • + P m = {x 1 + • • • + x m | x i ∈ P for all i}.
For nonnegative real numbers λ 1 , . . . , λ m , the function

f (λ 1 , . . . , λ m ) = Vol(λ 1 P 1 + • • • + λ m P m )
is known to be a homogeneous polynomial of degree n in the variables λ 1 , . . . , λ m . Hence there is a unique symmetric function Vol defines on n-tuples of polytopes in R n such that

f (λ 1 , . . . , λ m ) = m i1,...,in=1 Vol(P i1 , . . . , P in )λ i1 • • • λ in .
The number Vol(P 1 , . . . , P n ) is called the mixed volume of P 1 , . . . , P n . Mixed volumes of lattice polytopes have important connections to algebraic geometry, where they count the number of solutions to generic systems of polynomial equations; see [START_REF] Bernstein | The number of roots of a system of equations[END_REF]. If P 1 = • • • = P n = P , then Vol(P 1 , . . . , P n ) equals the ordinary volume Vol(P ). If P 1 , . . . , P m ⊂ R n+1 and each P i lies in a hyperplane

x 1 +• • •+x n+1 = α i for some α i ∈ R, then we define f (λ 1 , . . . , λ m ) = Vol(λ 1 P 1 + • • • + λ m P m )
in terms of the normalized volume defined previously, and we define the mixed volume Vol(P 1 , . . . , P n ) analogously.

Let c 1 , c 2 , . . . , c n be nonnegative integers such that c 1 + • • • + c n = n. We define

A c1,...,cn = n! Vol(∆ c1 1,n , ∆ c2 2,n , . . . , ∆ cn n,n )
where (∆ c1 1,n , ∆ c2 2,n , . . . , ∆ cn n,n ) denotes the n-tuple with c 1 entries ∆ 1,n , c 2 entries ∆ 2,n , and so on. The numbers A c1,...,cn are called mixed Eulerian numbers, and were introduced by Postnikov in [START_REF] Postnikov | Permutohedra, associahedra, and beyond[END_REF]. It is immediate that if c k = n and c i = 0 for all i = k, then A c1,...,cn = A(n, k). Furthermore, the result of Ehrenborg, Readdy, and Steingrímsson [START_REF] Ehrenborg | Steingrímsson, Mixed volumes and slices of the cube[END_REF] states that if c k-1 = r, c k = n -r, and c i = 0 for i = k -1, k, then A c1,...,cn equals the number of permutations w ∈ S n+1 with k -1 descents and w 1 = r + 1. The mixed Eulerian numbers satisfy many other remarkable properties; for example, we have

A 1,...,1 = n!, A k,0,...,0,n-k = n k , and A c1,...,cn = 1 c1 2 c2 • • • n cn if c 1 +• • •+c i ≥ i for all i.
These results were proven in [START_REF] Postnikov | Permutohedra, associahedra, and beyond[END_REF] using algebraic and geometric methods. Additional formulas involving mixed Eulerian numbers and their generalizations to other root systems were derived by Croitoru in [START_REF] Croitoru | Mixed volumes of hypersimplices, root systems and shifted Young tableaux[END_REF].

In this extended abstract, the main result is that each mixed Eulerian number enumerates a certain welldefine set of permutations in S n . (When c k = n and c i = 0 for all i = k, this set of permutations is precisely the set of permutations with k -1 descents.) We sketch how the above results arise from this combinatorial interpretation. We also give some new identities which follow from this interpretation. For example, we have that A c1,...,cn ≤ 1 c1 2 c2 • • • n cn for every mixed Eulerian number. We also give a simple formula for A c1,...,cn when the only nonzero c i are c 1 and c k for some k = 1. This number appeared in the work of Michałek et. al. [START_REF] Michałek | Exponential varieties[END_REF] during their study of exponential families arising from elementary symmetric polynomials.

In addition, we define the polytope Γ k,n ⊂ R n to be the convex hull of all points of the form

±e i1 ± e i2 ± • • • ± e i k where 1 ≤ i 1 < • • • < i k ≤ n.
In terms of Coxeter groups, the ∆ k,n correspond to the group A n and the Γ k,n correspond to the group B n . For nonnegative integers c 1 , . . . , c n such that

c 1 + • • • + c n = n, define B c1,...,cn = n! Vol(Γ c1 1,n , Γ c2 2,n , . . . , Γ cn n,n
). We refer to the B c1,...,cn as the type B mixed Eulerian numbers, whereas the A c1,...,cn are type A mixed Eulerian numbers. We give a combinatorial interpretation for the B c1,...,cn analogous to that of the A c1,...,cn and list several identities that follow from this interpretation.

The main theorem

Let n be a positive integer, and let S be a totally ordered set with |S| = n. Let C = (C 1 , . . . , C n ) be a sequence of n pairwise disjoint sets such that

• C 1 ∪ • • • ∪ C n = S, and
• s < t whenever s ∈ C i , t ∈ C j , and i < j.

We will call such a C a division of S. Let |C| denote the sequence

(|C 1 | , . . . , |C n |).
We say that an element s ∈ S is admissible with respect to C if either s is the smallest element of C 1 , s is the largest element of C n , or s ∈ C i for i = 1, n. Given an admissible element s, we define the deletion of s from C as follows. Let i be such that s ∈ C i , and let

C - i = {t ∈ C i | t < s} and C + i = {t ∈ C i | t > s}.
The deletion of admissible s from C results in a sequence of n -1 sets, denoted by C s = (C s 1 , . . . , C s n-1 ), given as follows:

• If i = 1, then C s = (C + 1 ∪ C 2 , C 3 , . . . , C n ). • If i = 1, n, then C s = (C 1 , . . . , C i-2 , C i-1 ∪ C - i , C + i ∪ C i+1 , C i+2 , . . . , C n ). • If i = n, then C s = (C 1 , . . . , C n-2 , C n-1 ∪ C - n ). In any case, C s is a division of S \ {s}.
Now suppose s 1 is admissible with respect to C, s 2 is admissible with respect to C s1 , s 3 is admissible with respect to (C s1 ) s2 , s 4 is admissible with respect to ((C s1 ) s2 ) s3 , and so on, until s n . We obtain a permutation s 1 s 2 . . . s n of S. Call any permutation constructed in this way a C-permutation. Note that the number of C-permutations depends only on |C|.

Example 2.1. Suppose n = 5 and C = ({1}, ∅, {2, 3}, {4}, {5}). The element 2 is admissible with respect to C, and C 2 = ({1}, ∅, {3, 4}, {5}). The element 3 is admissible with respect to C 2 , and (C 2 ) 3 = ({1}, ∅, {4, 5}). The element 1 is admissible with respect to (C 2 ) 3 , and ((C 2 ) 3 ) 1 = (∅, {4, 5}). The element 5 is admissible with respect to ((C 2 ) 3 ) 1 , and (((C 2 ) 3 ) 1 ) 5 = ({4}). The element 4 is admissible with respect to (((C 2 ) 3 ) 1 ) 5 . Hence 23154 is a C-permutation. The construction of this permutation is visualized below.

1 ∅ 23 4 5 1 ∅ 34 5 1 ∅ 45 ∅ 45 4
On the other hand, 23145 is not a C-permutation because 4 is not admissible with respect to ((C 2 ) 3 ) 1 = (∅, {4, 5}).

Example 2.2. Suppose C = ({1, . . . , n}, ∅, . . . , ∅). The only element admissible with respect to C is 1, and C 1 = ({2, . . . , n}, ∅, . . . , ∅). The only element admissible with respect to C 1 is 2, and so on. Thus the only C-permutation is 12 . . . n.

Similarly, if C = (∅, . . . , ∅, {1, . . . , n}), then the only C-permutation is n(n -1) . . . 1.

Example 2.3. Suppose C is a division of S and |C| = (1, . . . , 1). Then every element of S is admissible with respect to C. Moreover, for any element s ∈ S, C s satisfies |C s | = (1, . . . , 1). So by induction, every permutation of S is a C-permutation.

Example 2.4. Let C be a division of the form C = (C 1 , ∅, . . . , ∅, C n ). Then the only admissible elements with respect to C are the first element of C 1 and the last element of C n . Furthermore, when we delete either of these elements, the resulting sequence of sets is again of the form (C 1 , ∅, . . . , ∅, C n-1 ). So when we construct a C-permutation by successively deleting admissible elements, at each step we delete either the first element of the first set or the last element of the last set. Thus the C-permutations are the permutations where the elements of C 1 appear in ascending order and the elements of C n appear in descending order.

Example 2.5. We will see from Corollary 3.8 that if 1 ≤ k ≤ n and C is the division of {1, . . . , n} with C k = {1, . . . , n} and C i = ∅ for all i = k, then a permutation w ∈ S n is a C-permutation if and only if it has k -1 descents.

We are now ready to state the main result. Let

f n (λ 1 , . . . , λ n ) = Vol(λ 1 ∆ 1,n + λ 2 ∆ 2,n + • • • + λ n ∆ n,n ) = c1+•••+cn=n 1 c 1 ! • • • c n ! A c1,...,cn λ c1 1 • • • λ cn n so that A c1,...,cn = ∂ c1 1 • • • ∂ cn n f n .
The main idea is to write a recursive formula for f n . To do this, we use the following: Proposition 2.7. Let λ 1 , . . . , λ n be nonnegative real numbers. Fix a real number 0 ≤

x ≤ λ 1 + • • • + λ n , and let 1 ≤ i ≤ n be such that λ i+1 + • • • + λ n ≤ x ≤ λ i + • • • + λ n (where 0 ≤ x ≤ λ n if i = n). Set t = λ i + • • • + λ n -x. Consider the cross section of λ 1 ∆ 1,n + λ 2 ∆ 2,n + • • • + λ n ∆ n,n
with first coordinate equal to x. The projection of this cross section onto the last n coordinates is congruent to the following polytopes in the following cases:

• If i = 1, (t + λ 2 )∆ 1,n-1 + λ 3 ∆ 2,n-1 + • • • + λ n ∆ n-1,n-1 . • If 2 ≤ i ≤ n -1, λ 1 ∆ 1,n-1 + • • • + λ i-2 ∆ i-2,n-1 + (λ i-1 + λ i -t)∆ i-1,n-1 + (t + λ i+1 )∆ i,n-1 + λ i+2 ∆ i+1,n-1 + • • • + λ n ∆ n-1,n-1 . • If i = n, λ 1 ∆ 1,n-1 + • • • + λ n-2 ∆ n-2,n-1 + (λ n-1 + λ n -t)∆ n-1,n-1 .
This gives the following formula for f n :

Proposition 2.8. We have While C-permutations are defined recursively in general, there are certain cases where more explicit descriptions can be given. This allows us to calculate various formulas for mixed Eulerian numbers, as described in the next section.

Properties of mixed Eulerian numbers

Using algebraic and geometric techniques, Postnikov proved the following properties of mixed Eulerian numbers.

Theorem 3.1 (Postnikov [6]). The mixed Eulerian numbers have the following properties:

Theorem 2 . 6 .

 26 Let C = (C 1 , . . . , C n ) be a division with |C| = (c 1 , . . . , c n ). Then A c1,...,cn equals the number of C-permutations. Sketch of proof. It suffices to show that the mixed Eulerian numbers satisfy the recursion A c1,...,cn = s admissible w.r.t. C A |C s | .

f n (λ 1

 1 , . . . , λ n ) = λ1 0 f n-1 (t + λ 2 , λ 3 , . . . , λ n ) dt + 1 (λ 1 , . . . , λ i-2 , λ i-1 + λ i -t, t + λ i+1 , λ i+2 , . . . , λ n ) dt + λn 0 f n-1 (λ 1 , . . . , λ n-2 , λ n-1 + λ n -t) dt.Applying the operator ∂ c1 1 • • • ∂ cn n gives us the desired recursion.

  (a) The numbers A c1,...,cn are positive integers defined for c 1 , . . . ,c n ≥ 0, c 1 + • • • + c n = n.

Acknowledgments

The author would like to thank Alexander Postnikov for introducing this problem to him and for many useful discussions.

This material is based upon work supported by the National Science Foundation Graduate Research Fellowship under Grant No. 1122374. 1365-8050

Gaku Liu (b) We have A c1,...,cn = A cn,...,c1 .

(c) For 1 ≤ k ≤ n, the number A 0 k-1 ,n,0 n-k equals the usual Eulerian number A(n, k). Here, 0 l denotes a sequence of l zeroes.

(d) We have (f) For 2 ≤ k ≤ n and 0 ≤ r ≤ n, the number A 0 k-2 ,r,n-r,0 n-k is equal to the number of permutations w ∈ S n+1 with k -1 descents and w 1 = r + 1.

(g) We have A 1,...,1 = n!.

(h) We have A k,0,...,0,n-k = n k .

(i) We have A c1,...,cn = 1

Theorem 3.2 (Postnikov [6]). Let ∼ denote the equivalence relation on the set of nonnegative integer sequences (c 1 , . . . , c n ) with

is a cyclic shift of (c 1 , . . . , c n , 0). Then for a fixed (c 1 , . . . , c n ), we have

Note: There are exactly C n = 1 n+1 2n n equivalence classes. We now sketch how these properties arise from the combinatorial interpretation of mixed Eulerian numbers given by Theorem 2.6. We also give the following four additional properties.

Theorem 3.4. Let c 1 , . . . , c n be nonnegative integers such that c 1 + • • • + c n = n, and suppose there exists some 0

To state the third property, we introduce some terminology. Say a sequence a 1 , a 2 , . . . , a n is extreme if for each 1 ≤ i ≤ n, either a i = min(a 1 , . . . , a i ) or a i = max(a 1 , . . . , a i ). Let des(a 1 , . . . , a n ) denote the number of descents in the sequence a 1 , . . . , a n . Theorem 3.5. For 1 ≤ k ≤ n -2, 0 ≤ r ≤ n, and 0 ≤ s ≤ n, the number A 0 k-1 ,r,s,n-r-s,0 n-k-2 is equal to the number of permutations w ∈ S n+2 such that w 1 = r + 1, w 2 = r + s + 2, and if p is the largest integer such that w 1 , . . . , w p is extreme, then des(w 1 , . . . , w p ) + des(w p+1 , . . . , w n ) = k. Theorem 3.6. We have

where A(n, k) is defined to be 0 if k ≤ 0 or k > n.

The numbers A n-m,0 k-2 ,m,0 n-k appeared in [START_REF] Michałek | Exponential varieties[END_REF], where the authors used the recursions of [START_REF] Croitoru | Mixed volumes of hypersimplices, root systems and shifted Young tableaux[END_REF] to obtain the formula

.

We do not have a combinatorial proof of Theorem 3.1(d), which was proven using the volume of the permutohedron.

Sketch of proof of Theorem 3.1

Property (f), which is a generalization of property (c), follows from the following proposition, which is proved using induction.

be real numbers and let S = {x 1 , . . . , x n }. Let C be the division of S with |C| = (0 k-2 , r, n -r, 0 n-k ). Let λ be a real number such that x r < λ < x r+1 . Then a permutation w = w 1 . . . w n of S is a C-permutation if and only if the sequence λ, w 1 , . . . , w n has k -1 descents.

Corollary 3.8. Let 1 ≤ k ≤ n and let C = (∅ k-1 , {1, . . . , n}, ∅ n-k ). Then a permutation w ∈ S n is a C-permutation if and only if it has k -1 descents.

Proof. Take r = 0 or n in the previous Proposition.

Property (e) follows from Theorem 3.2 and the note afterwards. Property (g) follows from Example (2.3). Property (h) follows from Example (2.4). Finally, we prove Theorem 3.3, which implies (i). We first introduce some terminology. Fix a division

. Let I C w : S → N be the function which takes each s ∈ S to its index in w with respect to C. Note that if s ∈ C i , then I C w (s) ∈ {1, . . . , i}. We will call any function I : S → N which maps C i into {1, . . . , i} and index function of C. w is an injection from hte set of C-permutations to the set of index functions of C. This map is a bijection if and only if

Note that by Theorem 3.3 and Theorem 3.1(b), we also have

with equality if and only if

Proof of Theorem 3.2

Let n be a positive integer and let c 1 , . . . , c n be nonnegative integers with

We will describe a process which is a cyclic version of the construction of C-permutations. Arrange the numbers 1, . . . , n around a circle C clockwise in that order. We will define n + 1 "blocks" as follows: for each 1 ≤ i ≤ n + 1, block B i initially contains the elements of C i . We view B 1 , . . . , B n+1 as being arranged around C in that order, including the empty blocks; i.e. B i is viewed as being between B i-1 and B i+1 even if B i is empty. For any element s ∈ {1, . . . , n}, we define the deletion of s from C as follows. Suppose s ∈ B i . Let B - i be the set of elements in B i which are to the left of (counterclockwise from) s, and let B + i be the set of elements in B i which are to the right of (clockwise from) s. To delete s, we remove s and the block B i from C, put all the elements of B - i into the block to the left of B i , and put all the elements of B + i into the block to the right of B i . The order of the undeleted elements remains unchanged. We can then delete another element, and so on. After we delete all n elements, we are left with only one block, which is empty. Since a nonempty block remains nonempty until it is deleted, this final empty block was originally empty and remained so throughout the process.

Let w = w 1 . . . w n ∈ S n be a permutation. Let r(w) be the r such that B r is the final block that remains when we successively delete w 1 , . . . , w n from C. It is not hard to see that for each r with C r = ∅, the set of w such that r(w) = r is precisely the set of (C r+1 , C r+2 , . . . , C r-1 )-permutations, where the indices of the C i are taken modulo n+1 and the elements {1, . . . , n} are ordered starting from the first element of C r+1 and going cyclically to the last element of C r-1 . There are A cr+1,...,cr-1 such permutations. Hence we have

which is exactly what we wanted to prove.

Sketch of proof of Theorem 3.6

Given a set S, define a -permutation of S to be a finite sequence of elements of S and symbols such that every element of S appears exactly once. A descent of a -permutation s 1 s 2 . . . is a pair of elements (s i , s j ) such that i < j, s i > s j , and s k = for every i < k < j. Theorem 3.6 follows from the following.

Proposition 3.11. Let C be a division with |C| = (n -m, 0 k-2 , m, 0 n-k ). Then the C-permutations are in bijection with -permutations of C k for which the number of 's is at most n -m and the number of 's plus the number of descents is equal to k -1.

Sketch of proof.

Assume C is a division of S. Suppose s = s 1 s 2 . . . is a -permutation of C k satisfying the above conditions. Let i be the largest index such that s i is either a or (s i , s j ) is a descent for some j. We obtain a C-permutation from s as follows: Begin with the subsequence s 1 . . . s i , and replace the first with the first element of C 1 , the second with the second element of C 1 , and so on, until all 's are replaced. Call the new sequence w = w 1 . . . w i . Append to the end of w the elements of S \ {w 1 , . . . , w i } in ascending order. The result is a C-permutation. Now suppose w = w 1 . . . w n is a C-permutation. Replace all w i for which w i / ∈ C k with 's. Call the resulting -permutation s = s 1 s 2 . . . . Now, call an index i good if either s i = or (s i , s j ) is a descent of s for some j. Delete any 's in s which occur after the (k -1)-th good index. The result is a -permutation of C k satisfying the desired conditions.

In fact, the above argument extends easily to the following: Proposition 3.12. Let C be a division with |C| = (n -m, 0 k-3 , r, m -r, 0 n-k ), and let λ be a number such that λ > s for all s ∈ C k-1 and λ < s for all s ∈ C k . Then the C-permutations are in bijection with -permutations s 1 s 2 . . . of C k-1 ∪ {λ} ∪ C k for which s 1 = λ, the number of * 's is at most n -m, and the number of 's plus the number of descents is equal to k -1.

Type B mixed Eulerian numbers

We now give an analogous combinatorial interpretation for the numbers B c1,...,cn . Let C = (C 1 , . . . , C n ) be a division of a set S. We say that an element s ∈ S is type B admissible with respect t0 C if either s is the smallest element of C 1 or s ∈ C i for i = 1. Given a type B admissible element s, we now define the type B deletion of s from C, which by abuse of notation we denote by C s . Let i be such that s ∈ C i . If i = n, then we define C s to be the same as in the type A case. If i = n, then we define

Given these definitions of admissibility and deletion, we define a type B C-permutation analogously as in the type A case. (a) We have (i) For 1 ≤ k ≤ n -2, 0 ≤ r ≤ n, and 0 ≤ s ≤ n, the number A 0 k-1 ,r,s,n-r-s,0 n-k-2 is equal to the number of permutations w ∈ S n+2 such that w 1 = r + 1, w 2 = r + s + 2, and if p is the largest integer such that w 1 , . . . , w p is extreme, then des(w 1 , . . . , w p ) + des(w p+1 , . . . , w n ) ≤ k.