Pierre-Yves Boëlle

Guy Thomas

Resistance to antibiotics: limit theorems for a stochastic SIS model structured by level of resistance

) developed a stochastic SIS model, which was structured by the level of resistance of bacterial strains. Here we study the asymptotic properties of this model when the population size is large. To this end, we cast the model within the framework of measure valued processes, using point measures to represent the pattern of bacterial resistance in the compartments of colonized individuals. We first show that the suitably normalized model tends in probability to the solution of a deterministic differential system. Then we prove that the process of fluctuations around this limit tends in law to a Gaussian process in a space of distributions. These results, which generalize those of Kurtz (1981, chap. 8) on SIR models, support the validity of the deterministic approximation and quantify the rate of convergence.

Introduction

During the last decades, bacterial resistance to antibiotics has become a major Public Health concern (Cars et al, 2008). The case of S. Pneumoniae, a pathogen responsible for respiratory infections, otitis, and meningitis, is particularly illustrative of the phenomenon, with prevalences of intermediate or resistant pneumococcal strains exceeding 10% in 50%, and 25% in 15%, of European countries (EARS-Network, 2012). The selection of resistant strains is the result of several interacting processes. At the bacterial level, point mutations and horizontal transfers of genetic material lead to changes in antibiotic resistance with a large range of effects [START_REF] Martinez | Antibiotics and antibiotic resistance genes in natural environments[END_REF]. In the absence of exposure to antibiotics, it is thought that most mutated strains do not give birth to significant populations, since they are rapidly outgrown by competing antibiotic-sensitive strains because of differences in fitness [START_REF] Maher | The fitness cost of antibiotic resistance in streptococcus pneumoniae: insight from the field[END_REF]. In the presence of antibiotics however, antibiotic-sensitive strains are selectively eliminated, so that antibiotic-resistant strains will develop and colonize the available niche. Exposure to antibiotics in human or animal populations is widespread, occuring through prescribed or over-the-counter medical treatments, and other routes such as food consumption. Once an antibiotic-resistant strain has been selected in a host, it may disseminate across the population through inter-individual contacts. [START_REF] Temime | Bacterial resistance to penicillin G by decreased affinity of penicillin-binding proteins: A mathematical model[END_REF] developed a model which integrates the processes operating both at the bacterial and the human population levels, together with their interactions. The model was cast in the form of an SIS stochastic model whose compartments specify the status of individuals with respect to colonization by bacteria, and exposure to antibiotics. We use the term 'colonization' rather than 'infection', since asymptomatic carriage is frequent. Each colonized individual was further described by the level of resistance of the colonizing bacterial strain which, in the case of penicillins, spreads over a continuum of values. Because of this extra structure, the model belongs to the class of individual-based models, and its properties in the large population limit cannot be investigated following the approach of [START_REF] Kurtz | Approximation of population processes[END_REF], as e.g. in Andersson and Britton (2000, chap.5). Although a law of large numbers was suggested by heuristic reasoning [START_REF] Temime | Deterministic and stochastic modeling of pneumococcal resistance to penicillin[END_REF], no rigorously established result is yet available.

In this paper, we consider the SIS model of [START_REF] Temime | Bacterial resistance to penicillin G by decreased affinity of penicillin-binding proteins: A mathematical model[END_REF], and represent the resistance patterns of the colonized individuals by point measures. This formalism allows a thorough investigation of the large population limit using the approach of [START_REF] Fournier | A microscopic probabilistic description of a locally regulated population and macroscopic approximations[END_REF] and [START_REF] Méléard | Convergence of the fluctuations for interacting diffusions with jumps associated with Boltzmann equations[END_REF]. A similar line of attack was used in [START_REF] Clémençon | A stochastic SIR model with contact-tracing: large population limits and statistical inference[END_REF] to study an SIR epidemic model with an age structure. More specifically, we prove both a law of large numbers and a central limit theorem for the suitably normalized model as the population size tends to infinity.

Model and notations

The population is assumed to be of constant size n, and is partitioned into five compartments as depicted in Figure 1 (see [START_REF] Temime | Bacterial resistance to penicillin G by decreased affinity of penicillin-binding proteins: A mathematical model[END_REF] for details): 4 compartments characterize the status of individuals with respect to both colonization by S. Pneumoniae and exposure to antibiotics; the fifth compartment corresponds to the so-called refractory phase, during which individuals cannot be colonized due to acquired immunity with respect to the bacterial agent: such a phase follows spontaneous decolonization but is not observed when decolonization results from exposure to antibiotics [START_REF] Prellner | Immunization and protection in pneumococcal otitis media studied in a rat model[END_REF]. Uncolonized unexposed individuals are in compartment X u ; individuals in refractory phase following spontaneous decolonization are in compartment X r ; uncolonized ex-posed individuals are in compartment X e ; colonized unexposed individuals are in compartment Y u ; and colonized exposed individuals are in compartment Y e . Colonized individuals are further characterized by their level of resistance to antibiotics, modeled as a continuous positive variable. Uncolonized individuals may become colonized through contact with colonized individuals. Colonized individuals may undergo decolonization, either spontaneously or through exposure to antibiotics. If exposed, they may in addition undergo an increase of their resistance level by selection of a population of genetically altered bacterial strains.

Let δ m denote the Dirac measure at m, and M "

! ř k i"1 δ mi , m i ą 0, k ď n )
denote the set of point measures on R `" p0, 8q, with total mass bounded by n. For ν " ř k i"1 δ mi P M and f a measurable bounded function on R `, we set xν, f y " ş R`ν pdmqf pmq "

ř k i"1 f pm i q.
For all time t, the state of the population is described by the vector Zptq " pZ Xu ptq, Z Xr ptq, Z Xe ptq, Z Yu ptq, Z Ye ptqq, which takes values in the state space E " t0, 1, . . . , nu 3 ˆM2 . More specifically,

• Z Xu ptq is the number of uncolonized unexposed individuals,

• Z Xr ptq is the number of individuals in refractory phase,

• Z Xe ptq is the number of uncolonized exposed individuals,

• Z Yu ptq is the random measure ř N Yu ptq i"1 δ aiptq , where N Yu ptq " xZ Yu ptq, 1y is the number of colonized unexposed individuals, and a 1 ptq, . . . , a N Yu ptq ptq denote their respective resistance levels,

• Z Ye ptq is the random measure ř N Ye ptq j"1 δ bj ptq , where N Ye ptq " xZ Ye ptq, 1y is the number of colonized exposed individuals and b 1 ptq, . . . , b N Ye ptq ptq denote their respective resistance levels. The dynamics of the population are driven by the following transitions:

• An unexposed individual, colonized or uncolonized, may become exposed, at rate α on .

• An exposed individual, colonized or uncolonized, may become unexposed, at rate α end .

The rates α on and α end do not differ between colonized and uncolonized individuals, since the majority of S. Pneumoniae carriers are asymptomatic, and exposure to antibiotics thus mainly results from prescription for unrelated infectious diseases.

• A colonized unexposed individual may undergo spontaneous decolonization and enter the refractory phase, at rate λ u .

• An individual in the refractory phase may leave this state and enter the uncolonized unexposed compartment, at rate α r . • A colonized exposed individual, with resistance level b, may undergo decolonization, at rate λ e pbq. The rate of decolonization under exposure to antibiotics is the sum of the spontaneous rate λ u and the rate due to treatment effect. Since the latter rate decreases as the resistance level of the colonizing strain increases, λ e is a non increasing function on R `, tending to λ u at infinity. We let λe " λ e p0q.

UNEXPOSED

• An uncolonized unexposed individual may get colonized through contact with a given colonized individual, exposed or unexposed, at rate β u {n.

The newly colonized individual acquires the resistance level of his contaminating contact.

• An exposed uncolonized individual may get colonized through contact with a given colonized individual, exposed or unexposed, with resistance level m, at rate β e pmq{n. To reflect the fact that the rate of colonization by contact under exposure to antibiotics is higher when the resistance level of the colonizing strain is higher, β e is a nonnegative nondecreasing function on R `, assumed bounded from above by βe . The resistance level of the newly colonized individual is m.

The normalizations β u {n and β e {n are standard in the formulation of epidemic models [START_REF] Andersson | Stochastic epidemic models and their statistical analysis[END_REF], section 2.1). They correspond to the fact that in a mixing population of size n, each individual has probability 1{pn ´1q » 1{n of making contact with another given individual.

In [START_REF] Temime | Bacterial resistance to penicillin G by decreased affinity of penicillin-binding proteins: A mathematical model[END_REF], β e pbq ranges from 0 to 2β u as b ranges from 0 to infinity. This reflects the fact that: if the transmitted strain is not resistant to antibiotics, treatment hinders colonization of the antibiotic exposed individual; whereas if it is highly resistant, treatment facilitates colonization as it clears the endogenous bacterial flora.

• A colonized exposed individual may undergo a genetic event, at rate µ.

As a result, the individual's resistance level increases by an amount of h, where h is a positive random variable with distribution Rpdhq " ρphqdh.

While genetic events may lead to drops in resistance level, such events are not observed, since, under exposure to antibiotics, bacterial strains with lower resistance will not be able to establish as the dominant strain.

In the sequel, the components of a generic process U with values in E will be denoted by U Xu , U Xr , U Xe , U Yu , and U Ye , respectively. For f, g : R `Ñ R measurable and bounded, we define

U f,g " pU Xu , U Xr , U Xe , U f Yu , U g Ye q,
where U f Yu " xU Yu , f y and U g Ye " xU Ye , gy. For f : R Ñ R, and h ě 0, we let τ h f p¨q " f p¨`hq.

The notation 1 A stands for the indicator function of the set A.

Throughout the paper, K will denote a generic constant, whose meaning may change from line to line.

Poisson measure representation

The process pZptqq tě0 will be realized as the solution of a system of stochastic differential equations driven by Poisson measures. On a probability space pΩ, F, Pq we consider the following independent random elements:

1. The initial state Zp0q P E, which is specified through

• The vector pZ Xu p0q, Z Xr p0q, Z Xe p0q, N Yu p0q, N Ye p0qq, giving the initial numbers of individuals in the compartments, assumed to be multinomial with index n, and probabilities pθ Xu , θ Xr , θ Xe , θ Yu , θ Ye q.

• The initial point measures Z Yu p0q " ř N Yu p0q

i"1 δ ai and Z Ye p0q " ř N Ye p0q j"1 δ bj , where pa i q iě1 , and pb j q jě1 are two sequences of i.i.d. positive random variables with respective distributions P u and P e . where ds and dh are Lebesgue measure on R `, di and dj are counting measures on N ˚, and dv is Lebesgue measure on r0, 1s. We let pF t q tě0 denote the canonical filtration generated by Zp0q and the above Poisson measures.

The following

The model pZptqq tě0 is defined as the pF t q-Markov process, which is the unique solution of the system of stochastic differential equations

ż t 0 ż N ˚1tiďN Yu ps´qu δ aips´q Q decol Yu pds, diq I decol Ye ptq " ż t 0 ż N ˚ż 1 0 1 tiďN Ye ps´qu 1 tvďλepbips´qq{ λeu δ bips´q Q decol Ye pds, di, dvq I mut Ye ptq " ż t 0 ż N ˚ż 1 0 ż R`1
tiďN Ye ps´qu pδ bips´q`h ´δbips´q q Q mut Ye pds, di, dhq.

Note that the above representation of pZptqq tě0 yields a straightforward algorithm for pathwise simulation of the process.

Law of large numbers

To establish a law of large numbers, as n Ñ 8, we consider the normalized process Z n " 1 n Z, with state space E 1 " r0, 1s 3 ˆpM 1 q 2 , where M 1 is the space of subprobability measures on R `. On E 1 we put the product topology induced by the usual topology on r0, 1s, and the topology of weak convergence on M 1 .

If S is a metric space, and T ą 0, Cpr0, T s, Sq denotes the space of continuous functions from r0, T s to S, equipped with the topology of uniform convergence, and Dpr0, T s, Sq denotes the space of càdlàg functions from r0, T s to S, equipped with the Skorohod topology (see e.g. Jacod and Shiryaev (2003, Chaper VI)).

To keep notations as concise as possible, given f, g : R `Ñ R measurable and bounded, we define two mappings, Ψ f,g and Ψf,g .

The map Ψ f,g " pΨ Xu , Ψ Xr , Ψ Xe , Ψ f Yu , Ψ g Ye q, is defined from Dpr0, T s, E 1 q to Dpr0, T s, R 5 q as Ψ Xu pU qptq "

ż
ż 8 0 ρphqdhpτ h g ´gqy ¯.
The map Ψf,g is defined from Dpr0, T s, E 1 q into Dpr0, T s, S 5 q, where S 5 is the space of real 5 ˆ5 symmetric matrices, as follows. Let

˘(3.1)
Then ΨpU qptq " ş t 0 ψpU psqqds, where integration is componentwise. The following proposition states a semi-martingale decomposition which is at the heart of the proofs.

Proposition 3.1. For all measurable bounded f, g : R `Ñ R, we may write

Z n,f,g " V n,f,g `M n,f,g ,

where

V n,f,g ptq " Z n,f,g p0q `Ψf,g pZ n qptq is a continuous finite variation process, and M n,f,g is a bounded càdlàg martingale, with predictable quadratic variation process

! M n,f,g "" 1 n Ψf,g pZ n q (3.2)
Proof. Let L n denote the infinitesimal generator of the Markov process Z n . By Ethier and Kurtz (2005, Proposition 4.1.7), for all measurable bounded function φ : E 1 Ñ R, the process

φpZ n ptqq ´φpZ n p0qq ´ż t 0 L n φpZ n psqqds
is a bounded martingale. This implies in particular that M n,f,g is a martingale. Taking φpx 1 , x 2 , x 3 , ν 1 , ν 2 q " x 2 1 , the process is also a martingale. Since a finite variation predictable martingale is a.s. constant, we have U 2 ´U1 " 0 a.s., hence the expression for ă M n Xu ą. The other components are found likewise, using appropriate functions.

U 1 ptq " Z n

Convergence of the normalized process

Since the predictable variation process ! M n,f,g " is a.s. Op1{nq, it is reasonable to expect that the noise process M n,f,g will tend to 0 as n Ñ 8, and that the normalized sequence will tend to a deterministic limit.

First we note that the initial condition vector, as defined in Section 2.1, tends to a deterministic limit. Lemma 3.2. As n Ñ 8, Z n p0q tends to zp0q " pθ Xu , θ Xr , θ Xe , θ Yu P u , θ Ye P e q a.s. in E 1 .

Theorem 3.3. For all T ą 0, the sequence pZ n q ně1 converges in probability in Dpr0, T s, E 1 q to the deterministic process z " px u , x r , x e , ξ u , ξ e q P Cpr0, T s, E 1 q, which is the unique solution to the following equation

z f,g ptq " z f,g p0q `Ψf,g pzqptq (3.3)
for all measurable bounded functions f, g : R `Ñ R.

Proof. Since the limit is deterministic, it is enough to prove convergence in law. The uniqueness of solutions, and the characterization of limit points of pZ n q ně1 as solutions of (3.3) are established as in [START_REF] Fournier | A microscopic probabilistic description of a locally regulated population and macroscopic approximations[END_REF].

C-tightness (see Jacod and Shiryaev, 2003, Definition VI.3.25) of the sequence pZ n q ně1 in Dpr0, T s, E 1 q is equivalent to C-tightness of each component (Jacod and Shiryaev, 2003, Corollary VI.3.33).

1. By Aldous' criterion [START_REF] Aldous | Stopping times and tightness[END_REF], pZ n Xu q ně1 , pZ n Xr q ně1 , and pZ n Xe q ně1 are tight in Dpr0, T s, r0, 1sq. Since they have a.s. jumps of order 1{n they are C-tight (Jacod and Shiryaev, 2003, Proposition VI.3.26).

2. Similarly, for all f measurable and bounded, pxZ n Yu , f yq ně1 and pxZ n Ye , f yq ně1 are C-tight in Dpr0, T s, Rq. It follows that pZ n Yu q ně1 and pZ n Ye q ně1 are C-tight in Dpr0, T s, M v 1 q, where M v 1 denotes the space M 1 endowed with the vague topology (Roelly-Coppoletta, 1986).

Thus pZ

n q ně1 is C-tight in Dpr0, T s, E v 1 q, where E v 1 " r0, 1s 3 ˆpM v 1 q 2 . If Z is a limit point in law of pZ n q ně1 in Dpr0, T s, E v
1 q, it can be shown that x ZYu , 1y and x ZYe , 1y are limit points in law in Dpr0, T s, Rq of pxZ n Yu , 1yq ně1 and pxZ n Yu , 1yq ně1 , respectively. The method of proof, similar to that in [START_REF] Méléard | Slow and fast scales for superprocess limits of agestructured populations[END_REF], relies on approximating the functions 1 rk,`8q , k P N, by continuous functions such as

ψ k pmq " $ ' & ' % 0 if m ď k, m ´k if k ă m ď k `1, 1 if m ą k `1,
noticing that the functions 1´ψ k and ş 8 0 ρphqdhpτ h ψ k ´ψk q have compact support.

Theorem 3 in [START_REF] Méléard | Sur les convergences étroite ou vague de processus à valeurs mesures[END_REF] then implies C-tightness of pZ n Yu q ně1 and pZ n Ye q ně1 in Dpr0, T s, M 1 q.

Absolute continuity and densities

In this section, we assume that P u and P e , hence ξ u p0q and ξ e p0q, are absolutely continuous with respect to Lebesgue measure on R `. Under these conditions, Gronwall's lemma implies that ξ u ptq and ξ e ptq are absolutely continuous for all t, and we let π u pt, mq and π e pt, mq denote their respective densities with respect to Lebesgue measure. The following corollary then follows from (3.3) and Fubini's theorem.

Theorem 3.4. Assume that P u and P e are absolutely continuous with respect to Lebesgue measure on R `, and that β e , λ e and ρ are bounded and continuous. q. This is the system of ordinary and partial integro-differential equations that was derived heuristically in [START_REF] Temime | Deterministic and stochastic modeling of pneumococcal resistance to penicillin[END_REF]. The main interest of the system of equations (3.4), in contrast to equation (3.3), is that it can be integrated numerically, and used to analyze the long term trends of antibiotic resistance. In this respect, Theorems 3.3 and 3.4 confer theoretical validity to the simulations presented in Temime et al (2005, section 5.3), which show that, in the long term, the respective numbers of individuals in the five compartments tend to a steady-state equilibrium, whereas the level of resistance in colonized individuals increases with time. Likewise, system (3.4) may be used to estimate the distribution of resistance levels in colonized individuals after a given period of time [START_REF] Temime | Deterministic and stochastic modeling of pneumococcal resistance to penicillin[END_REF], section 6.2). The approximation of the stochastic model by its deterministic limit was observed to be acceptable for population sizes above 5000.

β u pπ u pt,

Central limit theorem

In this section, we prove a central limit theorem for the sequence pη n q ně1 " p ?

npZ n ´zqq ně1 of fluctuation processes. Writing as above η n " pη n Xu , η n Xr , η n Xe , η n Yu , η n Ye q, we note that η n Yu ptq and η n Ye ptq are signed measures, whose limits, as n Ñ 8, are not measure valued in general. Thus, following the approach of [START_REF] Métivier | Weak convergence of measure valued processes using Sobolevimbedding techniques[END_REF] and [START_REF] Oelschläger | Limit theorems for age-structured populations[END_REF], we are led to chose as state space for pη n ptqq tě0 a space of distributions which is the dual of a weighted Sobolev space.

Notations, definitions, and preliminary results

Although the level of resistance to antibiotics is a positive variable, it will be convenient to consider that the support of the measures η n Yu ptq and η n Ye ptq is the whole of R. Likewise, we shall also consider that the support of the measures P u , P e , Z n Yu ptq, Z n Ye ptq, ξ u ptq, and ξ e ptq is R, and that the functions λ e and β e are defined on R.

For k P N and γ ě 0, let W k,γ denote the space of functions φ : R Ñ R, such that

}φ} 2 W k,γ " ż R ÿ jďk |D j φpmq| 2 dm 1 `|m| 2γ ă `8,
where D j φ is the weak jth derivative of φ. Equipped with the ¨ W k,γ norm, W k,γ is a separable Hilbert space, whose dual is denoted by W ´k,γ , with dual norm ¨ W ´k,γ . In the present context, the space of compactly supported smooth functions on R is dense in W k,γ [START_REF] Kufner | How to define reasonably weighted Sobolev spaces[END_REF].

We also define the space C k,γ of k-times continuously differentiable functions on R, such that

lim |m|Ñ8 |D j f pmq| 1 `|m| γ " 0 @j ď k, with norm f C k,γ " ÿ jďk sup mPR |D j f pmq| 1 `|m| γ .
Clearly, if f P C k,γ , |f pmq| ď f C k,γ p1 `|m| γ q. Moreover for k ě 0, l ě 1, γ ě 0, and α ą 1{2, the inclusions

W k`l,γ ãÑ C k,γ ãÑ W k,γ`α (4.1)
are continuous [START_REF] Métivier | Weak convergence of measure valued processes using Sobolevimbedding techniques[END_REF], i.e. there exist constants K and L such that

}f } W k,γ`α ď K}f } C k,γ ď L}f } W k`l,γ .
The following Proposition is proved as in Méléard (1998, Lemma A, section 3.2), using the continuity of the inclusion W k,γ ãÑ C 0,γ for k ě 1.

Proposition 4.1. Let k P N ˚and γ ě 0. For fixed m P R and h ě 0, the mappings

D m : φ P W k,γ Þ Ñ φpmq D mh : φ P W k,γ Þ Ñ pτ h φ ´φqpmq
are linear and continuous. Moreover

D m W ´k,γ ď Kp1 `|m| γ q D mh W ´k,γ ď Kp1 `|m| γ `hγ q.
By Parseval's identity, we deduce Corollary 4.2. Let pφ j q jě1 be a Hilbert basis of W k,γ . Then ÿ jě1 φ j pmq 2 ď Kp1 `|m| 2γ q ÿ jě1 ppτ h φ j ´φj qpmqq 2 ď Kp1 `|m| 2γ `h2γ q.

For k P N and γ ě 0, let r E ´k,γ " R 3 ˆpW ´k,γ q 2 , equipped with the Hibertian product norm

ζ ´k,γ " `|ζ Xu | 2 `|ζ Xr | 2 `|ζ Xe | 2 ` ζ Yu 2 W ´k,γ ` ζ Ye 2 W ´k,γ ˘1{2 .
The motivation for considering weighted Sobolev spaces is that we need constant functions to belong to the space of test functions. While any γ ą 1{2 would thus be adequate, we have chosen γ " 1 for simplicity. The choice of k was guided by two considerations: first, k ě 1 is required for continuity of the map m Þ Ñ δ m ; and second, tightness of pη n q ně1 will be proved using the fact that the dual inclusion W ´k,γ`α ãÑ W ´pk`1q,γ is a Hilbert-Schmidt operator for α ą 1{2. Overall this led to the choice of W ´2,1 as state space for η n Yu and η n Ye , and of W ´1,1`α , α ą 1{2 as an auxiliary state space. Now, the measures Z n Yu ptq, Z n Ye ptq, ξ u ptq and ξ e ptq belong to W ´1,1`α as soon as they have moments of order q " 2p1 `αq ą 3. However, the existence of moments of order q ą 5 is needed in order to deal with the quadratic variation processes, whose expressions contain products of test functions.

For p ě 0, let χ p denote the function m Þ Ñ |m| p , and let ρ p " ş 8 0 h p ρphqdh. Hypothesis 1. There exists q ą 5 such that xP u , χ q y ă `8, xP e , χ q y ă `8, and ρ q ă `8.

The proof of the following Lemma, which shows that the above moment hypothesis propagates in time, is analogous to that of Theorem 3.1 in [START_REF] Fournier | A microscopic probabilistic description of a locally regulated population and macroscopic approximations[END_REF].

Lemma 4.3. Under Hypothesis 1, sup 0ďtďT xξ u ptq `ξe ptq, χ q y ă `8,

sup n E sup 0ďtďT xZ n
Yu ptq `Zn Ye ptq, χ q y ă `8.

The above lemma shows in particular that the mappings Ψ φ,ψ and Ψφ,ψ are well defined for all φ, ψ P W 1,q .

We have the semimartingale decomposition η n,φ,ψ ptq " r V n,φ,ψ ptq `Ă M n,φ,ψ ptq, where r V n,φ,ψ ptq " ? npV n,φ,ψ ptq ´zφ,ψ ptqq " η n,φ,ψ p0q `?npΨ φ,ψ pZ n qptq ´Ψφ,ψ pzqptqq is a continuous finite variation process, and Ă M n,φ,ψ ptq " ? nM n,φ,ψ ptq is a square integrable martingale with predictable quadratic variation process ! Ă M n,φ,ψ "" Ψφ,ψ pZ n q.

In obvious notations, we consider the processes r V n and Ă M n , which, as shown below, may be considered as r E ´1,q{2 valued processes.

The martingale term

Proposition 4.4. The process Ă

M n " p Ă M n Xu , Ă M n Xr , Ă M n Xe , Ă M n Yu , Ă M n
Ye q is a.s. a càdlàg square integrable martingale, taking its values in the Hilbert space r E ´1,q{2 . Moreover

sup n E ˆsup 0ďtďT } Ă M n ptq} 2 ´1,q{2 ˙ă `8. (4.2)
Proof. Let pφ j q jě1 be a Hilbert basis of W 1,q{2 . By Parseval's identity,

Ă M n ptq 2 ´1,q{2 " p Ă M n Xu ptqq 2 `p Ă M n Xr ptqq 2 `p Ă M n Xe ptqq 2 `ÿ jě1 Ă M n,φj Yu ptq 2 `ÿ jě1 Ă M n,

`Zn

Ye , χ q y ȃnd the last line is bounded uniformly in n, by Hypothesis 1 and Lemma 4.3. Moreover Ă M n is a martingale in the separable Hilbert space r E ´1,q{2 , since Ă M n,φj ,φ k is a martingale for all j and k, and Epsup j,k Ă M n,φj ,φ k R 5 q ă 8 (see [START_REF] Badrikian | Martingales hilbertiennes[END_REF]).

Finally, mimicking the proof of Ferland et al (1992, Proposition 3.5) shows that Ă

M n has a.s. càdlàg paths.

The definitions of the scalar Doob-Meyer processes of a square integrable Hilbert-valued martingale may be found in [START_REF] Badrikian | Martingales hilbertiennes[END_REF] or [START_REF] Métivier | Semimartingales[END_REF].

Proposition 4.5. The scalar Doob-Meyer process ć Ă M n č of the square integrable martingale Ă M n is given by

ć Ă M n č "ă Ă M n Xu ą `ă Ă M n Xr ą `ă Ă M n Xe ą `ć Ă M n Yu č `ć Ă M n Ye č (4.

The drift term

We now consider the continuous finite variation process Ṽ n .

The following assumption ensures that the mappings φ Þ Ñ λ e φ and φ Þ Ñ β e φ are bounded operators from W 1,q{2 to W 1,q{2 , and from W 2,1 to W 2,1 .

Hypothesis 2. The functions λ e and β e are twice differentiable, and their derivatives are bounded a.e. Lemma 4.6. Let T ą 0. Under Hypotheses 1 and 2, r V n ptq ´1,q{2 ď η n p0q ´1,q{2 `JT

ż t 0 ds η n psq ´1,q{2 (4.4)
where J T is a deterministic constant that is independent of n and t for t ď T .

Proof. Writing v n " r V n ´ηn p0q, we have

r V n ptq ´1,q{2 ď η n p0q ´1,q{2 ` v n ptq ´1,q{2 ď η n p0q ´1,q{2 `|v n Xu ptq| `|v n Xr ptq| `|v n Xe ptq| ` v n Yu ptq W ´1,q{2 ` v n Ye ptq W ´1,q{2 .
Let φ, ψ P W 1,q{2 . Since v n,φ,ψ " ? npΨ φ,ψ pZ n q ´Ψφ,ψ pzqq, we get

| v n
`αend η n Ye psq W ´1,q{2 φ W 1,q{2 `βu p η n Yu psq W ´1,q{2 ` η n Ye psq W ´1,q{2 q φ W 1,q{2 `βu |η n Xu psq| ξ u psq `ξe psq W ´1,q{2 φ W 1,q{2 |v n,ψ Ye ptq| ď ż t 0 ds ´αon η n Yu psq W ´1,q{2 ψ W 1,q{2 ` η n Ye psq W ´1,q{2 pα end `λe qψ W 1,q{2 `p η n Yu psq W ´1,q{2 ` η n Ye psq W ´1,q{2 q β e ψ W 1,q{2 `|η n Xe psq| ξ u psq `ξe psq W ´1,q{2 β e ψ W 1,q{2 `µ η n Ye psq W ´1,q{2 ż 8 ´8 ρphqdhpτ h ψ ´ψq W 1,q{2
¯.

Now observe that

• Constants β u and λ u belong to W 1,q{2 .

• By Hypothesis 2, functions β e and λ e belong to W 1,q{2 , and there exist constants K β and K λ such that β e φ W 1,q{2 ď K β φ W 1,q{2 and λ e φ W 1,q{2 ď K λ φ W 1,q{2 for all φ P W 1,q{2 .

• By Hypothesis 1, there exists a constant K q{2 such that

ş 8 ´8 ρphqdhτ h ψ W 1,q{2 ď K q{2 ψ W 1,q{2 for all ψ P W 1,q{2 .
• By continuity of the inclusion W 1,q{2 ãÑ C 0,q{2 and Hypothesis 1,

Ξ q{2 pT q " sup 0ďsďT ξ u psq `ξe psq W ´1,q{2 ă `8.
This proves the Lemma with e.g.

J T " 2pα on `αr `αend q `βu `β e `pβ u `Kβ qp1 `Ξq{2 pT qq ` β u W 1,q{2 ` β e W 1,q{2 `λu `Kλ ` λ u W 1,q{2 ` λ e W 1,q{2 `µp1 `Kq{2 q.
Lemma 4.7. Under Hypothesis 1, sup n Ep}η n p0q} 2 ´1,q{2 q ă `8 (4.5)

Proof. Let pφ j q jě1 be a Hilbert basis in W 1,q{2 . By independence of the number N Yu p0q of colonized unexposed individuals at time 0, and their resistance levels pa i q iě1 , we have

E}η n Yu p0q} 2 W ´1,q{2 " 1 n ÿ jě1 E " ´NYu p0q ÿ i"1 pφ j pa i q ´xP u , φ j yq ¯2* `nE `NYu p0q n ´θYu ˘2( ÿ jě1 xP u , φ j y 2
since the crossproducts are 0. Indeed

E " N Yu p0q ÿ i"1 `φj pa i q ´xP u , φ j y ˘`N Yu p0q n ´θYu ˘ı " n ÿ i"1 E ! E " 1 tiďN Yu p0qu `φj pa i q ´xP u , φ j y ˘`N Yu p0q n ´θYu ˘ıˇˇˇN Yu p0q ) " n ÿ i"1 E ! `NYu p0q n ´θYu ˘1tiďN
Yu p0qu E `φj pa i q ´xP u , φ j y ˘ı)

" 0. Now, by independence again,

1 n ÿ jě1 E " ´NYu p0q ÿ i"1 pφ j pa i q ´xP u , φ j yq ¯2* " 1 n ÿ jě1 E ˆNYu p0q ÿ i"1 pφ j pa i q ´xP u , φ j yq 2 ď E ÿ jě1 φ 2 j pa 1 q ď KEp1 `|a 1 | q q, and nE `NYu p0q n ´θYu ˘2( ÿ jě1 xP u , φ j y 2 " θ Yu p1 ´θYu q ÿ jě1 xP u , φ j y 2 ď θ Yu p1 ´θYu qE D a1 2 W ´1,q{2 .
A similar bound holds for E}η n Ye p0q} 2 W ´1,q{2 .

Uniform estimate for the fluctuation process

Proposition 4.8. Under Hypothesis 1,

sup n E sup 0ďtďT }η n ptq} 2 ´1,q{2 ă `8 (4.6) sup n E sup 0ďtďT }η n ptq} 2 ´2,1 ă `8 (4.7)
Proof. By convexity, we have from (4.4)

}η n ptq} 2 ´1,q{2 ď 2} r V n ptq} 2 ´1,q{2 `2} Ă M n ptq} 2 ´1,q{2 ď 4}η n p0q} 2 ´1,q{2 `4J 2 T t ż t 0 ds}η n psq} 2 ´1,q{2 `2} Ă M n ptq} 2 ´1,q{2 .
Thus, we get by Gronwall's lemma

sup n E sup 0ďtďT }η n ptq} 2 ´1,q{2 ď `4 sup n E}η n p0q} 2 ´1,q{2 `2 sup n E sup 0ďtďT } Ă M n ptq} 2 ´1,q{2 ˘e4J 2 T T 2
, which is finite by (4.2) and (4.5). Inequality (4.7) follows by continuity of the inclusion W ´1,q{2 ãÑ W ´2,1 .

4.5 Tightness in Dpr0, T s, r E ´2,1 q Proposition 4.9. The sequences pη n q ně1 and p Ă M n q ně1 are tight in Dpr0, T s, r E ´2,1 q.

Proof. To show that the sequences p Ă M n q ně1 and pη n q ně1 are tight in Dpr0, T s, r E ´2,1 q, we use the Aldous-Rebolledo-Joffe-Métivier criterion [START_REF] Joffe | Weak convergence of sequences of semimartingales with applications to multitype branching processes[END_REF][START_REF] Rebolledo | Central limit theorems for local martingales[END_REF] for Hilbert valued càdlàg semimartingales. First (4.2) and (4.6), together with the fact that the inclusion r E ´1,q{2 ãÑ r E ´2,1 is a Hilbert-Schmidt operator, imply that, for fixed t P r0, T s, p Ă M n ptqq ně1 and pη n ptqq ně1 are tight in r E ´2,1 . It remains to prove the Aldous conditions for ć Ă

M n č and for r V n , which are easily established using Lemma 4.3.

Limit points

Lemma 4.10. All limit points of pη n q ně1 and p Ă M n q ně1 are a.s. continuous.

Proof. Let ∆η n psq (respectively ∆ Ă M n,φ,ψ psq) denote the jump at s of η n,φ,ψ (respectively Ă M n,φ,ψ ). Since Ă M n and η n have the same jumps, it is enough to show that sup 0ďtďT η n ptq ´ηn pt´q ´2,1 tends to 0 in probability (Jacod and Shiryaev, 2003, Proposition VI.3.26).

The jumps of η n Xu , η n Xr , and η n Xe , are a.s. of size 1 ? n . Let now φ P W 2,1 , and consider the jumps of η n,φ Yu and η n,φ Ye . Given an arbitrary, but fixed, enumeration 1, ¨¨¨, n of the individuals in the population, we denote by m i,n ptq the resistance level of individual i at time t, setting m i,n ptq " 0 if i is uncolonized at t. Then, noting that a.s. two or more individuals cannot jump at the same time, the sizes of the possible jumps at time t of η n,φ Yu and η n,φ Ye are

• |φpm i,n pt ´qq|{ ? n if: individual i is colonized at t ´, and either undergoes decolonization, or a change in exposure status, or contaminates an uncolonized individual,

• |φpm i,n pt ´q`hq´φpm i,n pt ´qq|{ ? n: if individual i is exposed and colonized at t ´, and undergoes a mutation of amplitude h.

Thus in all instances the size of the jump is less than K ? n p1 `|m i,n ptq| `hq.

To obtain an uniform estimate for |m i,n ptq|, let 1 tjPCu ptq " 1 if individual j is in compartment C at time t, and 0 otherwise, and let k P t1, . . . , nu be an individual of the population. Taking in account only those transitions that increase the resistance level of individual k, we have where K is independent of n and i. The Markov inequality concludes the proof.

m 2 k,n ptq ď m 2 k,n p0q `ż t 0 ż N ˚żN ˚1tkPXu u ps ´q1 ti"ku 1 tjPYuYYeu ps ´q ˆ`m 2 j,
Proposition 4.11. For all T ą 0, the sequence p Ă M n q ně1 tends in law in Dpr0, T s, r E ´2,1 q to the process B " pB Xu , B Xr , B Xe , B Yu , B Ye q, which is a continuous centered square integrable Gaussian martingale, such that, for all φ, ψ P W 2,1 , the predictable quadratic variation process of B φ,ψ is ! B φ,ψ "" Ψφ,ψ pzq.

Proof. Let B be the limit in law in Dpr0, T s, Ẽ´2,1 q of a subsequence p Ă M n q ně1 . For φ, ψ P W 2,1 , the subsequence p Ă M n,φ,ψ q ně1 tends in law to B φ,ψ (by the continuous mapping theorem). Since the jumps of Ă M n,φ,ψ are those of η n,φ,ψ , we have

E sup sďt ∆ Ă M n,φ,ψ psq 2 R 5 ď p3 ` φ 2 W 2,1 ` ψ 2 W 2,1 qE sup sďt ∆η n psq 2 ´2,1
so that sup sďt }∆ Ă M n,φ,ψ psq} 2 R 5 Ñ 0 in probability for all t by (4.8). We deduce from (4.1) that φ, ψ P W 2,1 ñ φψ P W 1,q{2 for q ą 5. It then follows by (4.6) and Hypothesis 1 that ! Ă M n,φ,ψ " t tends in probability to Ψφ,ψ pzqptq for all fixed t. Now, the process Ψφ,ψ pzq is deterministic and 0 at time 0. Moreover the matrix Ψφ,ψ pzqptq ´Ψ φ,ψ pzqpsq is symmetric and nonnegative for all s ă t.

Therefore, using Pollard (1984, Theorem VIII.13), we conclude that p Ă M n,φ,ψ q ně1 tends in law to the continuous Gaussian centered martingale B φ,ψ with predictable quadratic variation ! B φ,ψ "" Ψφ,ψ pzq.

Next, the family t Ă M n ptq, t P r0, T s, n ě 1u of r E ´2,1 -valued random variables is uniformly integrable by (4.2). Therefore B is a martingale with respect to the filtration it generates (Jacod and Shiryaev, 2003, Propositions IX.1.1 and IX.1.12). Moreover, by Doob's inequality, E sup 0ďtďT Bptq 2 ´2,1 ă `8. Theorem 4.12. For all T ą 0, the sequence of processes pη n q ně1 tends in law in Dpr0, T s, r E ´2,1 q to the process η " pη Xu , η Xr , η Xe , η Yu , η Ye q P Cpr0, T s, r E ´2,1 q, On the one hand, η n,φ,ψ ´r V n,φ,ψ tends in law to B φ,ψ , by Proposition 4.11. On the other hand, since η is continuous (Lemma 4.10), η n,φ,ψ ´r Ψ φ,ψ pη n q tends in law to η φ,ψ ´r Ψ φ,ψ pηq by the continuous mapping theorem. We now show that r Ψ φ,ψ pη n q ´r V n,φ,ψ tends to zero in probability. Since r Ψ φ,ψ pη n qptq´r V n,φ,ψ ptq R Hence, η n,φ,ψ ´r V n,φ,ψ tends in law to η φ,ψ ´r Ψ φ,ψ pηq (Jacod and Shiryaev, 2003, Lemma VI.3.31). The lemma follows by uniqueness of the limit in law.

Lemma 4.14. ηp0q is defined uniquely as the limit in law in r E ´2,1 of pη n p0qq ně1 .

Proof. Since η is continuous, pη n p0qq ně1 is tight by projection at time 0. Now, for all φ, ψ P W 2,1 , straightforward calculations using characteristic functions show that any convergent subsequence pη n,φ,ψ p0qq ně1 tends in law to a Gaussian random variable with zero expectation and variance matrix

Ω φ,ψ " ¨θXu p1 ´θXu q ´θXu θ Xr ´θXu θ Xe ´θXu θ Yu φ ´θXu θ Ye ψ θ Xr p1 ´θXr q ´θXr θ Xe ´θXr θ Yu φ ´θXr θ Ye ψ θ Xe p1 ´θXe q ´θXe θ Yu φ ´θXe θ Ye ψ θ Yu p φ2 ´θYu φ2 q ´θXu θ Xe φ ψ θ Ye p ψ2 ´θYe ψ2 q ‹ ‹ ‹ ‹ '
where φ " xP u , φy, φ2 " xP u , φ 2 y, ψ " xP e , ψy, and ψ2 " xP e , ψ 2 y.

Lemma 4.15. Equation (4.9) has a unique solution in Dpr0, T s, r E ´2,1 q.

Proof. By Gronwall's lemma, the solution is pathwise unique. Next, by Jacod and Shiryaev (2003, Proposition IX.1.12), B is a martingale with respect to the filtration generated by pη, Bq, so that pathwise uniqueness implies uniqueness in law by the Yamada-Watanabe theorem (Revuz and Yor, 1991, Theorem IX.1.7).

Statistical applications

The application of Theorem 4.12 to practical statistical issues is far from straightforward, due to the fact that the limit processes zptq and ηptq have measure and distribution valued components, respectively. Moreover, although these processes may be converted to processes with real valued components by considering z φ,ψ and η φ,ψ for given φ, ψ P W 2,1 , the latter processes cannot be computed nor simulated from (3.3) and (4.9) in general. In this sense, theorems 3.3 and 4.12 are essentially existence theorems. However some results can be derived under more specific assumptions. Proof. It readily follows from the assumption and (4.9) that η 1,1 ptq is solution to the Langevin equation η 1,1 ptq " η 1,1 p0q `ż t 0 apsqη 1,1 psqds `B1,1 ptq, whose solution is (5.1).

Recall the definition of ψf,g pU psqq given by equations (3.1), and let A ˚denote the transpose of a square matrix A.

Corollary 5.2. If the functions λ e and β e are constant, ? npZ n,1,1 ptq ´z1,1 ptqq tends in law to the Normal distribution on R 5 with zero expectation and variance matrix Σ 1,1 ptq " Eptq ´Ω1,1 `ż t 0 Epsq ´1 ψ1,1 pzpsqqpEpsq ´1q ˚ds ¯Eptq ˚.

. For large n, under the conditions of Theorem 4.12 and Corollary 5.2, the distribution of ? npZ n,1,1 ptq ´z1,1 ptqq can be approximated by the Normal distribution N p0, Σ 1,1 ptqq. In particular, we have pV arZ Xu ptq, V arZ Xr ptq, V arZ Xe ptq, V arN Yu ptq, V arN Ye ptqq » npV arpη Xu ptqq, V arpη Xr ptqq, V arpη Xe ptqq, V arpη 1 Yu ptqq, V arpη 1 Ye ptqqq.

This shows that variability in the numbers of individuals in the compartments is of order ? n, explaining why simulations of the stochastic process in [START_REF] Temime | Bacterial resistance to penicillin G by decreased affinity of penicillin-binding proteins: A mathematical model[END_REF] show high variability.

If λ e or β e are not constant, but P u and P e are absolutely continuous with respect to Lebesgue measure, then px u ptq, x r ptq, x e ptq, π u pt, mq, π e pt, mqq can be computed via numerical integration of (3.4). Monte Carlo simulations of Z f n,f,g ptq ´zf,g ptq can thus be performed for any measurable and bounded functions f and g in order to evaluate quantities of interest, such as times of emergence of resistant bacteria.

It is well known that the existence of a central limit theorem with Gaussian limit is central to the validity of bootstrap methods [START_REF] Beran | Asymptotic theory for bootstrap methods in statistics[END_REF]. In a situation where model parameters (or at least some of them) are estimated from observed data (e.g. by conditional least-squares), Theorem 4.12 may thus be invoked to establish the consistency of bootstrap methods.

Discussion

The issue of bacterial resistance to antibiotics is critical from a Public Health perspective. Since the emergence of resistant strains and their dissemination throughout the population results from the complex interaction of several processes, it is natural to rely on mathematical modeling to investigate the possible time course evolutions and to evaluate the impact of potential interventions.

Realistic descriptions of biological and epidemiological processes are often conveniently achieved through individual based models involving chance events. Numerical simulation of the resulting stochastic models using standard simulation algorithms may quickly raise feasibility issues due to computational load as the population size is increased and the structure of the compartments is detailed. The large number of simulations required to fully explore the model variability makes the problem even worse. For such reasons, a common practice is to fall back on a deterministic model in place of the initial stochastic formulation. Here, we showed that using the formalism of measure valued processes, it it possible to fully justify the choice of a deterministic formulation as the limiting case of the initial stochastic model. This is clearly an improvement on heuristic arguments as in [START_REF] Temime | Deterministic and stochastic modeling of pneumococcal resistance to penicillin[END_REF].

The law of large numbers provides a firm theoretical foundation for the deterministic model in the limit of a large population. Under assumptions which allow for numerical evaluation of the deterministic limit, this limit provides a simple means for analyzing the temporal trends of bacterial antibioresistance.

In this respect, the law of large numbers provides a theoretical confirmation the validity of the long term trends described in [START_REF] Temime | Deterministic and stochastic modeling of pneumococcal resistance to penicillin[END_REF], such as convergence to a steady-state of the numbers of individuals in each compartment, together with an increase of the level of antibiotic resistance in carriers. This shows in particular that exposure to antibiotics is critical, and that the level of resistance will be increasing with time in the absence of intervention on this factor.

The central limit theorem specifies the rate of convergence and provides an explanation for the high variability of e.g. times to emergence of resistant bacterial strains described by [START_REF] Temime | Bacterial resistance to penicillin G by decreased affinity of penicillin-binding proteins: A mathematical model[END_REF], since it proves that for large n, the standard deviations of the process components are of order ? n. Regarding the particular model and situation studied in the present work, we note that the deterministic model used in [START_REF] Temime | Deterministic and stochastic modeling of pneumococcal resistance to penicillin[END_REF] provided good qualitative agreement with the time trends of antibiotic resistance in the field [START_REF] Temime | Bacterial resistance to penicillin G by decreased affinity of penicillin-binding proteins: A mathematical model[END_REF]. Including age structure in the model was worth considering, since the frequency of treatment by antibiotics, and the probability of colonization, depend on age, and since the analysis of a model with age structure raises no novel methodological issue. However, the model of [START_REF] Temime | Bacterial resistance to penicillin G by decreased affinity of penicillin-binding proteins: A mathematical model[END_REF] used an effective treatment frequency that was calculated by weighting observed frequencies of treatment with probabilities of colonization according to age.

Here, for the sake of definiteness, the initial numbers of individuals in the five compartments of the model were assumed to be multinomial. However, other distributions could be assumed instead. The proof of the law of large numbers carries over to any context in which Lemma 3.2 is valid, i.e. Z n p0q tends to a deterministic vector zp0q as n grows large. In particular, a deterministic initial configuration is suitable. If Z n p0q tends to a random limit, the convergence in Theorem 3.3 is in law rather than in probability.

The SIS model is a standard choice to study endemic diseases (Andersson and Britton, 2000, chapter 8). Alternatively one may use SIR models with demography. Technically, the issue of an open population of unbounded size is dealt with by using localizing stopping times, such as in [START_REF] Clémençon | A stochastic SIR model with contact-tracing: large population limits and statistical inference[END_REF] for instance. This option would yield similar results, but at the price of greater variability in the central limit theorem due to superimposed fluctuations in population size.

Figure 1 :

 1 Figure 1: Schematic description of the compartmental model, with the respective transition rates. The vertical axes in the Colonized compartments represent the scale of resistance levels, and the dots represent Dirac masses at the resistance levels of the colonized individuals.

  Poisson measures: • exposure onset Poisson measures Q on Xu pds, diq and Q on Yu pds, diq, with common intensity measure α on ds b di, • exposure ending Poisson measures Q end Xe pds, diq and Q end Ye pds, diq, with common intensity measure α end ds b di, • decolonization Poisson measures Q decol Yu pds, diq and Q decol Ye pds, di, dvq, with respective intensity measures λ u ds b di and λe ds b di b dv, • colonization Poisson measures Q col Xu pds, di, djq and Q col

	Xe pds, di, dj, dvq, n ds b di b dj and βe with respective intensity measures βu n ds b di b
	dj b dv,
	• a refractory phase exit Poisson measure Q exit Xr pds, diq, with intensity
	measure α r ds b di,
	• a mutation Poisson measure Q mut Ye pds, di, dhq, with intensity measure
	µds b di b ρphqdh,

  Xu ps´qu `1tjďN Yu ps´qu δ aj ps´q `1t1ďj´N Yu ps´qďN Ye ps´qu δ b j´N Yu ps ´q ps´q ˘Qcol Xu pds, di, djq

			ż t	ż
	I on Yu ptq "	0	N	˚1tiďN Yu ps´qu δ aips´q Q on Yu pds, diq
			ż t	ż
	I end Xe ptq "	0	N	˚1tiďZ Xe ps´qu Q end Xe pds, diq
			ż t	ż
	I end Ye ptq "	0	N	˚1tiďN Ye ps´qu δ bips´q Q end Ye pds, diq
			ż t	ż
	I exit Xr ptq "	0	N	˚1tiďZ Xr ps´qu Q exit Xr pds, diq
			ż t	ż
	I col Xu ptq " ˚1tiďZ I col 0 N ˚żN Xe ptq " ż t 0 ż N ˚żN ˚ż 1 0
	$ ' '	Z Xu ptq " Z Xu p0q ´Ion Xu ptq `Iend Xe ptq `Iexit Xr ptq ´xI col Xu ptq, 1y
	' ' ' ' '	Z Xr ptq " Z Xr p0q `xI decol Yu ptq, 1y ´Iexit Xr ptq
	&
		Z Xe ptq " Z Xe p0q `Ion Xu ptq ´Iend Xe ptq `xI decol Ye	ptq, 1y ´xI col Xe ptq, 1y	(2.1)
	'
	' ' ' '	Z Yu ptq " Z Yu p0q ´Ion Yu ptq `Iend Ye ptq ´Idecol Yu ptq `Icol Xu ptq
	'
	' %	Z Ye ptq " Z Ye p0q `Ion Yu ptq ´Iend Ye ptq ´Idecol Ye	ptq `Icol Xe ptq `Imut Ye ptq
	where
			ż t	ż
	I on Xu ptq "	0	N	˚1tiďZ Xu ps´qu Q on Xu pds, diq

1 tiďZ Xe ps´qu `1tjďN Yu ps´qu 1 tvďβepaj ps´qq{ βeu δ aj ps´q `1t1ďj´N Yu ps´qďN Ye ps´qu 1 tvďβepb j´N Yu ps ´qps´qq{ βeu ˆδb j´N Yu ps ´q ps´q ˘Qcol Xe pds, di, dj, dvq I decol Yu ptq "

  Xr psq `λu xU Yu psq, 1y Xu psq ´αend U Xe psq `xU Ye psq, λ e y ´UXe psqxU Yu psq `UYe psq, β e y

			ΨXr
	pU qptq "	ż t	ds ´´α r U ΨXe
		0
	pU qptq "	ż t 0	ds ´αon U Ψf

t 0 ds ´´α on U Xu psq `αr U Xr psq `αend U Xe psq ´βu U Xu psqxU Yu psq `UYe psq, 1y Yu pU qptq " ż t 0 ds ´´pα on `λu qxU Yu psq, f y `αend xU Ye psq, f y `βu U Xu psqxU Yu psq `UYe psq, f y Ψg Ye pU qptq " ż t 0 ds ´αon xU Yu psq, gy ´xU Ye psq, pα end `λe qgy `UXe psqxU Yu psq `UYe psq, β e gy `µxU Ye psq,

  ψXu pU psqq " α on U Xu psq `αr U Xr psq `αend U Xe psq `βu U Xu psqxU Yu psq `UYe psq, 1y ψXr pU psqq " α r U Xr psq `λu xU Yu psq, 1y ψXe pU psqq " α on U Xu psq `αend U Xe psq `xU Ye psq, λ e y `UXe psqxU Yu psq `UYe psq, β e y ψf Yu pU psqq " pα on `λu qxU Yu psq, f 2 y `αend xU Ye psq, f 2 y `βu U Xu psqxU Yu psq `UYe psq, f 2 y ψg Ye pU psqq " α on xU Yu psq, g 2 y `xU Ye psq, pα end `λe qg 2 y

	'
	%
	ψf,g "	¨ψ Xu ψXu,Xr ψXu,Xe	ψXu,Xr ψXu,Xe ψXr 0 0 ψXe	ψf Xu,Yu ψf Xr,Yu 0	0 0 Xe,Ye ψg	‹ ‹ ‹ ‹
		ψf Xu,Yu	ψf Xr,Yu	0	ψf Yu	ψf,g Yu,Ye	‹ '
		0	0	ψg Xe,Ye	ψf,g Yu,Ye	ψg Ye

`UXe psqxU Yu psq `UYe psq, β e g 2 y `µxU Ye psq, ż 8 0 ρphqdhpτ h g ´gq 2 y ψXu,Xr pU psqq " ´αr U Xr psq ψXu,Xe pU psqq " ´`α on U Xu psq `αend U Xe psq ψf Xu,Yu pU psqq " ´βu U Xu psqxU Yu psq `UYe psq, f y ψf Xr,Yu pU psqq " ´λu xU Yu psq, f y ψg Xe,Ye pU psqq " ´`xU Ye psq, λ e gy `UXe psqxU Yu psq `UYe psq, β e gy ψf,g Yu,Ye pU psqq " ´`α on xU Yu psq, f gy `αend xU Ye psq, f gy

  mq `πe pt, mqqdm

	dx r dt	" ´αr x r	0 `ż 8	λ u π u pt, mqdm
	dx e dt	" α on x u ´αend x e	0 `ż 8	λ e pmqπ e pt, mqdm
					ż 8
				´xe	β e pmqpπ u pt, mq `πe pt, mqqdm
					0
	B Bt	π

u pt, mq " ´pλ u `αon qπ u pt, mq `xu β u pπ u pt, mq `πe pt, mqq `αend π e pt, mq B Bt π e pt, mq " α on π u pt, mq ´pα end `λe pmqqπ e pt, mq `xe β e pmqpπ u pt, mq `πe pt, mqq `µ ż m 0 π e pt, hqρpm ´hqdh ´µπ e pt, mq (3.4) with initial condition pθ Xu , θ Xr , θ Xe , θ Yu dPu dm , θ Ye dPe dm

  φj Ye ptq 2 .

	By Doob's inequality, Proposition 3.1 and Corollary 4.2, we have
	E ´sup 0ďtďT	Ă M n Xu ptq 2 `sup 0ďtďT	Ă M n Xr ptq 2 `sup 0ďtďT	Ă M n Xe ptq 2
		`ÿ jě1	sup 0ďtďT	Ă M n,φj Yu ptq 2 `ÿ jě1	sup 0ďtďT	Ă M n,φj Ye ptq 2	ă
		4T `K1 `K2 E sup 0ďtďT	xZ n Yu

  n ps ´q ´m2 k,n ps

	Thus, by Lemma 4.3 and Gronwall's lemma,
						sup n	max 1ďkďn	Ep sup 0ďtďT	m 2 k,n ptqq ă `8,
	yielding
						E sup 0ďtďT	η n ptq ´ηn pt´q 2 ´2,1 ď	K n	,	(4.8)
						´q˘Q col Xu pds, di, djq
	`ż t 0	ż N ˚żN	0 ˚ż 1	1 tkPXeu ps ´q1 ti"ku 1 tjPYuYYeu ps ´q1 tvďβepmj,nps´q{ βeu
						ˆ`m 2 j,n ps ´q ´m2 k,n ps ´q˘Q col Xe pds, di, dj, dvq
	`ż t 0	ż N	0 ˚ż 8	1 tkPYeu ps ´q1 ti"ku 1 tvďµ Y pmj,nps´q{μu
						ˆ`pm 2 k,n ps ´`hq ´m2 k,n ps ´˘Q mut pds, di, dhq
	ď m 2 k,n p0q	`ż t 0	ż N ˚żN	˚1ti"ku 1 tjPYuYYeu ps ´qm 2 j,n ps ´qQ col Xu pds, di, djq
		`ż t 0	ż N ˚żN	0 ˚ż 1	1 ti"ku 1 tjPYuYYeu ps ´qm 2 j,n ps ´qQ col Xe pds, di, dj, dvq
		`K ż t 0	ż	N ˚ż 8 0	1 ti"ku pm 2 k,n ps ´q `h2 qQ mut pds, di, dhq,
	whence
	Ep sup 0ďtďT	m 2 k,n ptqq ď xθ Yu P u `θYe P e , χ 2 y
						`pβ u `β e q	ż T 0	dsE ´sup 0ďuďs xZ n Yu puq `Zn Ye puq, χ 2 y Kµ
						ż T
						0	dsEp sup 0ďuďs	m 2 k,n puqq `KµT ρ 2 .

  Proposition 5.1. Assume that the functions λ e and β e are constant. Then we have

	η 1,1 ptq " Eptq ´η1,1 p0q	`ż t	Epsq ´1dB 1,1 psq ¯.	(5.1)
				0
	where
	¨´α on ´βu νptq 0	α r ´αr	α end 0		´βu x u ptq λ u	´βu x u ptq 0	‹ ‹ ‹
	α on	0	´αend		´βe x e ptq	´βe x e ptq `λe	‹ ‹
	aptq "		´βe νptq	‹ ‹
	β u νptq	0	0		´pα on `λu q	α end	‹ ‹
						`βu x u ptq	`βu x u ptq	‹ ‹
	0	0	β e νptq		α on `βe x e ptq	´pα end `λe q	‹ '
							`βe x e ptq
	with νptq " xξ u ptq `ξe ptq, 1y, and Eptq " exp	`şt 0 apsqds ˘.

Acknowledgements

We wish to thank Michèle Thieullen for helpful discussions, and Viet Chi Tran for clarification of several points.

Then px u , x r , y u , π u , π e q is the solution to ρphqdhpτ h ψ ´ψqy ¯`B ψ Ye ptq, (4.9) for all φ, ψ P W 2,1 , where B φ,ψ is the Gaussian martingale defined in Proposition 4.11.

The proof rests on the three following lemmas. Lemma 4.13. All limit points in law of pη n q ně1 are weak solutions of the stochastic differential system (4.9).

Proof. Let η be the limit in law of a subsequence pη n q ně1 . For φ, ψ P W 2,1 , write (4.9) in the form η φ,ψ ptq " η φ,ψ p0q `ż t 0 dsL φ,ψ psqηpsq `Bφ,ψ ptq, and let the mapping r Ψ φ,ψ : Dpr0, T s, r E ´2,1 q Ñ Dpr0, T s, R 5 q be defined by r Ψ φ,ψ pζqptq " ζ φ,ψ p0q `ż t 0 dsL φ,ψ psqζpsq.

We have Ă M n,φ,ψ " η n,φ,ψ ´r V n,φ,ψ " pη n,φ,ψ ´r Ψ φ,ψ pη n qq `p r Ψ φ,ψ pη n q ´r V n,φ,ψ q.


Alberto Romele

The Ineffectiveness of Hermeneutics. Another Augustine's Legacy in Gadamer

Keywords: verbum in corde, hermeneutics, later Augustine, Gadamer, Grace

This article builds on Gadamer's rehabilitation of the Augustinian concept of inner word (verbum in corde). Unlike most interpretions, the thesis is that the Augustinian inner word does not show the potentialities, but rather the ineffectiveness of ontological hermeneutics. In the first section, it is argued that for the later Augustine the verbum in corde is the consequence of a Word-and Truth-event. In the second section, the author suggests that Gadamer has properly understood the verbum in corde as a matter of faith. In the third section, it is shown that Gadamer has found in the notion a paradigm for his philosophical and theological insights.

Concerning the former, he has always been fascinated by the evenemential character of the 'second' Heidegger's thought. Concerning the latter, Gadamer has explicitly accused Bultmann's demythologization of being 'human, all too human', and he has implicitly praised Barth's dialectical theology.

Introduction

This article builds on the chapter Sprache und verbum of Wahrheit und Methode, in which Gadamer rehabilitates the notion of 'inner word', and opposes it to the 'forgetfulness of language (Sprachvergessenheit)' that Plato would have inaugurated with the Cratylus. My hypothesis, as Jean Grondin and other scholars have already stated, is that the rehabilitation of 1 2 course, that part of the nature of experience is to be continually confirmed […]. But it is no longer a [hermeneutical] experience when it is repeated and confirmed.' The hermeneutical 4 experience, than, has not much to do with Aristotle's εµπειρία, nor with Hegel's Erfahrung.

According to the former, the universality of the concept is ontologically prior and what concerns him about experience is merely how it contributes to the formation of the concept. According to the latter, experience has the structure of a reversal of consciousness but, in the end, for Hegel, conscious experience should lead to a self-knowledge that no longer has anything other or alien to itself. Rather, the hermeneutical experience is represented by the Aeschylus' formula 'learning through suffering (πάθει µάθος/ durch Leiden lernen).' For the German philosopher, this phrase does not only mean that we become wise through suffering.

Aeschylus means more than this: 'He refers to the reason why this is so. What a man has to learn through suffering is not this or that particular thing, but insight into the limitations of humanity, into the absoluteness of the barrier that separates man from the divine. It is ulti-

mately a religious insight […].' 5
Hence, the notion of verbum in corde, both for Augustine and Gadamer, does not concern the philosophical explication of the linguistic and semantic structure of the human mind. This is the case, instead, of a long and almost parallel tradition that goes from Aristotle to J. Fodor. Isabelle Koch's recent interpretation of Augustine's inner word is inadequate, insofar it excludes the 'project of the intellectus fidei', to which, as she admits, 'all psychological and cognitive analysis of the De Trinitate conform.' Moreover, the notion of verbum in corde has 6 not much to do with the onto(theo)logical analogy of proportion between the Word in God and the incarnate Word on the one hand, and the inner and outer word in men on the other.

Several Greek and Latin Church fathers have used the classical Stoic distinction between λόγος ἐνδιάθετος and προφορικός in this way. Rather, according to a tradition that stretches from Paul to the Dialectical Theology of Karl Barth passing through Luther, the verbum in 7 corde, as thought by the late Augustine and rehabilitated by Gadamer, must be understood as a genuine theological concept. More precisely, the verbum in corde is the consequence of the donum dei, the gratuitous intervention of the grace on men, which is the sole source of good human words and actions. A certain affinity between ontological hermeneutics and Protestant theology in the twentieth century is attested among the scholars. The mutual influence between the earlier Heidegger and Bultmann is well known. Elsewhere, I have demonstrated the analogies between Paul Ricoeur's philosophical and biblical hermeneutics and Gerhard Ebel-! 2 the religious dimension in Gadamer's philosophy, arguing that there are several affinities between his hermeneutics and Barth's dialectical theology. Now, my hypothesis is precisely that Gadamer's non-philosophical and non-onto(theo)logical -or non-analogical -rehabilitation of the verbum in corde reflects an attitude towards Christology which is rather Barthian.

I will try to demonstrate my thesis in three steps. First of all, I will describe the development of the notion in the Augustinian corpus from the De dialectica to the XV Book of the De Trinitate. Secondly, I will hypothesize if Gadamer has properly understood Augustine's intentions with the use of the notion or rather, if he has betrayed them. Thirdly, I will compare Gadamer's rehabilitation of the notion with his philosophical and theological background.

Augustine's verbum in corde

Augustine inherits the notion of verbum in corde from two traditions. On one hand, the 'Greek' tradition of Plato, Aristotle and the Stoics. Plato and Aristotle used the concept to give an explication of the discursive thought and its logic and semantic properties. The Stoics 9 probably invented the terms λόγος ἐνδιάθετος and προφορικός in order to distinguish men from talking animals such as parrots and blackbirds. On the other hand, the 'Biblical' or, as transmitted. This definition is maintained in the De Magistro, written in 389: 'then a name (nomen) is also a word (verbum) whenever we see that it is uttered by an articulated sound accompanied by some significate.' In this phase, even written words are not properly con 13 sidered verba, but signorum signa or, at most, signa verborum . The devaluation of the outer 14 word seems radical, and many interpreters have unfortunately reduced the Augustinian theory of language to this early perspective. It must be stressed, however, that already in these early ! 3 writings Augustine developed some notions, which will play an important role in the formation of his concept of inner word. Among them, that of dicibile, which is defined in the De dialectica as what in the word is learned through the mind and not through the ears.

The verbum in corde explicitly appears in a second phase. Just as when we speak, in order that what we have in our minds may enter through the ear into the mind of the hearer, the word which we have in our hearts becomes an outward sound and is called speech; and yet our thought does not lose itself in the sound, but remains complete in itself, and takes the form of speech without being modified in its own nature by the change: so the Divine Word, though suffering no change of nature, yet became flesh, that He might dwell among us. 17 According to Graziano Ripanti, we assist in this context to a partial rehabilitation of the spoken word. In other words, for him Augustine did not use the distinction between inner and outer word in order to understand Christology, but rather the contrary. It is precisely in this 18 rehabilitation, he says, that one has to recognize the authentic Augustinian theory of language.

Language and communication are not the specific topics of the De Magistro, which are instead at the center of the De doctrina christiana, a sort of manifesto for philosophical hermeneutics. [START_REF] Ripanti | The most part of the contemporary interpretations of the De Magistro has been focused[END_REF] In a third phase, which starts with the VIII Book of the De Trinitate, probably composed around 417, the notion of verbum in corde finds its most complete and systematic exposition.

In particular, three uses of the notion between Books VIII and XV can be isolated. In Book VIII, Augustine describes a 'verbum apud me (word in me)':

And when indeed I wish to speak of Carthage, I seek within myself what to speak, and I find within myself a notion or image (phantasia) of Carthage […]. For its word is the image (phantasia) itself of it in my memory, not that sound of two syllables when Carthage is named, or even when that name itself is thought of silently from time to time, but that which I discern in my mind, when I utter that dissyllable with my voice, or even before I utter it. So also, when I wish to speak of Alexandria, which I never saw, an image (phantasma) of it is present with me […] and this is with me when I wish to speak of it, before I utter with my voice the five syllables which make the name that almost every one knows. [START_REF][END_REF] ! 4

The word is considered here as a (sensitive) remembered (phantasia) or imagined (phantasma) image. In both cases it is an inner word, i.e. a word independent from every spoken or written word. Yet, this kind of inner word is, for Augustine, still inadequate, because it depends upon the always-changing reality.

In Book IX, the bishop of Hippo describes a verbum, which is 'cum amore notitia (knowledge together with love)': 'A word, then, which is the point we wish now to discern and imitate, is knowledge together with love. Whenever, then, the mind knows and loves itself, its

word is joined to it by love.' This kind of inner word is adequate, insofar it does not depend 21 anymore upon exteriority, but upon the autoreflexive capacity of the human soul.

The inner word as cum amore notitia is neither the knowledge of a lower essence, as of the body, nor of a higher, as of God. Rather, this inner word is like the balanced love existing between true friends. Their relation is not similar to that between the color and the colored subject, 'since colored body is a substance, but color is in a substance; but as two friends are also two men, which are substances self what to answer. [START_REF] Grondin | Eine Biographie[END_REF] In other words, according to Augustine there is something in the human soul that constitutes and exceeds the soul itself. French phenomenologist Jean-Luc Marion, reflecting on the concept of memoria in Book X of the Confessiones, has called this foundational instance in man the 'immémorial (immemorial)': 'I live in a place, my self, where I do not find myself, where I am not in my place, where I am not myself: banished from the inside, I am not where I am.' 24 ! 5

Between Book X and XIV, Augustine presents three trinities in the human mind, which reflect with different degrees of perfection the Holy Trinity. These trinities are similar to the three genres of verbum in corde I am describing in this paragraph. In Book XI, a human trinity is presented, made of (1) the perceived body, (2) its impression in the gaze of the perceiving person and (3) the attention of the will, which connects the former to the latter. 25 This first trinity, still exterior, is replaced with a second trinity in man made of (1) the image of the perceived body in memory, (2) the form that reproduces such an image when the gaze of the subject is oriented to it, and (3) the attention of the will, which once again connects the former to the latter. The superiority of this trinity compared to the first one is evi 26 dent, since 'in these three there is no longer any diversity of substance. For neither is the sensible body there, which is altogether distinct from the nature of the living being, nor is the bodily sense there informed so as to produce vision, nor does the will itself perform its office of applying the sense, that is to be informed, to the sensible body, and of retaining it in it when informed.' Nevertheless, even this trinity, through memory contents, maintains a cer 27 tain link with exteriority. Then, in Book XII, Augustine presents a human trinity, which is completely detached from exteriority, oriented toward the eternal reasons: 'But it is the part of the higher reason to judge these corporeal things according to incorporeal and eternal reasons;

[…] But we judge corporeal things from the rule of dimensions and figures, which the mind knows to remain unchangeably.' This third trinity, however, cannot be considered the best 28 image in man of the Holy Trinity since, as Augustine says in Book XV recapitulating the previous analysis, human knowledge is inferior to the wisdom of man, 'which is granted him by God's gift in the partaking of the very God Himself.' 29

In Book XV, the bishop of Hippo returns to speak about the verbum in corde:

Whoever, then, is able to understand a word, not only before it is uttered in sound, but also before the images of its sounds are considered in thought […] whoever, I say, is able to understand this [word], is able now to see through this mirror and in this enigma some likeness of that Word of whom it is said, In the beginning was the Word, and the Word was with God, and the Word was God. 30

The two terms 'mirror (spaeculum)' and 'enigma (aenigma)' refer to the First Epistle to the Corinthians (13, 12): 'For now we see in a mirror (per spaeculum), in an enigma (in aenigmitate), but then face to face. Now I know in part, but then I will know fully, even as I was also fully known.' To explain these terms, Augustine uses the expression 'obscura allegoria (ob-! 6 scure allegory)': 'and hence, as every horse is an animal, but not every animal is a horse, so every enigma is an allegory, but every allegory is not an enigma. What then is an allegory, but a trope wherein one thing is understood from another? We have seen that the goal of all attempts to reach an understanding is agreement concerning the subject matter. Hence the task of hermeneutics has always been to establish agreement where there was none or

where it was disturbed in some way. The history of hermeneutics confirms this if, for example, we think of Augustine, who sought to mediate the Gospel with the Old Testament. [START_REF] Gadamer | [END_REF] Through the Augustinian theory of language, and in particular thanks to the notion of verbum in corde, Gadamer pretends to fight the Western oblivion of language. According to some interpreters, however, the German philosopher has used the notion as an instrument. Sebastiano

Vecchio, for instance, stated that the overabundance of meaning that hermeneutics attributes to the language is completely extraneous to Augustine. Johannes Brachtendorf said that not 40 one of the elements that for Gadamer contributes to the rehabilitation of language, is present in the Augustinian theory, neither its original character, nor its processuality, nor its immediate relation with reality. Since the verbum in corde represents the essence of things, it cannot be used to overcome the metaphysics of presence. According to Giorgio Santi, privileging a 41 hermeneutics of discourse over a hermeneutics of the verbum mentis, Gadamer hinders any truthful understanding of Being. Alfred Schindler's position is more tempered. On the one 42 hand, he says, in Augustine's theology of the Word, the language's devaluation that Gadamer refuses is present. On the other hand, however, for Augustine, human will has a central role in the generation of the word (generatio verbi), and this activity is similar to the evenemential character that Gadamer attributes to the language. [START_REF] Schindler | [END_REF] It is true that in several passages of the paragraph Sprache und verbum, Gadamer, reflecting on the Augustinian verbum in corde, seems to be concerned with the national and spoken ! 8 languages, as when he writes that 'the external word and with it the whole problem of the variety of languages, was explicitly devalued by Augustine, though he still discusses it.' [START_REF] Gadamer | Dialogischer Rückblick[END_REF] Nevertheless, the German philosopher is aware that for Augustine 'the fact that the verbum is spoken differently in different languages (Sprache), however, means only that it cannot reveal itself through the human tongue (Zunge) in its true being.' Similarly, in an interview 45 given to Jean Grondin in 1996, Gadamer confirms that 'the multiplicity of languages must not represent an insurmountable barrier for hermeneutics, since all languages can be learned.' 46 The notion of verbum in corde does not concern the multiplicity of human languages, but rather the 'infinite task', which 'lies in the experience of the limit of the word as such.' 47 According to other interpreters, Gadamer has simply accomplished Augustine's intentions.

Reasoning on the analogy of proportion between the Word in God and the incarnate Word on the one hand, the inner and the spoken word in man on the other hand, Gadamer would have fulfilled the partial rehabilitation of language already existing in the Augustinian theology of verbum. Jochem Hennigfeld stated that Gadamer has been the first to show that the Christian notion of incarnation is a valid alternative to the instrumental conception of language. For 48 Johann Kreuzer, therefore, Augustine's inner word does not limit the language in interiority, but rather opens it to its own most possibility of concreteness. Gadamer would have been able

to recognize in this notion a non-instrumental conception of language that is particularly useful for his hermeneutics. According to Graziano Ripanti, Augustinian theory of language is 49 at the same time Platonic and Biblical. From the former, Augustine would have inherited the idea of the mutability of being and of human action, and hence the consequent devaluation of language. From the latter, he would have inherited the idea that revelation is given through divine signs (signa divinitus data), and hence the resulting partial rehabilitation of language.

The signs of the Scriptures are an aid to human fallibility, just as Christ, becoming flesh, became a remedy for our wounds. Through the theory of the signa divinitus data, Augustine understands the 'deep connection between word and incarnation in the unity of the mistery of Christ.' Rehabilitating the notion of verbum in corde, Gadamer refers precisely to this Bibli 50 cal root, and in particular to the analogy of proportion which is implied.

It is true, here too, that in several passages of the paragraph Sprache und verbum, Gadamer seems concerned with the analogy of proportion, as when he says that 'exegesis interprets the speaking of the word to be as miraculous as the incarnation (Fleischwerden) of God.' Nev 51 ertheless, few lines below the German philosopher points out that:

! 9

The great miracle of language lies not in the fact that the Word becomes flesh and emerges in external being, but that that which emerges and externalizes itself in utterance is always already a word. That the Word is with God from all eternity is the victorious doctrine of the church in its defense against subordinationism, and it situates the problem of language, too, entirely within inner thought. 52 In other words, the mystery of Trinity is mirrored in the language insofar the word is nothing by itself, and does not seek to be anything: it has its being in its revealing. Gadamer seems to have perfectly understood, in sum, that Augustine's 'analogy' concerns just the Word of God and the inner word in man, the presence of the former and the latter as its effect on the human soul. Augustine's 'spaeculum (mirror)' is, in other terms, closed to Gadamer's 'speculative (spekulative)' structure of language, which has nothing to do with Hegel's description of speculative thought in his analysis of the logic of the philosophical proposition. Rather, it concerns the 'realization of meaning (Vollzug der Rede)' and the 'event of speech (Geschehen der Rede)', two expressions that Gadamer borrows from the 'first' Heidegger, but understands according to the the subjective genitive of the 'second ' Heidegger. 59 Before entering the third part of the article, there is still a point that must be clarified, i.e.

the fact that in the paragraph Sprache und verbum Gadamer discusses much more Thomas Aquinas' verbum mentis than Augustinian verbum in corde. According to Dominic Kaegi, when Gadamer treats the topic, he has not in mind the Augustinian relation between inner and outer word, as Jean Grondin states, but rather the Thomasian relation between word and thing. Far from being an extrinsic remark, this sounds as an objection that can invalidate the 60 entire article, along with the hundreds of pages that have been written on Gadamer's rehabilitation of the Augustinian notion of inner word. Now, it is true that when the German philosopher explains what hermeneutics can learn from the concept of inner word, he says that 'a person who thinks something -i.e., says it to himself -means by it the thing that he thinks.

His mind is not directed back toward his own thinking […]. Thus the word is not the expression of the mind but is concerned with the similitudo rei.' Yet according to Gadamer, this 61 similitudo has to be founded in its turn on the evenemential character of the word: 'Scholastic thought is not content with viewing concept formation as simply the reflection of the order of things.' Hence, Gadamer accepts Aquinas' ordinary adaequatio but subordinates it to the 62 Augustinian extraordinary presence -which is in-adaequatio -of the Word in the human soul. For this reason, Grondin's position seems reasonable, as he affirms that the interpretation of the inner word as a foundational instance beyond all dialogue is more Augustinian than Thomasian, and brings more justice to the Gadamerian hermeneutical project. Maybe

Gadamer had some other elements in mind, too, 'but I am sure to follow him, he says, connecting the original phenomenon of hermeneutics […] to the Augustinian notion of inner word.' 63

Gadamer's verbum in corde

! 11

For Ulrich Duchrow, 'it is surprising to see the extent to which Gadamer is ready to draw conclusions about the philosophy of language and ontology from Christology, though without considering the theological question further.' Therefore, Gadamer would have improperly 64 used some important dogmatic notions for its philosophical purposes. In my opinion, by contrast, when Gadamer treats the verbum in corde, he is consciously assuming a position which is closed to the early dialectical theology of the protestant theologian Karl Barth and to its Pauline and Lutheran sources. In these theological reflections, the German philosopher has found a mirror of his philosophical hermeneutics, which can be considered the result of a short circuit between the existential and hermeneutical approach of the 'first', and the evenemential and anti-hermeneutical thought of the 'second' Heidegger. Notoriously, the late Heidegger gives up with hermeneutics in the proper sense. In Unterwegs zur Sprache, in the dialogue between a Japanese and an Inquirer, he still uses the term, but he states that 'hermeneutics means not just the interpretation but, even before it, the bearing of message and tidings.' 65 The language is not much a dialogue, but rather a silent listening of Being: 'But language is monologue. This now says two things: it is language alone which speaks authentically; and, language speaks lonesomely.' 66 Heidegger seems aware of the radical incompatibility between his early and his late thought. On the contrary, Gadamer defends the dialogic character of language but, in the meantime, he confirms to have been radically fascinated by the evenemential thought of the 'second' Heidegger, as he said in an interview given to Italian interpreter Adriano Fabris in 1982. Of course, there are many passages of Wahrheit und Methode which are not extrane 67 ous to the personal perspective I am going to suggest in the conclusion of this article. This is the case, for instance, of the chapter devoted to the hermeneutical relevance of Aristotle, where Gadamer discusses the problem of the applicatio. These parts are perfectly coherent in themselves. Yet, the question here is how it is possible to articulate the dialogical aspect of his thought with the evenemential one. Far from being a partial objection to Gadamer's philosophy, I would say that partial is any perspective that considers his hermeneutics simply as a philosophy of the dialogue.

If on a philosophical point of view, Gadamer's position appears as quite paradoxical, on a theological point of view, instead, he assumes a more coherent perspective. His use of the In particular, the German philosopher is fascinated by the notion of grace. As he states in an article published in 1964, 'the task that Christianity assigns to hermeneutics is to ponder the strangeness (Fremdheit) and bewilderment (Befremdung) that dwells in the Christian message as such. This culminates in the attempt to think of redemption and faith as a divine grace in all respects, so that all capacity and all merit loose value.' Gadamer appreciates the Protestant 70 religious language to the extent that it is opened to an announcement, which 'goes against all human expectations.' For this reason, in Wahrheit und Methode, it is pointed out that 'cer 71 tainly preaching too is concerned with interpreting a valid truth, but this truth is proclamation;

and whether it is successful or not is not decided by the ideas of the preacher, but by the power of the word itself, which can call men to repentance even if the sermon is a bad one.' In a 72 text published in French in 1977, Herméneutique et théologie, Gadamer finally states that

The precondition of accepting the Christian message consists of opening the eyes to the experience of the divine, in a general sense. The conflict between law and grace remains essential. The idea of a dominant knowledge or technique […] is the inherent basis of all human beings. In the field of theology, however, it remains the biggest obstacle in the acceptance of the promise of the gospel, namely the opening to grace. [START_REF] Gadamer | Herméneutique et théologie[END_REF] This passage sounds really Barthian. To say the truth, however, Barth is named just once, in 74 a marginal footnote, in Wahrheit und Methode. The mention concerns Karl Barth's reflections on 'Feuerbach's reversal of the imago dei thesis of Genesis': 'This anthropological reversal and reinterpretation of religious experience, which became current in the nineteenth century, arises from the same subjectivism that lies at the basis of modern aesthetic thought.' This 75 brief statement is important, however, for at least three reasons.

! 13

First of all, because Gadamer opposes Barth's antisubjectivism to modern aesthetics, which is an object of criticism in the first part of Wahrheit und Methode. For Gadamer, beauty is not a matter of personal taste, not even that of the romantic genius. Rather, true beauty is universal, and 'the world of the work of art, in which play expresses itself fully in the unity of its course, is in fact a fully transformed world. In and through it everyone recognizes that that is how things are.' The same criticism is at the core of the second and third part of the 76 Gadamerian opus magnum. In the name of the 'wirkungsgeschichtliche Bewusstsein', which I propose to translate as 'conscience of the work of history', the German philosopher rehabilitates authority and tradition against Enlightenment's critics of prejudice: 'the fundamental prejudice of the Enlightenment is the prejudice against prejudice itself, which denies tradition its power.' In the aforementioned interview with A. Fabris, Gadamer affirms that the con 77 science of the work of history has nothing to do with the traditional concept of conscience, but rather with the late Heideggerian notion of Being. As we have seen in the course of this 78 article, the third part of Wahrheit und Methode is devoted to overcoming the instrumental notion of language. The brief reference to Karl Barth, thus, is mirrored the Gadamer's purpose to elevate hermeneutical experiences to the rank of truth events.

Secondly, the reference is important since Barthian religious antisubjectivism is opposite Protestant liberal theology, whose founder has been Schleiermacher. According to Gadamer, as Heidegger had already stated in his 1923 seminar Ontologie. Hermeneutik der Faktizität, Schleiermacher has privileged the technical and psychological dimension of hermeneutics.

Hence, he has exalted the subjective dimension of interpretation, i.e. the intentions of the author, to the expense of the claim for truth of the text -the thing of the text. However, Schleiermacher did not accomplish the technicist and psychologist orientation of the historical school and the romantic hermeneutics. His hermeneutics is still oriented toward the dogmatic interest for specific texts: 'his hermeneutical theory was still a long way from a historiology that could serve as a methodological organon for the human sciences. Its goal was the exact understanding of particular texts.' This is, after all, the intrinsic paradox of the entire liberal 79 theology's tradition. On the one hand, it aims to display the universal character of the proclamation. On the other hand, however, it remains within the specific proclamation of Jesus Christ. Liberal theology, then, cannot realize its proper claim for universality. Gadamer's critics to the Schleiermacherian hermeneutics can be understood as a consequence of the critics to the self-consciousness of faith, which is, according to him, the dogmatically dangerous ba-! 14 In conclusion, I can say that the Gadamerian use of the Augustinian verbum in corde,

properly understood as the consequence of the Word-and Truth-event of God on men, is the consequence of such a radical notion of grace, which is, in its turn, the adequate paradigm of the anti-methodological and anti-liberal ontological hermeneutics of Gadamer.

Conclusion

In the first section, I have demonstrated that the Augustinian notion of verbum in corde is an innovation with regards to both the 'Greek' and the 'Biblical' tradition. The bishop of Hippo is not asmuch interested into the linguistic structure of the human mind, nor in the relation between the inner and outer word, and its similitude with the Word of God. Rather, Augustine is concerned with the efficacy of the Word of God on men: the verbum in corde is the consequence of a Word-and Truth-event. If there is a partial rehabilitation of exteriority, as some interpreters have suggested, this is just a consequence of this more radical statement. In the second section, I have demonstrated that, through his rehabilitation, Gadamer did not misunderstand, nor accomplish Augustine's intentions. Rather, the German philosopher has properly understood the Augustinian notion of verbum in corde as a matter of faith, i.e. the consequence of a gift that cannot be returned. In his perspective, the importance of the notion 89 does not much reside in the rehabilitation of the spoken and national languages. More profoundly, the verbum in corde concerns the unique Word, which is behind -and supports -all genuine words. In the third section, I have suggested that Gadamer finds in the Augustinian notion a paradigm for both its philosophical and theological insights. Concerning the former, the German philosopher, although defending the dialogicity of language, has always been fas-

10 Claude

 10 Panaccio calls it, the 'Johannine' tradition of several Greek and Latin Church Fathers that used (or refused to use) the Stoic distinction to defend the Christian dogma of consub 11 stantiality. Yet, through the notion of verbum in corde Augustine also said something new. My thesis here is that the bishop of Hippo has been the first to think of the concept of inner word in light of the doctrine of grace.According to C. Panaccio, there are three phases in Augustine's use of the concept. Before 395, the concept is simply absent. In the De dialectica, probably composed in 387, Augustine affirms that 'a word (verbum) is the sign of some thing which can be understood by the hearer when pronounced by the speaker.' The verbum is just a means through which the meaning is 12

  […] but an enigma is, to explain it briefly, an obscure allegory[…].' As Jean Pépin has stated, allegorical interpretation and 31 allegorical expression represent two opposite intellectual processes. The former consists of the human capacity to go from the first (lower) to the second (higher) meaning of an allegorical expression. The latter corresponds to the creation or identification of signs with a double intentional structure, usually translating into images an abstract knowledge. The obscure 32 allegory is a particular kind of allegorical expression, whose purpose is not to conduct from the first to the second meaning, but rather to safeguard the distance between them. According to Tvetan Todorov, the function that Augustine attributes to allegorical expressions changed over the years. At the time of the first part of the De doctrina christiana, 33 which remained incomplete for almost thirty years, they were considered a sort of stratagem to draw towards the Scriptures the most clever minds, 'for the purpose of preventing a feeling of bore in the intellect, which generally holds in small esteem what is discovered without difficulty.' The 'traumatic' reception of the Pauline epistles, and in particular of the Pauline

  German term 'Wort (word)' to translate the Greek λόγος comes evidently from Luther. So his interest into Augustine's notion of verbum in corde is explained as well, since 'when Luther ! 12 translates the logos of the Prologue of the Gospel of John as 'Wort', he takes up the whole theology of the word, which goes back, at least, to the interpretations that St. Augustine gave of the Trinity.' In the aforementioned interview to RAI Italian television, Gadamer recalls 68 that [...] In the years when I was a student in Marburg (in the Protestant Marburg of the Lutheran church) and I attended the first university in the history of the Lutheran Reformation, there was an excellent library of philosophical texts. The Middle Age was represented by Augustine -nothing else! The modern era began with Galileo -nothing else! All that stood between Augustine and Galileo -Thomas, Duns Scotus, and more -was considered 'Catholic.'[START_REF] Gadamer | Ästetische und religiöse Erfahrung[END_REF]

  In the Epistulae ad Romanos incohata expositio, Augustine defines a 'verbum [...] quod corde conceptum (word conceived in the heart).' Speaking, he says, indicates 'not only what we shape with the tongue, but also 15 what is conceived in the heart and manifested through actions.' In the first part of the De 16 doctrina christiana, than, the bishop of Hippo uses the expression 'verbum quod corde gestamus (the word which we have in our heart)':

Interpreter of Augustine's verbum in corde

  tine understands the notion in the light of this evenemential and anti-hermeneutical perspective, which finds its roots in the doctrine of the gratuitous election to salvation.Gadamer's interest towards Augustine is focused on three topics: time, hermeneutics and language. Through his notion of time as 'form of human experience' and 'place of truth', Augustine would have somehow surpassed the Greek metaphysics of presence. In the De doctrina 38 christiana, Gadamer has seen a certain hermeneutical awareness, a methodological conscience and the effort to mediate between Old and New Testament. It is precisely in this effort of mediation that the German philosopher recognizes a central task for hermeneutics:

	Gadamer
	35
	hermeneutical position or, as Pier Cesare Bori said, from a 'joyful and playful (gioiosa e gio-
	cosa)' towards a 'narrow (angusta)' hermeneutics. The function of allegorical expressions 36
	no longer concerns the exercise of human intellect and free interpretation. Rather, allegorical
	expressions have the function to discriminate and exclude, 'to throw a veil over the minds of

34

doctrine of grace, would have radically affected this first hermeneutical perspective. Italian interpreter Gaetano Lettieri, as Kurt Flash had before him suggested in Logik des Schreckens, individuates in De diversis questionibus ad Simplicianum I, 2 a turning point: 'Therefore we must understand that the law was given not to introduce sin nor to extirpate it, but simply to make it known [...]. Sin cannot be overcome without the grace of God, so the law was given to convert the soul by anxiety about its guilt, so that it might be ready to receive grace.' After this crucial passage, allegorical expression is considered 'a sort of veil, which precludes to natural intelligence the true understanding [...] a veil that just the grace of God, and not human intelligence, can remove.' Augustine migrates from a hermeneutical towards an antithe godless either that they may be converted to piety or shut out from a knowledge of the mysteries.' In conclusion, comparing the verbum in corde with an obscure allegory, Augus 37 -! 7

  'Exactly the same thing, Gadamer says, is true of the mystery of the Trinity. Here too the important thing is not the earthly appearance of the Redeemer as such, but rather his complete divinity.' Similarly,

	I would like to recall, however, that Augustine has written over 15 books on the Trinity -the De Trinitate,
	which is his masterpiece. The Confessions are without any doubt the most famous work of world litera-
	ture (I am sure), but for philosophers these 15 books on the Trinity represent the core of his thought. How
	so? How is it possible? On this point it is absolutely necessary to be clear: Christianity would not be what

53

in a dialogue of 1998 with Gudrun Kühne-Bertram and Frithjof Rodi on the logical hermeneutics of Georg Misch and Hans Lips entitled Die Logik des verbum interius, Gadamer stresses that the important thing in what Germans call 'linguisticity (Sprachlichkeit)', and Stoics named λόγος ἐνδιάθετος, is that 'even the non perceptible meaning (vernehmbare Sinn) is already logos.'

[START_REF] Gadamer | Die Logik des verbum', 25. 55[END_REF]

Now, my thesis here is that in his rehabilitation of the verbum in corde Gadamer neither betrayed nor accomplished Augustine's intentions. Rather, the German philosopher has properly understood the notion. In an article published in 1993, Europa und die Oikoumene, Gadamer states that 'the descent of the Son from the Father' is similar to the 'emerging when something arrives to the language.' To define the notion of verbum in corde, Gadamer does 55 not use in this context the distinction between inner and outer word, but the Augustinian concept of distentio animi. The German philosopher defines elsewhere such a concept as the 56 'tension of the soul towards the future and, ultimately, towards the liberation from temporality through the intervention of the divine grace.'

[START_REF] Gadamer | Das Rätsel der Zeit[END_REF]

In one of the interviews given to RAI Italian television, Gadamer clearly explains the reason of his interest in the De Trinitate:

it is, if it would be a doctrine that can be taught, if the requirement of faith would not be lived just as faith, i.e. as a gift.

[START_REF] Gadamer | [END_REF]

! 10

  Gadamer affirms that 'despite the general aversion to methodological reflection, [the Römerbrief] is a sort of interpretation manifesto.' A quite strange statement, to tell the truth, since 81 one could say that it is precisely the lack of method that, for Gadamer, makes Karl Barth's exegesis a genuine hermeneutical practice: 'after all, he says, I do not propose any method, but I just describe what it is […] evidently I could not persuade Betti that a philosophical theory of hermeneutics, right or wrong, is not a doctrine of the method.' 82 Thirdly, the brief reference to Karl Barth in Wahrheit und Methode is important since it reminds the Gadamerian critics of Bultmann's demythologization. Jean Grondin, although 83 underestimating Gadamer's 'religious dimension', has opportunely stated that according to the German philosopher, Butlmann 'has remained too close to the (transcendental) conception of understanding presented in Being and Time', and that, for this reason, 'he could not consider the thought of the last Heidegger[…]. For Gadamer, instead, this thought could have been striking for theology, since it looks for a more adequate language to the expression of the divine.' According to Gadamer, in fact, Bultmann's affirmation in Glauben und Verstehen that 84 'the interpretation of the biblical writings is subject to exactly the same conditions as any oth-As interpreter Giovanni Moretto has suggested, 'Gadamer call than Bultmann back to the Protestant orthodoxy, i.e. to a theology that, even if does not consider anymore, as Luther did, the human reason as something diverted and prostituted […], still deny the possibility of an interpretation of Biblical texts external to the context of faith.' 88

	er literature' is ambiguous. Theologian's interpretation, although scientific as Bultmann's 85
	demythologization pretends to be, 'must never forget that Scripture is the divine proclamation
	of salvation.' On the one hand, Bultmann recognizes, as Heidegger did, that all interpreta 86 -
	tion presupposes a vital relation of the interpreter with the interpreted text, and a preliminary
	relation with the thing (Sache), which is communicated through the text. On the other hand,
	Bultmann does not see that such a pre-comprehension is not the result of a scientific process
	of interpretation, but rather is 'presupposed as already given'. The prerequisite, which is
	opened to interpretation, consists of being touched by the problem of God. And such a pre-
	requisite is not given by the simple fact of being a man, as liberal theology explicitly sug-
	gests, and Bultmann's demythologization implicitly admits; rather, Gadamer affirms, 'it is
	first from God -i.e. from faith.' Only the grace of God, which is a truth event, opens for the 87
	German philosopher the space for genuine interpretation.
	The Catholic Church, with its predilection for the works, would not have understood, ac-

cording to Gadamer, this principle of ontological hermeneutics. Bultmann, in his turn, has forgotten that the demythologization is already enclosed in this first prerequisite, exactly as Heideggerian analytics of the Dasein is enclosed into the ontological question on the sense of ! 15

Being.

  him at least twice. First of all, because the verbum in corde does not set in motion, as the French interpreter affirms, any circularity between the said and the unsaid. The Augustinian notion does not much suggest that 'there is something like a dialogue 'behind' or better 'with' every enunciation.' Rather, the verbum in corde suggests that, behind all true dia 90 logue, there is just a monologue. Hence, the notion does not charge the language of its essential hermeneutical circularity, but it seems rather to 'anesthetize' it. The verbum in corde, therefore, makes hermeneutics somehow ineffective. Secondly, because it is not true that the Augustinian notion represents a paradigm for all hermeneutics aiming at truth and universality. In Paul Ricoeur, for instance, truth and method do not represent any alternative. More 91 over, I believe that hermeneutics should now question its own concepts of meaning, truth and being without fearing to be considered nihilistic. Between the substantialization of meaning and the Nietzschean formula 'there are no facts, only interpretations', there is room for a 'lesser hermeneutics', which does not consider understanding as a Meaning-, Truth-or Beingevent, but rather as the always-renegotiable conviction of having reached a little bit of meaning, truth or beings.Notes1. See Grondin, 'Gadamer und Augustin'; L'universalité de l'herméneutique; 'L'universalité de l'herméneutique et de la réthorique: ses sources dans le passage de Platon à Augustin dans Vérité et Méthode'. 2. See Oliva, Das innere verbum; Artos, The inner Word. 3. In Wahrheit und Methode, Gadamer referers just once to the 'verbum cordis' -as he calls it -of De Trinitate XV,10-15. 4. Gadamer, Wahrheit und Methode, 359 (Truth and Method, 348). 5. Ibid., 361 (351). 6. Koch, 'Le verbum in corde', 3. 7. For the notion of inner word in Luther, see zur Mühlen, Nos extra nos; Beutel, In dem Anfang, 372-406. For these authors, Luther has never completely renounced the classical distinction between inner and outer word. Rather, he has radically reinterpreted it in light of the distinction between litera and spiritus. In other words, Luther has not been concerned with the ontical or ontological status of the word, but with its (theological) effectiveness (Wirksamkeit/Wirkungscarachter). 8. See Romele, 'La parole efficace chez Gerhard Ebeling et Paul Ricoeur'. 9. For Plato see Thaet., 189e, Phil. 38d-e and Soph., 263e; for Aristotle see An.Post. 10, 76b and Cat. 6, 4b 34- 35. Verità e interpretazione, 168. 79.Gadamer, Wahrheit und Methode, 201 (195). 80. Moretto, La dimensione religiosa, 112. 81.Gadamer, 'Hermeneutik und Historismus', 391. 82. Ibid., 394-395. 83. For a similar critics to Bultmann see alsoRicoeur, 'Préface à Bultmann'. 84. Grondin, 'Gadamer et Bultmann', 126. 85. Gadamer, Wahrheit und Methode, 336 (327). 86. Ibid.. 87. Ibid., 337 (327). 88. Moretto, La dimensione religiosa, 98-99. 89. Jean-Luc Marion has well shown the incompatibility of donation and interpretation. See, for instance, Marion, Dieu sans l'être, 212. 90. Grondin, L'universalité de l'herméneutique, 38. 91. See Ricoeur, 'La fonction herméneutique'.Beutel, Albrecht. In dem Anfang war das Wort. Studien zu Luthers Sprachverständnis [In the Beginning was the Word. Studies into Luther's Language Understanding]. Tübingen: Mohr Siebeck, 1991. Bori, Pier Cesare. "Figure materne e Scrittura in Agostino." Annali di storia dell'esegesi 9, no. 2 (1992): 397-420. Brachtendorf, Johannes. Die Struktur des menschlichen Geistes nach Augustinus. Selbstreflexion und Erkenntnis Gottes in De Trinitate [The Structure of the Human Soul according to Augustine. Self-reflection and Knowledge of God in De Trinitate]. Hamburg: Felix Meiner, 2000. Chiesa, Curzio. "Le problème du langage intérieur chez les stoïciennes." Revue internationale de philosophie 178, 1991: 301-321. Cipriani, Nello. "L'altro Agostino di G. Lettieri." Revue d'Etudes Augustiniennes 48, 2002: 249-266. Duchrow, Ulrich. Sprachverständnis und biblisches Hören bei Augustin [Language Understanding and biblical Hearing in Augustine]. Tübingen: Mohr Siebeck, 1965. Flasch, Kurt. Logik des Schreckens. Aug. von Hippo De diversis questionibus ad Simplicianum I, 2 [The Logic of the Terror. Augustine of Hippo's De diversis questionibus ad Simplicianum I, 2]. Mainz: Dieretich, 1990. Gadamer, Hans-Georg. Agostino. http://www.emsf.rai.it/gadamer/interviste/08_agostino/ agostino.htm#5. Accessed October 23, 2014. Gadamer, Hans-Georg. "Ästetische und religiöse Erfahrung." In Ästethik und Poetik I: Kunst als Aussage. Gesammelte Werke8, 143-155. Tübingen: Mohr Siebeck, 1993.

	77. Ibid., 275 (273).
	cinated by the evenemential character of the late Heidegger's thought. Concerning the latter, he has explicitly accused Bultmann's demythologization of being 'human, all too human', and he has implicitly praised Barth's dialectical theology. ! 17 78. Gadamer, ! 22

Now, for what I have demonstrated in these pages, Jean Grondin's hypothesis, according to which the verbum in corde is a paradigm for hermeneutics and a privileged point of view to test its effectiveness, can be considered valid. Yet, for the same reason, it is possible to dis-! 16 agree with ! 20