Charlotte Dion

Sarah Lemler

Eva Löcherbach

Exponential ergodicity for diffusions with jumps driven by a Hawkes process

Keywords: AMS Classification: 60J35, 60F99, 60H99, 60G55 Diffusions with jumps, nonlinear Hawkes process, Piecewise deterministic Markov processes (PDMP), Ergodicity, Foster-Lyapunov drift criteria, β-mixing

In this paper, we introduce a new class of processes which are diffusions with jumps driven by a multivariate nonlinear Hawkes process. Our goal is to study their long-time behavior. In the case of exponential memory kernels for the underlying Hawkes process we establish conditions for the positive Harris recurrence of the couple (X, Y ), where X denotes the diffusion process and Y the piecewise deterministic Markov process (PDMP) defining the stochastic intensity of the driving Hawkes. As a direct consequence of the Harris recurrence, we obtain the ergodic theorem for X. Furthermore, we provide sufficient conditions under which the process is exponentially β-mixing.

Introduction

Motivation

In this work, we introduce a new class of continuous time processes X = (X t ) t≥0 taking values in R, with jumps driven by a multivariate nonlinear Hawkes process N, satisfying the following equation

X t = X 0 + t 0 b(X s )ds + t 0 σ(X s )dW s + t 0 a(X s-) M j=1 dN (j) s , (1.1)
where W is the standard one-dimensional Brownian motion and N is a multivariate Hawkes process having M components with intensity process λ. N is supposed to be independent from W. We shall specify the dynamics of N later on, in particular, the precise form of the stochastic intensity process λ is given in (2.1) below.

The jumps of the Hawkes process impact the dynamic of the diffusion process. In this model, the structure of the jumps is different from classical jump-diffusion processes with Lévy jumps. Indeed: 1) the intensity of the jump process N does not depend on the dynamic of X, 2) the Hawkes process has a special structure of time dependency: the state of the intensity at time t depends on the entire past, 3) the multidimensional nature of the Hawkes process can be interpreted as the influence of M subjects communicating one with each other and impacting the dynamic of X along time.

Our main motivation is to be able to describe systems of interacting neurons, where we are in particular interested in modeling the membrane potential of a single neuron together with the sequence of spike times of its presynaptic neurons. Indeed, neurons are communicating through the emission of electrical signals, forming a network. When a (presynaptic) neuron sends a signal to another (postsynaptic) neuron, the membrane potential of the neuron increases sharply, and we say that the neuron spikes at this time. If we are looking at one fixed neuron at the heart of this network, its activity between two spikes of one of its presynaptic neurons can be designed by the present model. Indeed, during this interval the membrane potential can be described by a diffusion process influenced by the spikes coming from the presynaptic neurons which are modeled by the Hawkes process. Hawkes processes have been studied recently in the context of neurosciences [see [START_REF] Reynaud-Bouret | Inference of functional connectivity in neurosciences via Hawkes processes[END_REF][START_REF] Hodara | Hawkes processes with variable length memory and an infinite number of components[END_REF] to model the interactions between the neurons within their sequences of spikes. The new model we propose allows to use both informations: the continuous membrane potential for a fixed neuron together with the spikes of several other neurons around it.

The aim of the present article is to study the longtime behavior of the process X. Since X is not Markov on its own, we propose a study of the couple of processes (X, λ) in the case when the intensity λ of N can be described in terms of a piecewise deterministic Markov process (PDMP) Y, see (2.4) below. Such a description is possible when the memory kernels are exponential. We prove the positive Harris recurrence of (X, Y ) together with ergodicity, and we establish the speed of β-mixing. These are important probabilistic properties which are interesting in its own right but which are also the key ingredients for the theoretical study of estimators of the model parameters. The estimation of the coefficients of the process is the issue of the companion paper [START_REF] Dion | Nonparametric drift estimation for diffusions with jumps driven by a Hawkes process[END_REF].

State of the art

Time-homogeneous diffusion processes are well known Markov processes that have been widely studied. Let us cite for example [START_REF] Ito | Diffusion processes and their sample paths[END_REF] which has initiated the work on Markovian properties of diffusion processes or the by now classical references [START_REF] Ikeda | Stochastic Differential Equations and Diffusion Processes[END_REF][START_REF] Revuz | Continuous martingales and Brownian motion[END_REF] in which all properties of diffusions are summarized. Ergodicity and strong mixing have been established in the pioneering papers [START_REF] Yu | On polynomial mixing bounds for stochastic differential equations[END_REF][START_REF] Sa | On subexponential mixing rate for Markov processes[END_REF]. Then, jump-diffusion processes have been introduced, for example in the context of risk management [see 36], either driven by a Poisson process or more generally by a Lévy process [see 1, for a review]. Their ergodicity and mixing properties have been studied from a probabilistic point of view by [START_REF] Masuda | Ergodicity and exponential β-mixing bounds for multidimensional diffusions with jumps[END_REF].

On the other hand, Hawkes processes [START_REF] Hawkes | Spectra of some self-exciting and mutually exciting point processes[END_REF] have been studied a lot lately. These point processes generalize the Poisson processes by introducing what is commonly called "self-excitation": Present jumps are able to trigger (or to inhibit) future jumps, and in this way, correlations between successive jump events are introduced. [START_REF] Daley | An introduction to the theory of point processes: volume II: general theory and structure[END_REF] gives a precise synthesis of earlier results published by Hawkes. Important probabilistic results for these mutually exciting point processes, such as stability, limit theorems and large population limits, have been obtained in [START_REF] Brémaud | Stability of nonlinear Hawkes processes[END_REF][START_REF] Delattre | Hawkes processes on large networks[END_REF][START_REF] Bacry | Some limit theorems for Hawkes processes and application to financial statistics[END_REF][START_REF] Bacry | Second order statistics characterization of Hawkes processes and nonparametric estimation[END_REF][START_REF] Clinet | Statistical inference for ergodic point processes and application to limit order book[END_REF][START_REF] Raad | Renewal time points for Hawkes processes[END_REF]. Ergodicity for general non-linear Hawkes processes with general memory kernels has been recently established by [START_REF] Raad | Renewal time points for Hawkes processes[END_REF]. For linear Hawkes processes, [START_REF] Graham | Regenerative properties of the linear Hawkes process with unbounded memory[END_REF] proves exponential ergodicity, giving very precise estimates on the regeneration times. Finally [START_REF] Clinet | Statistical inference for ergodic point processes and application to limit order book[END_REF] establishes exponential ergodicity in the particular case of linear Hawkes processes with exponential memory kernels.

Relying on one of these recent results on ergodicity of Hawkes processes, it is therefore reasonable to argue that the solution of (1.1) should be ergodic itself, provided we are able to ensure that the diffusion part of (1.1), that is the dynamic (1.1) with a(•) ≡ 0, driving the motion in between successive jumps, is ergodic.

However, no such general criterion, ensuring that the ergodicity of the underlying jump process implies the ergodicity of the associated diffusion with jumps, exists, at least not to our knowledge.

The present paper proposes therefore a short study of the ergodicity of diffusions with jumps driven by Hawkes processes with exponential memory kernels. In this case, the associated stochastic intensity processes of the jump part can be expressed in terms of particular Markov processes, PDMP's, see e.g. [START_REF] Duarte | Stability, convergence to equilibrium and simulation of non-linear Hawkes processes with memory kernels given by the sum of Erlang kernels[END_REF]. For these PDMP's, ergodicity has been established by [START_REF] Clinet | Statistical inference for ergodic point processes and application to limit order book[END_REF] in the case of linear Hawkes processes, and by [START_REF] Duarte | Stability, convergence to equilibrium and simulation of non-linear Hawkes processes with memory kernels given by the sum of Erlang kernels[END_REF] in the case of non-linear Hawkes processes with Erlang kernels. Here, we extend these results to the general non-linear case, including also inhibitions1 , and more importantly, we prove the joint ergodicity as well as the β-mixing property for the couple (X, Y ) in the case of exponential kernel functions. Finally, we give criteria ensuring that the mixing is exponential.

Main contributions and plan of the paper

We propose a general study of the longtime behavior of a jump diffusion process, where the jumps are driven by a multivariate non-linear Hawkes process in dimension M, having exponential memory kernels. Our proof of the Harris recurrence of (X, Y ) follows a well-known scheme, see e.g. [START_REF] Meyn | Stability of Markovian processes I : Criteria for discrete-time chains[END_REF][START_REF] Meyn | Stability of Markovian processes III : Foster-Lyapunov criteria for continuous-time processes[END_REF].

Firstly, we establish a local Döblin lower bound for the transition semigroup of the process (X, Y ), based on a lower bound of the joint transition density of (X, Y ). This is the content of Theorem 3.1. The ideas that we use to obtain this lower bound are very natural, and they have appeared independently of our work already in [START_REF] Clinet | Statistical inference for ergodic point processes and application to limit order book[END_REF]. They have also been exploited in [START_REF] Duarte | Stability, convergence to equilibrium and simulation of non-linear Hawkes processes with memory kernels given by the sum of Erlang kernels[END_REF] and in [START_REF] Löcherbach | Absolute continuity of the invariant measure in piecewise deterministic Markov Processes having degenerate jumps[END_REF]: more precisely, we use M 2 consecutive jumps which are carefully chosen to produce a part of Lebesgue density for the transition of Y. Once the jump part of the process possesses some part of Lebesgue density, we have to ensure that this density is preserved by the stochastic flow. This part of the proof relies on fine support properties of (X, Y ) and provides, as a by-product, a simulation algorithm for the process. In particular we need to localize the possible positions of the diffusion part, conditionally on the jump times of the underlying Hawkes process that have already been chosen.

In a second step, a Foster-Lyapunov condition (see Proposition 3.4) ensures the control of the return times to some compact set K. We then use control arguments to deduce from this a precise control of the hitting times of the set C where the transition densities are lower-bounded according to the local Döblin lower bound. Both results together imply the positive Harris recurrence of the couple (X, Y ), as well as the ergodic theorem. This is our first main result, stated in Theorem 3.6.

Finally, following the steps of [START_REF] Masuda | Ergodicity and exponential β-mixing bounds for multidimensional diffusions with jumps[END_REF] where the ergodicity and exponential β-mixing bounds are established for diffusions with jumps driven by a Lévy process and the general framework of [START_REF] Michajlovič | Introduction to Ergodic rates for Markov chains and processes with applications to limit theorems[END_REF], our second main theorem 3.8 states the mixing property of the process together with a control on the rate of convergence. To conclude, our Theorem 3.10 provides sufficient criteria implying the exponential control of the β-mixing coefficient of (X, Y ).

Finally, let us mention that it is possible to extend our results to multivariate diffusions driven by Hawkes processes. However, the primary aim of our article is to provide a theoretical probabilistic foundation for a statistical application where we want to deal with data describing the membrane potential of one single neuron together with the spike trains (the point process data associated to the subsequent times of action potentials) of its presynaptic neurons. This is why we concentrate on a one-dimensional diffusion model, driven by a multivariate Hawkes process in this paper. Indeed, in this modeling context, the joint evolution of the membrane potentials of, say, d neurons, is independent, conditional on the evolution of the spike trains of their presynaptic neurons. This is due to the fact that the membrane potential processes of different neurons within a network do only interact via the incoming stimuli, that is, the incoming spike trains of the presynaptic neurons. In other words, their driving Brownian motions are independent -no interaction is present in their drift nor in their diffusion coefficient.

Our paper is organised as follows. We introduce our model together with all necessary notation and assumptions in Section 2. Section 2.3 provides a simulation algorithm of the process which is interesting in its own right. In Section 3, we prove the ergodicity and finally the exponential β-mixing of the coupled process (X, Y ).

2 Presentation of the model and first properties

The model

Throughout this paper we work on a filtered probability space (Ω, F, F). We start by introducing the jump part of our process, that is, the multivariate Hawkes process.

This process is defined in terms of a collection of jump rate functions f i : R → R + , 1 ≤ i ≤ M, and in terms of interaction functions (also called memory kernels) h ij : R + → R, 1 ≤ i, j ≤ M, which are measurable functions. Let moreover n (i) , 1 ≤ i ≤ M, be discrete point measures on R -satisfying that

R - |h ij |(t -s)n (j) (ds) < ∞ for all t ≥ 0.
We interpret them as initial condition of our process. Assumption 2.1. We suppose that each f i , 1 ≤ i ≤ M, is Lipschitz continuous with Lipschitz constant γ i . Definition 2.2. A non-linear Hawkes process with parameters (f i , h ij ) 1≤i,j≤M , and with initial condition n (i) , 1 ≤ i ≤ M, is a multivariate counting process N t = (N (1) t , . . . , N (M ) t

), t ≥ 0, such that 1. P-almost surely, for all i = j, N (i) and N (j) never jump simultaneously, 2. for all 1 ≤ i ≤ M, the compensator of N (i) t is given by t 0 λ (i) s ds, where

λ (i) t = f i   M j=1 t- 0 h ij (t -u)dN (j) u + M j=1 0 -∞ h ij (t -u)dn (j) u   . (2.1)
If the functions h ij are locally integrable, the existence of the process (N t ) t≥0 with the prescribed intensity (2.1) on finite time intervals follows from standard arguments, see e.g. [START_REF] Brémaud | Stability of nonlinear Hawkes processes[END_REF] or [START_REF] Delattre | Hawkes processes on large networks[END_REF].

We consider a jump diffusion process X = (X t ) t≥0 taking values in R, whose jumps are governed by N. More precisely, X is solution of the stochastic differential equation

X t = X 0 + t 0 b(X s )ds + t 0 σ(X s )dW s + t 0 a(X s-) M j=1 dN (j) s , (2.2)
where X 0 is a random variable independent of W and of N. Here W is the standard Brownian motion in dimension one, independent of the multivariate Hawkes process N having intensity (2.1).

Proposition 2.3. Suppose that Assumption 2.1 holds, that the coefficients b, σ are globally Lipschitz continuous, that a is measurable and that the memory kernel functions h ij are locally integrable. Then equation (2.2) admits a unique strong solution.

Proof. Theorem 6 of [START_REF] Delattre | Hawkes processes on large networks[END_REF] implies that for any T > 0, almost surely, N t has only a finite number of jumps on [0, T ]. We can therefore work conditionally with respect to the choice of these jumps and construct the solution of (2.2) by pasting together trajectories of strong solutions to the diffusion equation (that is, the solution of (2.2) with a(•) ≡ 0) at the successive jump times (see [START_REF] Ikeda | A construction of Markov processes by piecing out[END_REF] for related ideas).

In the sequel we shall also need estimates on the transition densities of the diffusion process underlying (2.2). We want to be able to deal with a general drift term b that is not bounded in order to include Ornstein-Uhlenbeck type processes to the class of models that we consider. Unbounded drift coefficients add however a substantial difficulty when one wishes to apply techniques from Malliavin calculus to obtain bounds on the transition densities. Therefore, we impose the following additional assumptions which are inspired by [START_REF] Gobet | Lan property for ergodic diffusions with discrete observations[END_REF].

Assumption 2.4.
1. a is continuous.

2. b and σ are of class C 2 .

3. There exist positive constants c, q such that for all

x ∈ R, |b (x)| + |σ (x)| ≤ c and |b (x)| + |σ (x)| ≤ c(1 + |x| q ).
4. There exist strictly positive constants σ 0 and σ 1 such that σ 0 ≤ σ 2 (x) ≤ σ 1 for all x ∈ R.

Remark 2.5. We do only need the above assumption to obtain good lower bounds on the transition densities of the underlying non-jumping diffusion process, see (3.3) below.

To study the longtime behavior of X and to ensure its ergodicity, we introduce two additional conditions which are classical in the study of diffusion processes and which are derived from [START_REF]Has'minskii. Stochastic stability of differential equations[END_REF], see also [START_REF] Yu | On polynomial mixing bounds for stochastic differential equations[END_REF].

In the following, we assume we are in one of the following frames.

Assumption 2.6. Exponential frame. There exist d ≥ 0, r > 0 such that for all x satisfying |x| > r, we have xb(x) ≤ -dx 2 . Moreover, one of the two following conditions holds true.

1. For all |x| > r, 2xa(x) + a 2 (x) ≤ 0.

2. For all |x| > r, |a(x)| ≤ C|x| η for some η < 1.

Assumption 2.7. Polynomial rate. There exist γ > σ 1 /2, r > 0 and m ∈]2, 1 + 2γ

σ 2 1 [, such that for all x satisfying |x| > r, we have xb(x) ≤ -γ and |x + a(x)| m -|x| m ≤ 0.

Markovian framework

Our study relies on the general theory of Markov processes. To be able to work within a Markovian framework, we impose a special structure on the interaction functions and suppose that Assumption 2.8. For all

1 ≤ i, j ≤ M, h ij (t) = c ij e -α ij t , c ij ∈ R, α ij > 0.
(2.3)

In this case we may introduce the auxiliary Markov process

Y = Y (ij) , Y (ij) t = c ij t 0 e -α ij (t-s) dN (j) s + c ij 0 -∞ e -α ij (t-s) dn (j) s , 1 ≤ i, j ≤ M. (2.4)
The intensities can be expressed in terms of sums of these Markov processes, that is, for all

1 ≤ i ≤ M, λ (i) t = f i   M j=1 Y (ij) t-   .
Notice that

Y (ij) 0 = y (ij) 0 = c ij 0 -∞ e α ij s dn (j)
s , and dY

(ij) t = -α ij Y (ij) t dt + c ij dN (j)
t .

(2.5)

In particular, the process

(X t , Y (ij) t , 1 ≤ i, j ≤ M )
is a Markov process. Its longtime behavior is determined on the one hand by the longtime behavior of the underlying Hawkes process N t and on the other hand on the one of the continuous diffusion process with drift b and diffusion coefficient σ. Note also that N t is an autonomous process, that is, N t does not depend on X.

2.3 First properties of the process (X, Y ) and an associated stochastic flow.

In the sequel, we shall denote the whole process by Z

t := (X t , Y t ). It takes values in R × R M ×M .
We write (P t ) t≥0 for its associated transition semigroup and z = (x, y), y = (y (ij) ) 1≤i,j≤M for generic elements of its state space.

Proposition 2.9. Grant Assumptions 2.1 and 2.4. Then the process Z is a Feller process, that is,

P t Φ belongs to C b (R × R M ×M ) whenever Φ ∈ C b (R × R M ×M ).
The proof of this result follows from classical arguments, see e.g. the proof of Proposition 4.8 in [START_REF] Höpfner | Statistical models for Birth and Death on a Flow: Local absolute continuity and likelihood ratio processes[END_REF], or [START_REF] Ikeda | A construction of Markov processes by piecing out[END_REF]. In what follows we shall give an alternative proof relying on an explicit construction of the process Z as a stochastic flow. This construction is interesting in its own right and based on the fact that Z is a piecewise continuous Markov process, that is, a generalization of the piecewise deterministic Markov processes to those traveling in between successive jumps according to stochastic flows instead of deterministic ones, see [START_REF] Ikeda | A construction of Markov processes by piecing out[END_REF]. We start by giving the principal elements needed to construct this flow.

The associated stochastic Brownian flow. Thanks to our assumptions on b and σ, by Theorem 4.2.5 of [START_REF] Kunita | Stochastic flows and stochastic differential equations[END_REF], there exists a unique stochastic flow of homeomorphisms

Φ t (x) for 0 ≤ t < ∞, x ∈ R, such that Φ t (x) = x + t 0 b(Φ u (x))du + t 0 σ(Φ u (x))dW u .
(2.6)

In particular, we have that

(t, x) → Φ t (x) is continuous (0≤t<∞, x ∈ R).
The stochastic flow Φ describes the evolution of X in between successive jumps of X (or, equivalently, of N ).

The associated deterministic flow. In between successive jump events of N, the process Y evolves according to the deterministic flow

ϕ t (y) = (ϕ (ij) t (y)) 1≤i,j≤M , where ϕ (ij) t (y) = e -α ij t y (ij) , 1 ≤ i, j ≤ M.
(2.7)

We are now ready to give the simulation algorithm for Z.

Simulation algorithm for Z. We propose a simulation algorithm for Z for any family of starting configurations z ∈ K 1 × K 2 , where K 1 ⊂ R, K 2 ⊂ R M ×M are compact sets. The first step is to construct an upper bound Nt on the number of jumps of Z, that is, of N, during some finite time interval [0, t]. To do so, observe that by Lipschitz continuity of f i with Lipschitz constant γ i , we have for

1 ≤ i ≤ M , f i (x) ≤ f i (0) + γ i |x|. Let γ := max i γ i , c := sup 1≤i,j≤M |c ij |. (2.8)
Consider now the one-dimensional auxiliary linear Hawkes process N * t having intensity λ * t , t ≥ 0, where

λ * 0 := sup y∈K 2 sup t≥0 M i=1 f i ( M j=1 e -α ij t y (ij) )
and

λ * t = λ * 0 + M γc t- 0 dN * u = λ * 0 + M γcN * t-, t ≥ 0.
Observe that N * is a one dimensional linear Hawkes process with memory kernel

h(t) = M γc1 R + (t).
Obviously, for all t ≥ 0,

M i=1 N (i) t ≤ N * t .
Moreover, for all t ≥ 0, N * t < ∞ almost surely. In what follows, we shall write T * 1 < T * 2 < . . . < T * n < . . . for the successive jump times of N * .

We work conditionally on the realization of N * on [0, t], that is, on the event {N * t = n}, and on a realization of the associated jump times 0 < t 1 < t 2 . . . < t n < t. Our goal is to construct a version of Z, conditionally on these choices, which is continuous in the starting point z. This construction relies on the classical thinning method, also known as acceptance-rejection method.

During this construction, we choose successively random variables U 1 , . . . , U n taking values in {0, 1, . . . , M } and define a process z s = z s (z, t n 1 , U n 1 ), depending on these choices, for 0 ≤ s ≤ t. Here, t n 1 = (t 1 , . . . , t n ) denotes the choices of the successive jump times and U n 1 = (U 1 , . . . , U n ) the sequence of indices of jumping particles. This process is defined recursively as follows. Firstly, we put

z s = (x s , y s ) = (Φ s (x), ϕ s (y)) for all 0 ≤ s < t 1 .
Then, conditionally on y t 1 -= y 1 , we choose a random variable U 1 ∈ {0, 1, . . . , M } with law q(y 1 , t 1 , •), where

q(y, t, {i}) = f i ( j y (ij) ) λ * t , 1 ≤ i ≤ M, q(y, t, {0}) = 1 - M i=1
q(y, t, {i}).

(2.9)

U 1 gives the label of the particle (or component) that will jump at time t 1 , if U 1 = 0, no jump happens at this time. More precisely, on {U 1 ≥ 1}, we put

x t 1 := x 1 + a(x 1 ), y (ij)
t 1 := y (ij) 1 + c ij 1 {U 1 =j} , for all 1 ≤ i, j ≤ M, and
z t 1 := z 1 on {U 1 = 0}.
Then we put

x s = Φ s-t 1 (x t 1 ), y s = ϕ s-t 1 (y t 1 ) for all t 1 ≤ s < t 2 ,
and we proceed iteratively by choosing, conditionally on y t 2 -= y 2 , a random variable

U 2 ∼ q(y 2 , t 2 , •),
and so on. Finally, we obtain a terminal value x t = Φ t-tn (x tn ) and y t = ϕ t-tn (y tn ). It is easy to check that L(z t (z,

t n 1 , U n 1 )) = L(Z t |N * t = n, T * 1 = t 1 , . . . , T * n = t n ).
The important point is that the above construction ensures the continuity of the application

z → z t (z, t n 1 , U n 1 ).
Alternative proof of Proposition 2.9. Noticing that the law of N * does not depend on the starting point z but only on an upper bound of the associated intensities, the above construction implies that for any

Φ ∈ C b (R × R M ×M ), the mapping K z → P t Φ(z) ∈ R
is continuous, implying the assertion of Proposition 2.9. Indeed, we can write

P t Φ(z) = n≥0 P(N * t = n) [0,t] n L(T * 1 , . . . , T * n |N * t = n)(dt 1 , . . . , dt n ) n i=1 M u i =0 P(U 1 = u 1 , . . . , U n = u n |T 1 = t 1 , . . . , T n = t n , Z 0 = z)E W Φ(z t (z, u n 1 )),
where the last expectation E W is only taken with respect to the underlying Brownian motion W.

Suppose now that (z k ) k ⊂ K 1 × K 2 is a sequence of initial configurations converging to z as k → ∞.
Then, by the structure of (2.9),

P(U 1 = u 1 , . . . , U n = u n |T 1 = t 1 , . . . , T n = t n , Z 0 = z k ) → P(U 1 = u 1 , . . . , U n = u n |T 1 = t 1 , . . . , T n = t n , Z 0 = z),
as k → ∞, by continuity of f i , 1 ≤ i ≤ M, and of z → z s (z, t n 1 , U n 1 ), for all s ≤ t. Moreover, using dominated convergence, E W Φ(z t (z k , u n 1 )) → E W Φ(z t (z, u n 1 )), as k → ∞, implying that P t Φ(z k ) → P t Φ(z), as k → ∞, where we have used dominated convergence once more.

3 Ergodicity of the process (X, Y )

In this section, we start proving the positive Harris recurrence of Z t = (X t , Y t ) using the regeneration technique.

A Döblin type lower bound

The following theorem is the main result of this section. It states a local Döblin lower bound for the transition semigroup of the process, which is the main ingredient towards ergodicity of the process (Z t ).

Theorem 3.1. Grant Assumptions 2.4 and 2.8 and suppose moreover that f i (x) > 0 for all 1 ≤ i ≤ M, for all x ∈ R. Then there exists T > 0 such that for all z * = (x * , y * ) ∈ R × R M ×M and for all x * * ∈ R the following holds. There exist R > 0, open sets I 1 ⊂ R and I 2 ⊂ R M ×M with strictly positive Lebesgue measure such that

x * * ∈ I 1 , (3.1)
and a constant β ∈ (0, 1), depending on I 1 , I 2 , R and the coefficients of the system with

P T (z, dz ) ≥ β1 C (z)ν(dz ), (3.2)
where C = B R (z * ) is the (open) ball of radius R centred at z * , and where ν is the uniform probability measure on I 1 × I 2 .

Remark 3.2. 1. Together with a Lyapunov argument implying that the process Z comes back to the set C infinitely often, the above result will imply the ν-irreducibility of the sampled chain (Z kT ) k≥0 .

We want to stress that the above construction implies that for any given x * * ∈ R and some ε > 0 we can construct ν such that B ε (x * * ) lies in the support of the projection on the x-variable of ν. Of course, this property is related to the support properties of the underlying diffusion process, granted by Assumption 2.4. It will imply that the invariant density of X is bounded away from 0 on each compact set, see Proposition 3.7 below. 2. A lower bound of the type (3.2) is trivial if we are only interested in the transitions of the process X. Indeed, in this case it is sufficient to consider the event where no jump has appeared up to time T and to use known lower bounds on the transition densities of the diffusion part. However, since X is not Markov on its own, we do need to work with the couple of processes (X, Y ), and therefore have to establish such lower bounds for the joint transition of X and Y.

The main idea of the proof is to use the jump noise of M2 successive jumps of the underlying Hawkes process to create firstly a Lebesgue density for the process Y. Such ideas have already been exploited in [START_REF] Clinet | Statistical inference for ergodic point processes and application to limit order book[END_REF] and in [START_REF] Duarte | Stability, convergence to equilibrium and simulation of non-linear Hawkes processes with memory kernels given by the sum of Erlang kernels[END_REF]. The important second step is then to use density estimates of the underlying diffusion to prove that a joint Lebesgue density of (X T , Y T ) exists which is strictly lower-bounded on C.

The proof is done in several steps which are the subject of the next subsection.

Some useful properties of the underlying stochastic flow and proof of Theorem 3.1

We start by collecting useful properties of the stochastic flow governing the evolution of X in between successive jumps. Transition densities. Due to our assumptions on b and σ, the stochastic flow Φ t (x) given by Equation (2.6), possesses a transition density x → p t (x, y) with explicit lower bounds. More precisely, Proposition 1.2 of [START_REF] Gobet | Lan property for ergodic diffusions with discrete observations[END_REF] implies that, for some suitable constants c, C,

c -1 t -1/2 e -C(x-y) 2 /t e -Ctx 2 ≤ p t (x, y) ≤ ct -1/2 e -1 C (x-y) 2 /t e Ctx 2 . (3.3)
Here, the constants c and C do only depend on the coefficients b and σ. 2 Support properties. We will use the control theorem which goes back to [START_REF] Stroock | On the support of diffusion processes with applications to the strong maximum principle[END_REF] [START_REF] Stroock | On the support of diffusion processes with applications to the strong maximum principle[END_REF], see also Millet and Sanz-Sole (1994) [START_REF] Millet | A simple proof of the support theorem for diffusion processes[END_REF], Theorem 3.5. For some time horizon T 1 < ∞ which is arbitrary but fixed, write H for the Cameron-Martin space of measurable functions h : [0, T 1 ] → R having absolutely continuous components h(t) = t 0 ḣ(s)ds with

T 1 0 ( ḣ) 2 (s)ds < ∞. For x ∈ R and h ∈ H, consider the deterministic flow (ϕ (h) t (x)) t≥0 solution to dϕ (h) t (x) = b(ϕ (h) t (x))dt + σ(ϕ (h)t (x)) ḣ(t)dt, with ϕ (h) 0 (x) = x, (3.4
(x) = b(x) - 1 2 σ(x)σ (x).
Denote by Q T 1 x the law of the solution (Φ t (x)) 0≤t≤T 1 . Then for any x ∈ R and h ∈ H,

ϕ (h) t (x) |t∈[0,T 1 ] ∈ supp Q T 1 x . (3.5)
We are now able to give the proof of Theorem 3.1.

Proof. We suppose without loss of generality that for all 1 ≤ j ≤ M,

c ij = 0 for all i, α ij = α kj , for all 1 ≤ i, k ≤ M, i = k. 3
In what follows, we impose first M consecutive jumps of N (1) , followed by M jumps of N (2) etc up to M consecutive jumps of N (M ) , and we suppose that they all happen before time T. Then we can lower-bound, for all

A ∈ B(R), B ∈ B(R M ×M ), z ∈ R × R M ×M , P T (z, A × B) ≥ E z 1 A (X T )1 B (Y T ), N (i) T = M, ∀1 ≤ i ≤ M, T (1) 1 < T (1) 2 < . . . < T (1) M < T (2) 1 < . . . < T (2) M < . . . < T (M ) 1 < . . . T (M ) M < T ,
where

T (i) k , k ≥ 1, are the successive jump times of N (i) , 1 ≤ i ≤ M.
In what follows we shall write

A T := {N (i) T = M, ∀1 ≤ i ≤ M, T (1)
1 < T (1) 2 < . . . < T (1)
M < T (2) 1 < . . . < T (2) M < . . . < T (M ) 1 < . . . T (M ) M < T }.
Step 1. We first deal with the process Y t . This part of the proof follows well-known arguments that have been developed independently from our work in [START_REF] Clinet | Statistical inference for ergodic point processes and application to limit order book[END_REF], and that we have also already used in [START_REF] Duarte | Stability, convergence to equilibrium and simulation of non-linear Hawkes processes with memory kernels given by the sum of Erlang kernels[END_REF] and [START_REF] Löcherbach | Absolute continuity of the invariant measure in piecewise deterministic Markov Processes having degenerate jumps[END_REF]. We reproduce the main arguments here since the second step will be a control of the diffusion part conditionally on the results of this first step.

Recall that in between successive jump events of N, the process Y evolves according to the deterministic flow ϕ t (y) introduced in (2.7) above. Thus, on the event A T , starting from Y 0 = y ∈ R M ×M , we first let the flow ϕ evolve starting from y up to some first jump time t 1 . At that jump time each particle having index (i1) gains an additional value c i1 . We then successively choose the following inter-jump waiting times t 2 , . . . , t M 2 under the constraint t 1 + . . . + t M 2 < T. We write

s 1 = T -t 1 , s 2 = T -(t 1 + t 2 ), . . . , s M 2 = T -(t 1 + . . . + t M 2 ).
Conditionally on Y 0 = y, the successive choices of s = (s 1 , . . . , s 2 M ) as above, the position of

Y (ij) T is given by γ (ij) (y, s) = e -α ij T y (ij) + c ij [e -α ij s jM -M +1 + . . . + e -α ij s jM ].
In what follows we work at fixed y and we write γ y : s → γ(y, s) = (γ (11) , . . . , γ (M 1) , γ (12) , . . . , γ (M 2) , . . . , γ (1M ) , . . . , γ (M M ) )(y, s).

We will use the jump noise which is created by the M 2 jumps, i.e., we will use a change of variables on the account of s 1 , . be the Jacobian matrix of the the map s → γ y (s). This matrix does not depend on the initial position y. Indeed, one easily finds that

∂γ y (s) ∂s = C(s) =      C (1) 0 • • • 0 0 0 C (2) 0 • • • 0 . . . . . . . . . . . . • • • 0 • • • 0 0 C (M )      (s),
where for each 1 ≤ j ≤ M , C (j) (s) is the M × M matrix given by

C (j) (s) = -      c 1j (α 1j ) -1 e -α 1j s jM -M +1 . . . c 1j (α 1j ) -1 e -α 1j s jM c 2j (α 2j ) -1 e -α 2j s jM -M +1 . . . c 2j (α 2j ) -1 e -α 2j s jM . . . . . . . . . c M j (α M j ) -1 e -α M j s jM -M +1 . . . c M j (α M j ) -1 e -α 2j s jM     
.

We have to prove that each C (j) (s) is invertible. This is difficult for general choices of s. But if we choose t 0 such that M 2 t 0 < T and then

s * 1 = M 2 t 0 , s * 2 = (M 2 -1)t 0 , . . . , s * M 2 = t 0 such that s k -s k+1 = t 0 for all 1 ≤ k < M 2 , then det C (j) (s * ) = M i=1 c ij (α ij ) -1 e -α ij s * jM det V (j) (t 0 ), where V (j) (t 0 ) =     
e -(M -1)α 1j t 0 . . . e -2α 1j t 0 e -α 1j t 0 1 e -(M -1)α 2j t 0 . . e -2α 2j t 0 e -α 2j t 0 1 . . . . . . . . . . . . . . .

e -(M -1)α M j t 0 . . . e -2α M j t 0 e -α M j t 0 1

     .
Therefore, det V (j) (t 0 ) is a Vandermonde determinant which is different from 0 since by assumption all e -α ij t 0 and e -α kj t 0 are different, for all i = k.

By continuity we therefore have that for any choice of t 0 < T /M 2 there exists ε > 0 such that for all s ∈ B ε (s * ), ∂γy(s) ∂s is invertible. It will be proved now that this uniform invertibility of the Jacobian matrix of the map s → γ y (s) implies the first part of inequality (3.2). For that sake, we shall also need the following notation. For each couple (y, s), we write y 0 = y, and then define recursively for all jM -M + 1 ≤ k ≤ jM, where j varies between 1 ≤ j ≤ M, y

s k-1 -s k (y k-1 ) + c ij 1 { =j} , (i ) k = ϕ (i )
for all 1 ≤ i, ≤ M, where we put s 0 = T. The sequence y 1 , . . . , y M 2 corresponds to the positions of the process Y right after the successive jumps, starting from the initial location y ∈ R M ×M , induced by the inter-jump waiting times Ts 1 , s 1 -s 2 , . . . , s M 2 -1 -s M 2 which are determined by s.

Introduce now for each y ∈ R M ×M the total jump rate

f (y) := M i=1 f i   j y (ij)  
and for each t ≥ 0, the survival rate e(y, t) = exp -

t 0 f ϕ s (y) ds . (3.7)
We define for each couple (y, s) (recall that s 0 = T ),

q y (s) =   M j=1 jM -1 k=jM -M f j (ϕ s k -s k+1 (y k )) e(y k , s k -s k+1 )   e(y M 2 , s M 2 ). (3.8)
Since f (y) > 0 for all y ∈ R M ×M and from the definition of e(•, •), we deduce that for any couple (y * , s * ) there are neighborhoods W s * and U y * of s * and y * respectively such that inf (y,s)∈U y * ×W s * q y (s) > 0.

(3.9)

Let us now fix (y * , s * ) such that the matrix

W y → I 2 s → γ y (s)
is a diffeomorphism, with W y ⊂ W s * , and also

inf y∈B R (y * ) inf s∈Wy det ∂γ y (s) ∂s -1 > 0. (3.10)
Reducing (if necessary) R, we may assume also that B R (y * ) ⊂ U y * . Thus we have that by (3.9) and (3.10),

β 1 := inf y∈B R (y * ) inf s∈Wy q y (s) det ∂γ y (s) ∂s -1 > 0. (3.11)
Once we have done all these steps we can conclude with the following preliminary result. For any measurable B ∈ B(R M ×M ) and for any z = (x, y) such that y ∈ B R (y * ), using the change of variables y = γy (s), we obtain the lower bound

E y 1 B (Y T ), N (i)
T = M, ∀1 ≤ i ≤ M, T (1)
1 < . . . < T

(1)

M < T (2) 1 < . . . < T (M ) M < T ≥ Wy q y (s)1 B (γ y (s))ds 1 . . . ds M 2 ≥ β 1 I 2 ∩B dy 1 . . . dy M 2 . (3.12)
This would establish the desired result if we were only interested in the autonomous process Y.

Step 2. We now deal with the process X. Of course, we still work conditionally on the choice of s 1 , . . . , s M 2 of the first step. Analogously to (3.6) we therefore introduce the successive jump positions of the process X which are given by x 0 = x and then for all

1 ≤ k ≤ M 2 , x k = Φ s k-1 -s k (x k-1 ) + a(Φ s k-1 -s k (x k-1 )), (3.13)
where s 0 = T as before.

Conditionally on X 0 = x, Y 0 = y, the successive choices of s = (s 1 , . . . , s M 2 ) as above, the position of X T is then given by

Γ(z, s) = Φ s M 2 (x M 2 ),
where z = (x, y). Notice that x M 2 depends on the choices of s 1 , . . . , s M 2 and of course on the evolution of the stochastic flow between the successive jump times.

Step 3. Therefore, conditioning with respect to X T -s M 2 = x M 2 , we obtain

P T (z, A × B) ≥ E z (1 A (X T )1 B (Y T ), A T ) x ≥ Wy q y (s)1 B (γ y (s)) E A p s M 2 (x M 2 , u)du ds 1 . . . ds M 2 , (3.14)
where p s M 2 (x M 2 , u) is the transition density of (3.3). Notice that in the last line, expectation E is taken with respect to Brownian motion only, that is, with respect to the law of X T -s M 2 under the Wiener measure, conditionally on the choices s 1 , . . . , s M 2 .

The lower bound estimate given in (3.3) implies that for fixed s M 2 and x M 2 , u → p s M 2 (x M 2 , u) is lower bounded in small neighborhoods of x M 2 . Therefore, in what follows we need to localize the possible positions of x M 2 , conditionally on the choices of the jumps times s 1 , . . . , s M 2 .

Step 4. Localization of x M 2 .

We apply the support theorem, that is, (3.5), to our process X t in between the successive jump times 0, T 1 := t 1 , T 2 := t 1 +t 2 , . . . , T M 2 := t 1 +. . . t M 2 , by choosing on each time interval [T n , T n+1 [, n = 0, . . . , M 2 -1, the control function h ≡ 0. Consequently, introducing

x1 := ϕ (0) t 1 (x) + a(ϕ (0) t 1 (x)), . . . , xM 2 := ϕ (0) s M 2 -1 -s M 2 (x M 2 -1 ) + a(ϕ (0) s M 2 -1 -s M 2 (x M 2 -1 )),
there exists an open neighborhood U xM of xM 2 , such that

P(X T -s M 2 ∈ U xM 2 ) > 0.
Notice that xM 2 is a continuous function of the starting point x and of s; that is, xM 2 = F (x, s) for some continuous function F. This implies that for any starting point x * and for R > 0 sufficiently small, reducing W s * if necessary, there exists a compact K = K(x * , s * ) such that F (x, s) ∈ K for all x ∈ B R (x * ) and for all s ∈ W s * , whence

inf x∈B R (x * ) inf s∈W s * P(X T -s M 2 ∈ J 1 ) > 0,
where

J 1 = xM 2 ∈K U xM 2 .
Notice that J 1 has compact closure. Let now x * * ∈ R be arbitrarily chosen. The lower bound of (3.3) implies that there exists an open interval I 1 x * * , such that inf

x∈J 1 ,y∈I 1 ,s∈W s * p s M 2 (x, y) > 0.
Therefore,

β 2 := inf x∈B R (x * ) inf y∈B R (y * ) inf s∈Wy inf u∈J 1 ,v∈I 1 p s M 2 (u, v)P x (X T -s M ∈ J 1 ) > 0. (3.15)
Therefore, coming back to (3.14), for all z ∈ B R (x * , y * ),

P T (z, A × B) ≥ E z 1 A (X T )1 B (Y T ), N (i)
T = M, ∀1 ≤ i ≤ M, T (1)
1 < T (1) 2 < . . . < T (M ) M < T ≥ Wy q y (s)1 B (γ y (s)) E 1 {X T -s M 2 ∈J 1 } A p s M 2 (X T -s M 2 , u)du ds 1 . . . ds M 2 ≥ Wy q y (s)1 B (γ y (s)) E 1 {X T -s M 2 ∈J 1 } A∩I 1 inf x∈J 1 p s M 2 (x, u)du ds 1 . . . ds M 2 ≥ β 2 λ(A ∩ I 1 ) Wy q y (s)1 B (γ y (s))ds 1 . . . ds M 2 ≥ β 2 λ(A ∩ I 1 )β 1 λ M ×M (B ∩ I 2 ),
where c 1 , c 2 > 0 and where K 1 ⊂ R M ×M is some compact. Moreover, writing for short f (y) = i f i ( j y (ij) ) for the total jump rate,

A Z 1 V (x, y) = 2xb(x) + σ 2 (x) + (2xa(x) + a(x) 2 ) f (y). (3.19)
Thanks to Assumption 2.7 and bound of σ 2 from Assumption 2.4,

2xb(x) + σ 2 (x) ≤ -c 3 x 2 + c 4 1 K 2 (x), (3.20)
where K 2 ⊂ R is a compact set. Now, suppose that Assumption 2.6 is satisfied. Then for all |x| > r, (2xa(x) + a(x) 2 ) f (y) ≤ 0 (recall that all f is non-negative) implying that

A Z 1 V (x, y) ≤ -c 3 x 2 + c 4 1 K 2 (x) + c 5 f (y). This implies (3.17) since f (y) ≤ c 6 + c 7 i,j |y (ij) | ≤ c 6 + c7 log(V 2 (y))
which is thus negligible with respect to the negative term -c 1 V 2 (y) of (3.18).

If only Assumption 2.7 holds, then we upper bound the jump part of (3.19) by

(2xa(x) + a(x) 2 ) f (y) ≤ C f (y) + C|x| 1+η f (y).
Choose 1 < p < 2 and q > 2 such that (1 + η)p < 2 and 1 p + 1 q = 1. Then

|x| 1+η f (y) ≤ 1 p |x| p(1+η) + 1 q f q (y).
Since (1 + η)p < 2, the first term

C 1 p |x| (1+η)p
will again be negligible with respect to the negative term -c 3 x 2 of (3.20). Finally, the second term C f (y) + C 1 q f q (y) is treated as previously. This concludes our proof.

A Z V ≤ d 1 -d 2 V 1-α . (3.21)
Proof. Using the same arguments as in the proof of Proposition 3.4, we have that

A Z 2 V (x, y) = A Z 2 V 2 (y) ≤ -c 1 V 2 (y) + c 2 1 K 1 (y). Moreover, for all |x| ≥ r, since |x + a(x)| m -|x| m ≤ 0, A Z 1 V (x, y) = b(x)V 1 (x) + 1 2 σ 2 (x)V 1 (x) + M j=1 f j M k=1 y (jk) [|x + a(x)| m -|x| m ] ≤ b(x)V 1 (x) + 1 2 σ 2 (x)V 1 (x).
A standard calculus, see e.g. [START_REF] Yu | On polynomial mixing bounds for stochastic differential equations[END_REF] or [START_REF] Löcherbach | Polynomial deviation bounds for recurrent harris processes having general state space[END_REF], shows that, for suitable constants e 1 , e 2 > 0,

b(x)V 1 (x) + 1 2 σ 2 (x)V 1 (x) ≤ e 1 -e 2 V 1 (x) 1-α . Using that V 1 (x) ≥ 1 such that (V 1 (x) + V 2 (y)) 1-α ≤ V 1 (x) 1-α + (1 -α)V 2 (y) ≤ V 1 (x) 1-α + V 2 (
y), we deduce from this the drift condition (3.21) as in the proof of Proposition 3.4.

Harris recurrence of Z

We do now possess all ingredients to obtain our main results.

Theorem 3.6. Grant Assumptions 2.4, 2.6, 2.8 and 3.3 and suppose that for all 1 ≤ i ≤ M, f i (x) > 0 for all x ∈ R. Then (Z t ) t≥0 is positive Harris recurrent with unique invariant measure π. In particular, for any starting point z and any positive measurable function g : R × R

M ×M → R + , as T → ∞, P z -almost surely, 1 T T 0 g(Z s )ds → π(g).
Proof of Theorem 3.6. 1) We fix any x * ∈ R and we wish to apply Theorem 3.1 with y * = 0 and x * = x * * . Let R be the associated radius. By Proposition 3.4 or Proposition 3.5, we know that for a suitable compact set

K = K 1 × K 2 , with K 1 ⊂ R, K 2 ⊂ R M ×M , Z comes back to K infinitely often almost surely. Moreover, sup y∈K 2 ,t≥0 ϕ t (y) 1 := F < ∞ and sup y∈K 2 ϕ t (y) 1 → 0
as t → ∞, by the explicit form of the flow in (2.7). Therefore there exists t * such that ϕ t (y) ∈ B R (0) for all t ≥ t * , for all y ∈ K 2 .

Applying once more the support theorem for diffusions and observing that σ is strictly positive, Equation (3.5) implies that inf

x∈K 1 P(Φ t * +s (x) ∈ B R (x * ), 0 ≤ s ≤ 2T ) > 0 and thus inf z∈K P z (X t * +s ∈ B R (x * ), Y t * +s ∈ B R (0), 0 ≤ s ≤ 2T ) > 0.
Consequently, using a conditional version of the Borel-Cantelli lemma, the sampled Markov chain (Z kT ) k∈N visits B R (x * , 0) infinitely often almost surely.

2) The standard regeneration technique (see e.g. [START_REF] Löcherbach | On Nummelin splitting for continuous time Harris recurrent Markov processes and application to kernel estimation for multi-dimensional diffusions[END_REF]) allows to conclude that (Z kT ) k∈N and therefore (Z t ) t are Harris recurrent. This concludes the proof.

The following by-product of the above result will prove to be useful when dealing with statistical inference within this new model class. Proposition 3.7. Grant all assumptions of Theorem 3.6 and write π X for the projection of the invariant measure π onto the X-coordinate, that is, π X (dx) = R M ×M π(dx, dy). Then π X possesses a Lebesgue density which is bounded away from zero on each compact of R.

Proof. Let A ∈ B(R). Then for any t > 0, π X (A) = R×R M ×M π(dz)E z [1 A (X t )]. (3.22)
Let L t := sup{s ≤ t : ∃j : ∆N (j) s = 1} be the last jump time of the process before time t, L t = 0 if there is no such jump. Then by Fubini,

E z [1 A (X t )] = A E z [p t-Lt (X Lt , y)]dy,
where p t (x, y) is the transition density of (3.3). This implies the existence of the density π X (x) which is given by π

X (x) = R×R M ×M π(dz)E z [p t-Lt (X Lt , x)]
for any t > 0.

Notice that we do not dispose of any regularity results of π X (x) with respect to x. Indeed, the upper bound in (3.3) does not allow to conclude that the almost sure continuity in x of p t-Lt (X Lt , x) survives the integration π(dz)E z (. . .).

We are however able to prove that π X is lower bounded on compacts K ⊂ R. For that sake, fix any x * * ∈ K and apply (3.2) to (x * , y * ) ∈ supp(π) such that π(B R (z * )) = π(C) > 0 and to x * * . Then for any measurable A ⊂ I 1 = I 1 (x * * ), applying the lower bound of (3.2) to (3.22) and taking t = T,

A π X (x)dx ≥ βπ(C) 1 λ(I 1 ) A dx implying that inf x∈I 1 π X (x) ≥ βπ(C) 1 λ(I 1 ) > 0.
Therefore we have just shown that for all x * * ∈ K, there exists an open interval I 1 = I 1 (x * * ) containing x * * ∈ I 1 such that π X is strictly lower bounded on I 1 . Since we can cover the compact K by a finite collection of such open intervals I 1 (x * * ), this implies the desired lower bound of π X on compacts.

In the sequel, following [START_REF] Meyn | Stability of Markovian processes III : Foster-Lyapunov criteria for continuous-time processes[END_REF], in any of the two frames (Assumption 2.6 or 2.7) and for the choice of V as in Proposition 3.4 or Proposition 3.5, we introduce µ V := sup It is now straightforward to obtain our second main result. Theorem 3.8 (Ergodicity). Grant all assumptions of Theorem 3.6. Then there exist c 1 , c 2 > 0 such that for all z ∈ R × R M ×M , under Assumption 2.6,

P t (z, •) -π V ≤ c 1 V (z)e -c 2 t , (3.23)
and under Assumption 2.7,

P t (z, •) -π T V ≤ c 1 V (z)t 1 α -1 , (3.24)
where α is as in Proposition 3.5.

Proof. The sampled chain (Z kT ) k≥0 is Feller according to Proposition 2.9. Moreover it is ν-irreducible, where ν is the measure introduced in Theorem 3.1, associated with the point (x * , 0) and x * * , for any choice of x * , x * * ∈ R, used in the proof of Theorem 3.6. Since ν is the uniform measure on some open set of strictly positive Lebesgue measure, the support of ν has non-empty interior. Theorem 3.4 of [START_REF] Meyn | Stability of Markovian processes I : Criteria for discrete-time chains[END_REF] implies that all compact sets are 'petite' sets of the sampled chain. Under Condition 2.6, the Lyapunov condition established in Proposition 3.4 allows to apply Theorem 6.1 of [START_REF] Meyn | Stability of Markovian processes III : Foster-Lyapunov criteria for continuous-time processes[END_REF] which implies the assertion in the exponential frame. In the polynomial frame, the assertion follows from Theorem 3.2 of [START_REF] Douc | Subgeometric rates of convergence of f-ergodic strong markov processes[END_REF] or from Theorems 1.19 and 1.23 in [START_REF] Michajlovič | Introduction to Ergodic rates for Markov chains and processes with applications to limit theorems[END_REF].

Remark 3.9. In the above Theorem 3.8, we do only treat two cases for the rate of convergence to equilibrium: the exponential and the polynomial one. Using slightly different Lyapunov functions, we could also deal with more general sub-exponential rates of convergence, as they have for example been considered in Theorem 1.20 in [START_REF] Michajlovič | Introduction to Ergodic rates for Markov chains and processes with applications to limit theorems[END_REF]. We do not detail these calculations here since we are mostly interested in exponential rates of convergence. Finally, notice that it is also possible to interpolate between the total variation norm considered in (3.24) and the weighted total variation norm • V considered in (3.23), however at the cost of slower rates of convergence. Details can be found in the very instructive paper [START_REF] Douc | Subgeometric rates of convergence of f-ergodic strong markov processes[END_REF].

3.5 Exponential β-mixing for Z = (X, Y ).

It is now easy to deduce from the above results the exponentially β-mixing property of the process, under Condition 2.6. Recall that the β-mixing coefficient of Z is given by β Z (t) = sup denotes the total variation distance. Notice that if µ = π, then the process is in its stationary regime, and β Z (t) = P t (z, •) -π T V π(dz).

Theorem 3.10. Grant all assumptions of Theorem 3.6 and suppose that Condition 2.6 holds. Then Z is exponentially β-mixing, that is, there exist constants K, θ > 0 such that for any initial law µ with µ(V ) < ∞, β Z (t) ≤ Ke -θt .

Proof. Suppose firstly that µ = π. Then Theorem 4.3 of [START_REF] Meyn | Stability of Markovian processes III : Foster-Lyapunov criteria for continuous-time processes[END_REF] implies that V dπ < ∞ such that we are able to integrate (3.23) against π(dx) to obtain β Z (t) ≤ c 1 π( V )e -c 2 t .

Putting K := c 1 and θ = c 2 , this implies the result in this case.

In order to deal with the general non-stationary process, we apply Lemma 3.9 of [START_REF] Masuda | Ergodicity and exponential β-mixing bounds for multidimensional diffusions with jumps[END_REF] with h = V , δ(t) = c 1 e -c 2 t and κ = sup s≥0 E( V (Z s )),

to deduce that β Z (t) ≤ 2c 1 κe -c 2 t .

Putting K := 2c 1 κ and θ = c 2 , this implies the result, if we have already shown that κ is finite. This last fact follows immediately from (3.17), following the first lines of the proof of Theorem 6.1 of [START_REF] Meyn | Stability of Markovian processes III : Foster-Lyapunov criteria for continuous-time processes[END_REF]. Indeed, we have by Dynkin's formula that

e αt E z (V (Z t )) ≤ V (z) + β α e αt ,
implying that E z (V (Z t )) ≤ e -αt V (z) + β α .

Integrating this last inequality with respect to µ(dz) implies the result.

  ) on [0, T 1 ]. In the above formula (3.4), b is Stratonovich drift given by b

  ∂γ y * (s * ) ∂s is invertible. By Lemma 6.2 of [4], there exist an open neighborhood B R (y * ) of y * , an open set I 2 ⊂ R M ×M , and for any y ∈ B R (y * ), an open set W y such that γy (s) :

Proposition 3 . 5 .

 35 Grant Assumptions 2.4, 2.7 (Polynomial frame) and 3.3. Let V 1 (x) = 1 + |x| m and α = 2/m ∈]0, 1[. Then there exist positive constants d 1 , d 2 such that the following Foster-Lyapunov type drift condition holds

s≥0P

  t (z, •) -µP s+t (•) T V µP s (dz),where µ = L(Z 0 ) is the law of the initial configuration and whereµ T V := sup g:|g|≤1 µ(g)

  . . , s M 2 . Therefore, let

	∂γ y (s) ∂s	=	∂γ y (s) ∂s 1	, . . . ,	∂γ y (s) ∂s M 2

3

Otherwise, if αij = α kj , we introduce the new process Y (i+k),j t

:= Y ij t + Y kj t

which is Markov again.

Notice that[START_REF] Chen | The multivariate Hawkes process in high dimensions: Beyond mutual excitation[END_REF] has obtained a control of the τ -mixing coefficient in the inhibitory case -but this does not imply the β-mixing property of the process.

It is possible to replace our Condition 2.4 by any other condition ensuring that pt(x, y) is strictly lower bounded on compact sets for any t > 0 .

where we have finally applied (3.12), and where λ and λ M ×M denote the Lebesgue measure on R, R M ×M , respectively. This implies the desired result putting β := β 1 β 2 λ(I 1 )λ M ×M (I 2 ) and ν := U I 1 ×I 2 , the uniform probability law on I 1 × I 2 .

A Foster-Lyapunov type condition

In order to prove the positive Harris recurrence of the process Z, we need of course a stability condition which is a Lyapunov type condition. Notice that the process Z t = (X t , Y t ) has the following extended generator, defined for sufficiently smooth test functions g by

with

To obtain stability we first introduce the following classical stability assumption for Hawkes processes. Recall that γ i denotes the Lipschitz constant of the rate function f i .

Under the above stability condition, let κ ∈ R M + be a left eigenvector of H, associated to the eigenvalue ρ and having non-negative components, that is, for all j, i κ i H ij = ρκ j ≥ 0. Such a vector κ exists thanks to the theorem of Perron-Frobenius. Following [START_REF] Clinet | Statistical inference for ergodic point processes and application to limit order book[END_REF], we introduce a Lyapunov function by defining m ij = κ i α ij for all 1 ≤ i, j ≤ M and V : R × R M ×M → R + by

where V 1 : R → R + will be chosen in the sequel. Notice that V (x, y) ≥ 1 for all x, y.

Proposition 3.4. Grant Assumptions 2.4, 2.6 (Exponential frame) and 3.

Then there exist positive constants d 1 , d 2 such that the following Foster-Lyapunov type drift condition holds

is the diffusion part and A Z 2 V = A Z V -A Z 1 V is the jump part of the generator. The arguments of the proof of Proposition 4.5 of [START_REF] Clinet | Statistical inference for ergodic point processes and application to limit order book[END_REF] imply that


Evangelia Adamou

Background

Linguists are increasingly paying attention to the ways bilinguals may alternate between two languages in one setting and with a single interlocutor, a phenomenon referred to as code-switching. Research in contact linguistics has shed light on the effects of code-switching on the linguistic structures of each of the two languages in the long run. It has been shown, for example, that in some extreme cases code-switching may lead to radical changes through the creation of mixed languages (see [START_REF] Mcconvell | Gurindji Kriol: A mixed language emerges from code-switching[END_REF]; O'Shannessy 2012). Torres Cacoullos and Travis (2018) present strong evidence situated on the other end of the spectrum, supporting the view that despite frequent code-switching the two grammars remain intact. These novel data are of particular interest in the context of the most recent investigations on the cognitive effects of bilingualism (see Jarvis [START_REF] Pavlenko | Cross-linguistic influence in language and cognition[END_REF][START_REF] Kroll | Bilingualism, mind, and brain[END_REF][START_REF] Schmid | The relevance of first language attrition to theories of bilingual development. Epistemological keynote paper with commentaries[END_REF] Köpke 2017 for an overview). More specifically, current research highlights the fact that transfer is a dynamic and bidirectional process, constantly taking place at the level of the individual speaker. Even though it starts as an online, transient process, it can eventually modify underlying representations in a more permanent manner and lead to change.

The focus in Bilingualism in the community. Code-switching and grammars in contact is on the relatively homogenous bilingual networks comprising Spanish-English speakers who reside in various traditionally bilingual communities of New Mexico in the United States of America, both in rural areas and in cities such as Albuquerque, Santa Fe, Los Lunas, Taos, and Las Vegas. As the authors argue, the New Mexican communities differ from recent migrant communities in that the use of the two languages for several generations is maintained whereas among migrants shift to the dominant language is rapidly taking place within just a few generations [START_REF] Poplack | Sometimes I'll start a sentence in Spanish y termino en espanol[END_REF][START_REF] Silva-Corvalán | Bilingualism and language change: The extension of estar in Los Angeles Spanish[END_REF][START_REF] Backus | Patterns of language-mixing. A study in Turkish-Dutch bilingualism[END_REF]. Although shift to English is also widespread among the New Mexican populations under study, New Mexicans were most likely bilinguals in English and in Spanish over the past 150 years, long enough for contact-induced changes to occur at the level of the community.

From a methodological point of view, the present study is based on the quantitative analysis of an original spoken corpus. Although corpus-driven research has become a very dynamic paradigm in contact linguistics with the development of computerized corpora, Torres Cacoullos and Travis (henceforth TC & T) follow the long established corpus-driven methods of the variationist framework, pioneered among others by Labov (e.g., 1971Labov (e.g., , 1984)). More generally, linguists interested in bilingualism and its short-and long-term effects have often compiled or relied on existing spoken corpora in order to document the most ordinary way of speaking (see [START_REF] Poplack | Sometimes I'll start a sentence in Spanish y termino en espanol[END_REF][START_REF] Myers-Scotton | Duelling Languages: Grammatical Structure in Codeswitching[END_REF][START_REF] Silva-Corvalán | Language contact and change: Spanish in Los Angeles[END_REF][START_REF] Gardner-Chloros | Code-switching[END_REF][START_REF] Herring | I saw the madre': evaluating predictions about codeswitched determiner-noun sequences using Spanish-English and Welsh-English data[END_REF]. TC & T argue that the analysis of natural, ecologically-valid data offers the best possible evidence to understand bilingualism, while pointing to the shortcomings of other methods, such as labbased research using experimental data, intuitive judgments, introspection or anecdotal evidence collected in the field.

TC & T are prominent researchers in the field of variationist linguistics. In this perspective, patterns in language use are examined by paying close attention to factors such as frequency and priming [START_REF] Poplack | Sometimes I'll start a sentence in Spanish y termino en espanol[END_REF][START_REF] Poplack | Myths and facts about loanword development[END_REF][START_REF] Torres Cacoullos | Two languages, one effect: structural priming in spontaneous code-switching[END_REF]. In addition, TC & T take into consideration the social factors that might be shaping code-switching and language change (also see Stell & Yakpo 2015). Variationist as well as usage-based approaches contrast with top-down approaches that aim to test the validity of theoretical models, such as the Matrix Language Frame (MLF) model (Myers-Scotton & Jake 2017) or the Minimalist approach to code-switching (MacSwan 2016).

In section 2, I start by presenting the outline of the book. In section 3, I discuss methodological issues and, in Section 4, I focus on the main topic of the book, convergence or absence of it, in language contact.

Outline of the book

Bilingualism in the community. Code-switching and grammars in contact is organized into 11 chapters. In addition, it includes a list of figures and tables, two appendices, bibliographical references and an index.

Chapter 1 is the introductory chapter, entitled "Language contact through the lens of variation" (pp. 1-12). TC & T challenge previous research in which "convergence has often been hastily diagnosed on the basis of perceived departures from an idealized monolingual norm" (p. 3). Against this shortcoming, TC & T propose that linguists should always base their claim about contactinduced change following comparison between the bilingual speech and the monolingual equivalents (see section 1.1, "Benchmarks and comparisons"). The present study relies on an original bilingual corpus of 29 hours of transcribed recordings, or approximately 300,000 words. The authors then systematically compare the bilingual productions with various monolingual benchmarks. In addition, they remind their readers that variation does not always lead to language change, but is an inherent feature of both bilingual and monolingual speech. In section 1.2, TC & T invoke the importance of probabilistic, quantitative methods and criticize research that relies solely on overall rates, even when such rates reach statistical significance.

Chapter 2, "The community basis of bilingual phenomena" (pp. 13-34), starts with a presentation of the contact site, New Mexico, where Spanishspeaking populations were settled as early as the sixteenth century, and of the Spanish contact variety, which is close to Mexican Spanish. Following this introduction, TC & T explain how they constituted "a principled speaker sample" (pp. 24-26), comprising 40 participants, most of them being over 60 and residing in rural localities, with a variety of occupations. They conclude that this is the most appropriate sample to examine the outcomes of language contact, which is "not afforded by studies of immigrant communities or studies without community footing, such as those based on assorted university students" (p. 34). One notes nonetheless that this is not a representative sample of a single "community", but of speakers from different communities.

Chapter 3, "Good data: capturing language use" (pp. focuses on the quality of the dataset, which according to TC & T needs to rely on spontaneous speech recorded among individuals who reside in communities with similar profiles. It is argued that "bilingual patterns cannot be detected with speaker samples consisting of university students at hand nor with corpora amalgamating texts culled from a range of dialects and contact sites" (p. 39).

Following the variationist methodological principles, in-group community members who were all students at the University of New Mexico recorded the data through sociolinguistic interviews. Transcription methods are presented in section 3.3, opting for an orthographic transcription. Community members provide the first draft that the team verifies through a minimum of five rounds of revisions. Access to the data is restricted to researchers "familiar with the speech community" (p. 48). In sections 3.4, 3.5, and 3.6, authors motivate the use of a prosodically-based transcription into intonation units (based on the annotator's perception using the software ELAN) and stress its significance for the study of code-switching and syntax. Interestingly, a first analysis of the data shows that the effect of alignment to the interlocutor's language choice is not borne out in the New Mexican corpus, that is, speakers do not switch from one language to the other following their interlocutor's switching. Put differently, language choice does not take place at the level of turn taking, but at the level of the conversation.

In Chapter 4, entitled "Characterizing the bilingual speaker" (pp. 57-73), TC & T use three types of data: self-reports based on a questionnaire, content analysis of the sociolinguistic interviews, and production data. Particularly informative is the comparison that shows absence of a correlation between scores of preferred language and self-rating (collected through questionnaires) with rates of subject pronouns in production. In Section 4.3, the authors consider that it is "misleading to transpose into the setting of the bilingual community constructs that are applied to university students learning a second language or individual nuclear families moving to a new country" (p. 63). They claim, in particular, that dichotomies such as L1 vs. L2, simultaneous vs. sequential bilingualism, early vs. late do not account for the complex reality of bilingualism as apparent in the New Mexican data. In my view, the authors correctly draw attention to methodological difficulties and the increased attention required in complex bilingual settings. However, I think that, quite on the contrary, the above-mentioned dichotomies are most relevant as they allow for distinguishing between speaker profiles that differ maximally, for example, New Mexicans with two L1s, acquired early and simultaneously, and bilinguals who acquired an L2 later in life, for example in the classroom or following migration at some point in their lives. In section 4.4, TC & T present the results of an analysis of the corpus based on the language of the verb. This analysis reveals an even distribution of the clauses between the two languages, which leads TC & T to conclude that code-switching in New Mexico is a discourse mode, a type of "intra-situational code-switching" (p. 68), which is not locally motivated by the change of the interlocutor, situation, or topic as code-switching is. Though TC & T do not use this term, I suggest that this discourse mode is a good example of "code-mixing" in the continuum proposed by [START_REF] Auer | From code-switching via language mixing to fused lects: Toward a dynamic typology of bilingual speech[END_REF]. This would allow us to understand the differences between the New Mexican (code-mixing) data and other classic codeswitching data reported in the literature where one of the two languages may be numerically dominant, as well as differences with other more advanced outcomes on the continuum, such as "fused-lects". In section 4.5, TC & T test a novel method to establish language dominance for individual speakers as an alternative to experimental psycholinguistic measures by exploiting the rates of use of Spanish and English clauses in their corpus (i.e., coding of the language of the clause based on the language of the finite verb). Results indicate that the frequency of use of the finite verb in English or Spanish does not correlate with self-report of language preference or self-rating of language ability, two measures that TC & T criticize for providing "a superficial view of linguistic experience for members of this bilingual community" (p. 71).

TC & T then choose to focus on subject pronoun expression for detailed examination. A variety of reasons motivates this choice. On the one hand, this choice is particularly relevant for the languages under study: subject pronoun expression is a well-investigated topic in both English and Spanish and is a potential candidate for grammatical convergence as the two languages exhibit an "overlap in their person-number categories and deictic meaning" (p. 111). On the other hand, this choice is driven by the contradictory theoretical claims that can be found in the literature (p. 136). For [START_REF] Heine | Language contact and grammatical change[END_REF], for example, contact-induced grammatical change is instantiated through differences in rates, in particular, with the extension of a minor pattern into a major pattern. For [START_REF] Sorace | Internal and external interfaces in bilingual language development: Beyond structural overlap[END_REF], overextension of subject pronouns is a default strategy among bilinguals independent of the languages they speak. For Silva-Corvalán (1994), simplification is likely to occur among bilinguals through loss of pragmatic functions, leading to the reduction of subject pronoun use. Beyond these predictions about rates, most authors predict that Spanish bilinguals will become less sensitive to the constraints of subject pronoun use due to their intensive contact with a non-null-subject language such as English. Turning to the more recent literature though, I would like to add the possibility that transfer will most likely be bidirectional. Though not explicitly stated, in practice, bidirectionality is tested in the various analyses conducted by TC & T. Finally, against the "convergence" scenario, where the two languages grow similar, TC & T consider the possibility that no change will take place in either of the languages.

Chapter 5, "Subject pronoun expression: reconsidering the constraints" (pp. 74-110), introduces variable subject expression in Spanish. The focus is on the most well established constraints in the literature: contrast and emphasis (section 5.1), ambiguity resolution (section 5.2), accessibility as subject continuity (section 5.3), coreferential subject priming (section 5.4), clause type (section 5.5), tense-aspect and temporal sequencing (section 5.6), verb class (sections 5.7 and 5.8), and grammatical person differences (section 5.10).

Chapter 6 is entitled "Cross-language comparisons: Reconsidering language types" (pp. 111-135). In this chapter, TC & T propose a variationist typology based on structure of variability (section 6.2) with the goal to compare null subject languages, like Spanish, and non-null subject languages, like English (section 6.1). Their approach goes beyond any "theory-internal convictions" (p.

115) and relies strongly on the analysis of empirical, conversational data. For the monolingual benchmark varieties, they draw on existing English and Spanish monolingual spoken corpora (section 6.3). The English monolingual data are from the Santa Barbara Corpus of Spoken American English with recordings from various locations. The Spanish monolingual data are from the Corpus of Conversational Colombian Spanish (Cali, Colombia). One can object that these corpora are not ideal benchmarks, i.e., an English corpus from speakers of similar socio-economic and educational background from New Mexico would have been more adapted; a corpus of Mexican Spanish would have matched better the Spanish data given the above-mentioned similarity between the two varieties.

From a practical viewpoint, however, these two corpora present the advantage of using the same prosodic transcription as the New Mexican bilingual corpus does.

Analysis of the two monolingual corpora shows that, when taking into consideration both syntax and prosody, English and Spanish are similar at the level of subject expression in coordination (section 6.4). In particular, the analysis reveals that VP coordination with the conjunction and exhibits the highest rate of unexpressed subjects (as opposed to other conjunctions or asyndetic coordination) and that the tighter the prosodic linking (measured through boundaries and contours of prosodic units), the higher the likelihood that subjects will be unexpressed. Analysis of the Spanish monolingual data points to a similar trend, leading TC & T to conclude that English does not differ from Spanish in coordinate clauses. Analysis of the data also shows that a prosodic restriction in subject pronoun expression is characteristic of English: null subjects occur only in prosodic-initial position. In contrast, Spanish null subject pronouns can appear in both prosodic initial and non-initial position. TC & T argue that this is a crucial difference of subject pronoun use between English and Spanish and is a potential "conflict site" for the bilinguals, where the two languages could potentially converge to match one another (section 6.5). They then identify the various constraints that the two languages share, restricted to the contexts within which the speakers can vary in their use or not of subject pronouns, and measure their strength (section 6.6). The constraints examined are accessibility (measured through the degree of syntactic or prosodic linking to the preceding subject), coreferential subject priming, verb class, and tense-aspect. Logistic regression analyses reveal that the probabilistic constraints are similar for the two languages, i.e., null subject pronouns occur when linked to the preceding coreferential subject, when the previous coreferential subject is unexpressed, and with dynamic verbs. However, analysis also shows that some constraints are stronger than others within each language. For example, in English, accessibility is a stronger constraint than priming and verb class, while this is not the case in Spanish. Moreover, a fine-grained analysis at the level of lexical constructions allows for the identification of more subtle differences between the two languages (section 6.7). First, Spanish speakers disfavor null subject pronouns with cognition verbs, whereas English speakers exhibit more variation at that level.

Second, English speakers favor null subjects with the use of a motion verb, such as go and come, followed by the conjunction and and a verb of speech, such as say and tell. While Spanish speakers follow a similar trend, it is less frequent than in English.

Once TC & T have described subject pronoun use for the monolingual varieties of Spanish and English using the variationist method, they turn to examine subject pronoun use in the bilingual variety of Spanish as observed in the New Mexican data; see chapter 7, "Assessing change and continuity" (pp. 136-159). They first compare the contemporary New Mexican data with earlier stages of the New Mexican variety based on data drawn from a corpus of recordings from the 1990s. These earlier recordings document the speech of the parental generation in relation to the speakers recorded in the contemporary study (section 7.4). Second, TC & T compare the rates of subject pronoun use in the New Mexican data with those reported for two monolingual Spanish varieties, from Mexico City, Mexico and from Madrid, Spain. In addition, they compare the linguistic conditioning of subject pronoun use in the New Mexican data with that reported for the variety of San Juan, Puerto Rico and Madrid, Spain. Note that comparison of the bilingual data with the monolingual benchmark from Colombia is presented in Chapter 8, but it is not clear why such a comparison is not conducted in this chapter too. The analysis between the old and recent New Mexican data provides evidence for continuity in the constraints of subject pronoun use. In addition, comparison between the bilingual and monolingual datasets shows that the relevant constraints affect subject continuity in the bilingual Spanish productions in the same way they affect subject continuity in the monolingual Spanish productions (i.e., priming, tense, and genre). In sum, the data do not provide any evidence for convergence between the two languages.

TC & T relate this finding to the fact that New Mexican speakers have been using their two languages in equal rates, most likely for more than three generations (section 7.6).

Comparison between the English and Spanish bilingual productions of New Mexicans and the English and Spanish monolingual data from the United States of America and Colombia is conducted in Chapter 8, "The most intimate contact: The bilinguals' two languages" (pp. 160-173). First, authors compare rates of subject pronoun use in bilingual New Mexican English and in monolingual English, and find no differences ("The NMSEB English rate, however, are not higher than the corresponding figures of 2 percent and 5 percent for the monolingual SBCSAE corpus", p. 161-162). However, at that point, they do not conduct the same comparison to examine how the bilinguals' Spanish might be influenced by English. This is a gap given that the predictions in the literature are that the Spanish rate of subject pronoun use is affected, not the English rate. TC & T then turn to examine linguistic conditioning in those contexts where variation is possible in the two languages, namely coordinate clauses, where the two languages are similar, and prosodic position, where the two languages differ. Against the convergence scenario, one discovers that New Mexican bilinguals respect the patterns of subject expression following the patterns occurring among monolingual speakers, in English and in Spanish.

However, based on visual inspection of the graph on the use of subject pronouns by the bilinguals in both prosodically initial and non-initial position, it seems to me that simplification may be ongoing as Spanish bilinguals use less pronouns when compared to Spanish monolinguals. Transfer from English does not predict this direction, as the authors correctly observe, since it should have led to the extension of subject pronoun use. In that case, why would Spanish bilinguals use less subject pronouns when compared to Spanish monolinguals? In non-initial intonation units this difference is numerically important, with 20% of expressed pronouns among the bilinguals and 38% among the monolinguals, and possibly also for initial intonation units (the exact rates for each type are difficult to extract from figure 8.3 and are not provided in the text or index either). I think this point deserves some discussion. In section 8.3, TC & T focus on subject expression in relation to the degree of syntactic and prosodic linking comparing the bilingual with the monolingual corpora. Again, they find no evidence for convergence between the two languages. Such results are of great interest, in view of the fact that the bilingual data come from conversations during which speakers alternated between the two languages freely, in a single setting and with the same interlocutor, therefore favoring at least online, transient processes of transfer. Finally, in 8.4, TC & T test the relevance of usage-based accounts that consider specific constructions as the most appropriate point of comparison to examine language change (see [START_REF] Backus | Codeswitching and language change: One thing leads to another?[END_REF]. Here, they compare the frequency of use of constructions such I went and got in New Mexican English and find similar rates to those in monolingual English (as predicted). In addition, they do not find an increase in such constructions in the Spanish bilingual data.

Similarly, they find no differences for constructions in Spanish with cognition verbs. Following usage-based approaches, they conclude that the high frequency of specific constructions leads to their entrenchment, which disfavors change. In accordance, one would like in the future to know more about the results of comparison for less frequent constructions.

In Chapter 9, entitled "Code-switching without convergence" (pp. 174-187), TC & T discuss the role of cognitive costs in code-switching and the ways in which these may lead to contact-induced change. In particular, they review the literature that suggests that convergence of the two grammars could help reduce the cognitive load among the bilinguals, possibly in the context of code-switching.

To test this hypothesis, TC & T examine whether the rate of subject pronouns is affected more strongly when English is used in the vicinity. Interestingly, they find no differences in subject pronoun rates whether English code-switching or single words are present in the proximity or not (section 9.3). In sections 9.4 and 9.5 the same results are obtained when examining the more fine-grained linguistic constraints. The present findings may therefore indicate that the two languages are "on" throughout the entire conversation. This is an interesting contribution to the literature on bilingualism, where evidence from natural conversations is still lacking. In particular, evidence from bilinguals who codeswitch frequently as a default mode (code-mixing) is rare. It could illustrate how frequent code-switchers may rely more heavily on the joint activation of the two languages rather than on their language control network as other bilinguals most likely do [START_REF] Green | Language control in different contexts: the behavioural ecology of bilingual speakers[END_REF].

Cross-language structural priming is examined in Chapter 10, "Codeswitching and priming" (pp. 188-202), a topic that has mainly been addressed in laboratory research. It is shown, in particular, that coreferential subject priming is affected by code-switching, although there is greater within-language than cross-language priming (section 10.2). It appears that "a previous English pronoun and a previous Spanish pronoun significantly increase the likelihood of a pronominal subject as compared with a previous unexpressed subject" (p. 191).

More specifically, the use of Spanish pronouns favors the use of coreferential Spanish pronouns in subsequent clauses, e.g., yo primes yo, and the use of null Spanish pronouns favors the use of null Spanish pronouns. English pronouns favor the use of Spanish pronouns, though to a lesser extent, e.g., I primes yo, and English null pronouns, which are very rare, do not prime Spanish null pronouns. In consequence, TC & T suggest that priming is not just a binary phenomenon, but its strength is variable both within and across languages.

Lastly, in Chapter 11, "Bilingualism in its linguistic and social context" (pp. 203-210), TC & T offer a summary of the main arguments put forward in this book. They conclude that the hypothesis of convergence between two languages in contact is not borne out in the New Mexican bilingual data. Although TC & T have illustrated this finding through subject pronoun use, they cite similar results for other phenomena such as SV order or differential object marking in the same population (section 11.1). To account for the counter-examples in the literature that document convergence between Spanish and English in the United States of America, TC & T suggest that these findings come from different bilingual populations, namely from second-generation immigrants. In particular, they argue that for a linguistic change to take place we need to examine what happens beyond the first two generations of bilinguals, something that is possible in the New Mexican community where speakers are minimally bilinguals for the past three generations (section 11.2). In addition, they stress the importance of relying on a solid methodology, based on the analysis of spontaneous speech, the use of quantitative methods, the analysis of linguistic conditioning and not mere rates, and of course the comparison of bilingual data with monolingual benchmarks (section 11.3). Finally, the hypothesis according to which frequent code-switching may determine convergence is rejected. Rather, they point to the fact that priming, which is a general mechanism in language production, is a better predictor (section 11.4). For example, cross-language priming between English I and Spanish yo is what might eventually lead to a change in the rates of Spanish subject pronouns. TC & T nonetheless stress that cross-language priming is less strong as within-language priming is and one should treat its effects with caution. In conclusion, TC & T suggest that even when bilinguals use two languages frequently and in the same discourse, and even though online cross-language effects take place, bilinguals ultimately keep their two grammars separate.

Discussion of methodology

As discussed in the previous section, one of the strengths of this study is its basis on spoken corpora. Spoken corpora are of special interest to linguists studying language contact as they potentially include a wide range of contact phenomena and may reveal information about the cognitive processes involved in the production and comprehension of bilingual speech. For corpus-driven studies to be valid, corpora need to be "big enough". This is a well-known challenge regarding spoken bilingual corpora since automated techniques are not available yet and considerable manual coding is still needed. TC & T meet this challenge by building a novel 29-hour-long corpus, from 40 bilingual speakers, targeting on average an hour of speech per speaker. The size of the New Mexican corpus is comparable to corpora from other classic studies in the field, such as Myers-Scotton's study (1993) with 20 hours of speech for each corpus from more than 100 speakers, [START_REF] Poplack | Sometimes I'll start a sentence in Spanish y termino en espanol[END_REF] on Puerto Rican code-switching, with 66 hours and a minimum of two hours of speech per speaker. [START_REF] Poplack | Myths and facts about loanword development[END_REF] remains an exception for the study of loanwords among French-English bilinguals in Canada relying on two corpora, each consisting of more than one million words, and a corpus from the nineteenth century with 500,000 words. Unfortunately, what all these studies also have in common is lack of accessibility of the original data, or even parts of the data, to the wider research community (NB: TC & T restrict access to researchers who are familiar with the community).

Access to the corpora is a point that deserves some discussion. It is clear that the dynamism of corpus-driven research is due to a large extent to the fact that it addresses public and political concerns for accountability and replicability in science. Indeed, there is increasing institutional support for FAIR data (Findable, Accessible, Interoperable, and Re-usable data) and one can regret that a contemporary corpus, elaborated with an NSF grant, does not meet these standards. TC & T motivate restricting access to researchers who are familiar with the community in the following way: "researchers unfamiliar with the speech community may unknowingly misunderstand linguistic material" (p. 48).

For example, TC & T express their concern that listeners unfamiliar with New Mexican Spanish may not understand that muchito means 'kid', following contraction from muchachito. Or that naïve listeners may perceive a preterit dijimos 'say-PRET.1PL' ([diximos]) instead of present tense dec-imos 'say.PRES.1PL' with aspirated onset /s/ ([deximos] vs. standard [desimos]).

However, a variety of materials may supplement raw data (if not directly annotated data with glosses) to prevent other researchers from arriving to the wrong conclusions. In any case, the risks identified by the authors are worth taking when compared to the benefit of open access corpora (see among others [START_REF] Adamou | A corpus-driven approach to language contact[END_REF][START_REF] Guzmán | Simple tools for exploring variation in code-switching for linguists[END_REF]. Of course, providing access to the data raises ethical questions, but TC & T already implement the necessary precautions, e.g., respect of the privacy of the speakers through anonymizing the parts of the sound files and transcripts that include personal information.

I would now like to address the claim in this book that although all methods are welcome in contact linguistics, "the ultimate yardstick must be the sociolinguistically constructed corpus" (p. 13). TC & T suggest that experimental data produced in controlled environments are not satisfactory for the study of non-standard bilingual varieties as "test situations and lab-based activities are associated with institutions that have excluded these varieties" (p. 39). However, recent studies have successfully adapted experimental and, more specifically, psycholinguistic methods in the field (see O'Shannessy & Meakins 2012; Lipski 2016; Adamou & Shen 2017). In addition, several authors regard corpus and experimental data as complementary: for example, [START_REF] Adamou | There are no language switching costs when codeswitching is frequent[END_REF] and [START_REF] Johns | Mixing things up: How blocking and mixing affect the processing of codemixed sentences[END_REF] showed that language switching costs align with the code-switching habits of the community as documented in corpus studies.

TC & T also suggest that "speaker judgments are unreliable to test hypotheses" (p. 39). Although [START_REF] Labov | When intuitions fail[END_REF] has shown that respondents' opinions regarding their linguistic behavior are not necessarily in agreement with the way they speak, [START_REF] Bresnan | Is Syntactic Knowledge Probabilistic? Experiments with the English Dative Alternation[END_REF] demonstrated that where the corpus predicts low probabilities, respondents also do the same and concludes that intuitive judgments reflect probability rather than grammaticality. Research has also shown that discrepancy between the way one speaks and the way one judges correctness is more pronounced when non-standard varieties are involved and that judgments regarding non-standard varieties are more fluid than judgments of standard varieties [START_REF] Henry | Non-standard dialects and linguistic data[END_REF]. It is therefore possible to take into consideration sociolinguistic norms in the analysis of judgments. More generally, acceptability judgment tasks are considered a reliable tool in linguistic research [START_REF] Sprouse | A comparison of informal and formal acceptability judgments using a random sample from Linguistic Inquiry 2001-2010[END_REF], in particular when researchers work with a large sample of participants [START_REF] Gibson | The need for quantitative methods in syntax and semantics research[END_REF].

To conclude, in my view, a variety of approaches are needed to offer insights for understanding language contact and bilingualism, ranging from experimental to naturalistic corpus data and intuitive judgments. In that sense, the variationist analysis of corpora in this book presents invaluable insights that can fruitfully combine with other methods.

Convergence or not?

The main finding in Torres Cacoullos and Travis ( 2018) is that Spanish-English bilinguals from the bilingual community of New Mexico in the United States of America do not exhibit any contact-related changes in either of their two everyday languages. In particular, quantitative analysis of the speech of New Mexican bilinguals reveals the use of similar amounts of Spanish pronouns, under the same conditions as Spanish monolinguals do. In parallel, New Mexican bilinguals use similar amounts of English pronouns, under the same conditions as English monolinguals do. TC & T argue that these results are due to the specificity of a bilingual community in a long-term contact setting.

However, researchers have identified contact-induced changes in a historical perspective relying on the comparative method, a method that has been solidly established in linguistics. The fact that we sometimes may only have access to the end-result does not affect the validity of the observation, in particular, as we are dealing with categorical changes in the grammatical categories rather than with rates and linguistic constraints as in the book under discussion. In my areas of expertise, for instance, it is uncontroversial that Slavic languages in the Balkans have developed definite articles as opposed to all other Slavic languages outside the Balkans through internal processes of grammaticalization from demonstratives, as attested in written sources, but also through contact with Greek, a language with definite articles [START_REF] Mladenova | Definiteness in Bulgarian[END_REF].

Even more uncontroversial is the fact that, Romani, an Indic language, exhibits several syntactic features that have been shaped through contact with non-Indic languages during the past ten centuries. Romani developed a definite article during the Byzantine times in contact with Greek [START_REF] Matras | Romani: A linguistic introduction[END_REF]): a first contactinduced change. Then, the Romani groups who migrated to countries where the new contact languages did not have grammaticalized definite articles gradually lost the Romani definite articles: a second uncontroversial contact-induced change. This process is ongoing in some varieties while in others loss of definite articles is completed. See for example Romani spoken in Poland, currently in contact with Polish, Romani spoken in Russia, currently in contact with Russian, and Romani spoken in Finland, currently in contact with Finnish (elicited data whereas in all Romani varieties from Europe a single copula is used and no variation at the level of the copulas can be found or reconstructed.

In sum, although the New Mexican data are valuable, they are not relevant on whether contact-induced change takes place in general. Rather these new data document the cases where contact-induced change does not take place.

Conclusion

To conclude, Torres Cacoullos' and Travis' book is a timely contribution to the fields of bilingualism and language contact. Beyond the interest of the data, one of the major strengths of the book is that it offers a solid methodology based on the variationist framework that the authors present in a comprehensive and sophisticated way. The book will appeal to anyone with an interest in language contact, bilingualism, language change, variationist linguistics, and Spanish in the United States of America. While it is a must-read for well-established researchers in the field, it is also accessible to students as it is written in a clear and pedagogical way, accompanying the reader gradually as the arguments and the findings unfold.

  romani.humanities.manchester.ac.uk/rms/; more specifically see the samples with the codes PL 003, RUS 005, FIN-002). In contrast, Romani groups that were in contact with languages with definite articles have maintained the use of the Romani definite articles. Another example comes from the Slavic variety of Molise, in Italy, which developed an indefinite article in contact with Italian as opposed to the Croatian variety in the Dalmatian coast, which did not develop an indefinite article[START_REF] Breu | Language contact of minority languages in Central and Southern Europe: a comparative approach[END_REF]. Moreover,[START_REF] Adamou | Replicating Spanish estar in Mexican Romani[END_REF] shows that Mexican Romani speakers who migrated to Mexico 150 years ago replicated Spanish estar, thus creating variation in the domain of attributive clauses,