Florentina Nicolau

Hugues Mounier

Flatness of networks of two synaptically coupled excitatory-inhibitory neural modules with maximal symmetry*

We consider networks of two synaptically coupled excitatory-inhibitory neural modules with maximal symmetry of the connection strengths, and for which the nonlinearities are described by a logistic sigmoidal function. It has been shown that the connection strengths may slowly vary with respect to time and that they can actually be considered as inputs of the network. In the recent publication [13], we considered the case of two synaptically coupled subnetworks and studied the problem of determining which connection strengths should be modified (in other words, which connection strengths should be considered as inputs), in order to achieve flatness for the resulting control system when no relation between the connection strengths (in particular, no symmetry) is assumed. In this paper, we consider a similar problem but under the assumption that all interactions (interactions between subnetworks and local interactions within the same subnetwork) are always symmetric.

I. INTRODUCTION

In this paper, we consider population-based networks of N synaptically coupled excitatory-inhibitory neural modules or subnetworks (simply, EI networks) of the following form [START_REF] Wilson | Excitatory and inhibitory interactions in localized populations of model neurons[END_REF], [START_REF] Wilson | A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue[END_REF] (see also [START_REF] Bressloff | Waves in neural media[END_REF], [START_REF] Ermentrout | Mathematical foundations of neuroscience[END_REF]):

τ e ẋn e =-x n e + F e N m=1 (w ee (n, m)x m e -w ei (n, m)x m i ) τ i ẋn i =-x n i + F i N m=1 (w ie (n, m)x m e -w ii (n, m)x m i ) , (1) with x n e , x n i ∈ R, and 1 ≤ n ≤ N . Each subnetwork is labeled by the discrete index n and contains a pair of mutually coupled local populations of excitatory and inhibitory neurons. The equations describing the dynamics of each subnetwork are usually called Wilson-Cowan equations for cortical dynamics. The states of the system x n e and x n i are, resp., the proportion of excitatory and inhibitory cells of the n-th subnetwork which become active per unit time. We will always use lower indices a ∈ {e, i} and b ∈ {e, i} to refer to the excitatory or inhibitory character of the considered objects, and integers 1 ≤ n ≤ N and 1 ≤ m ≤ N as upper indicies to label the subnetworks. The gain functions F e and F i are nonlinear, typically sigmoidal functions, τ e and τ i are time constants, and w ab (n, m), where a, b ∈ {e, i} and 1 ≤ n, m ≤ N , denote the strengths of connections between x n a and x m b (w ab (n, m) enters into the dynamics of x n a and characterizes the action of x m b on x n a ). It has been shown, see for instance [START_REF] Dayan | Theoretical neuroscience: computational and mathematical modeling of neural systems[END_REF] and the references therein, that (some of) the connection strengths w ab (n, m) may slowly vary with respect to time and can actually be considered as inputs of dynamical system [START_REF] Bressloff | Waves in neural media[END_REF]. Therefore, activity-dependent synaptic *Work partially supported by ANR-17-CE40-0005 MindMadeClear. 1 Florentina Nicolau is with Quartz EA7393 Laboratory, ENSEA, France. 2 Hugues Mounier is with Université Paris-Saclay, CNRS, CentraleSupélec, L2S, France. florentina.nicolau@ensea.fr, hugues.mounier@l2s.centralesupelec.fr plasticity is believed to play a crucial role in the development of neural circuits and it is thus possible to control a network of synaptically coupled excitatory-inhibitory modules through the connections strengths. An important question is which connection strengths should be modified (equivalently, which connection strengths should be considered as inputs of the network), in order to achieve a desired property for the resulting control system.

A property that is very useful in applications (e.g., for trajectory tracking, constructive controllability or trajectory generation) is that of flatness (see [START_REF] Fliess | Flatness and defect of non-linear systems: introductory theory and examples[END_REF], [START_REF] Fliess | A Lie-Bäcklund approach equivalence and flatness of nonlinear systems[END_REF]). In the recent paper [START_REF] Nicolau | Flatness of networks of two synaptically coupled excitatory-inhibitory neural modules[END_REF], we considered for the case of N = 2 synaptically coupled subnetworks, the question of which connection strengths among w ab (n, m), for a, b ∈ {e, i}, 1 ≤ n, m ≤ 2, should be modeled as control variables in order to obtain flatness for the resulting control system when no relation between the connection strengths (in particular, no symmetry) is assumed. In this paper, we study a similar problem but under the assumption that all involved interactions (that is, interactions between subnetworks and local interactions within the same subnetwork) are symmetric. This translates into relations between the connection strengths of the type w ab (n, m) = w ba (m, n), for all a, b ∈ {e, i} and 1 ≤ n, m ≤ N . A network of synaptically coupled excitatory-inhibitory neural modules for which all those relations hold will be called with maximal symmetry. Contrary to the case when no relation between the connection strengths is assumed (for which the control network becomes, after the application of an invertible static feedback transformation, a control-affine system), in general, control networks with maximal symmetry are nonlinear with respect to the control (and cannot be rendered control-affine via static feedback transformations) which considerably complicates flatness analysis. In general, flatness imposes particular configurations of the interactions between the subnetworks or even for the local interactions within a subnetwork and our goal is to identify network configurations that are flat for one of the simplest possible networks with maximal symmetry (i.e., for which N = 2).

We will study the following question: for flatness, are there interactions between the subnetworks or local interactions within the same network that are not allowed (translating into zero w's) or, on the contrary, that necessarily have to take place (translating into nonzero w's)? In the first case, we will talk about structural conditions, while in the second case we will call them regularity conditions. Of course, flatness of a control network depends on the number of inputs. Similarly to [START_REF] Nicolau | Flatness of networks of two synaptically coupled excitatory-inhibitory neural modules[END_REF], we will show that if enough connection strengths can be considered as inputs (at least 2 for a network of 2 synaptically coupled subnetworks) and if we have enough freedom in choosing which connection strengths are modeled as inputs, then the control network can be rendered flat without any restriction on the interactions between the two subnetworks (so without structural conditions). Contrary to [START_REF] Nicolau | Flatness of networks of two synaptically coupled excitatory-inhibitory neural modules[END_REF], there are cases when the network is never flat or for which we are not able to characterize flatness in general (in those cases, we can however state that the control network does not admit any flat output depending on the state and the control variables only, so if the control network is flat, then its flat output necessarily involves derivatives of the control).

The problem of deciding which connection strengths have to be considered as inputs in order to achieve flatness for the control network is related to that of constructing flat inputs for a dynamical system [START_REF] Fritzsche | Nonlinear control of nonobservable non-flat mimo state space systems using flat inputs[END_REF], [START_REF] Nicolau | Flat inputs: Theory and applications[END_REF], [START_REF] Waldherr | Flat inputs in the mimo case[END_REF], but several major differences can be listed and we send the reader to [START_REF] Nicolau | Flatness of networks of two synaptically coupled excitatory-inhibitory neural modules[END_REF] for discussion and comparison.

II. DEFINITIONS AND PROBLEM STATEMENT A. Flatness and linearization

Consider the nonlinear control system Ξ : ẋ = f (x, u), where x ∈ X ⊂ R s , u = (u 1 , . . . , u c ) ∈ R c . The dynamics f are C ∞ -smooth and rk ∂f ∂u = c. Fix an integer l ≥ -1 and denote ūl = (u, u, . . . , u (l) ). For l = -1, the sequence ū-1 is empty.

Definition 1: The system l+1) , for l ≥ -1, if there exist a neighborhood O l of (x 0 , ūl 0 ) and c smooth functions φ i = φ i (x, u, u, . . . , u (l) ), 1 ≤ i ≤ c, defined in O l , having the following property: there exist an integer r and smooth functions γ i , 1 ≤ i ≤ s, and δ j , 1 ≤ j ≤ c, such that x i = γ i (φ, φ, . . . , φ (r-1) ) and u j = δ j (φ, φ, . . . , φ (r) ) for any C l+r -control u(t) and corresponding trajectory x(t) that satisfy (x(t), u(t), . . . , u (l) (t)) ∈ O l , where φ = (φ 1 , . . . , φ c ) and is called a flat output. If a flat system admits a flat output whose components are functions of x only (resp., of (x, u) only), i.e., we have φ i = φ i (x) (resp., φ i = φ i (x, u)) for all 1 ≤ i ≤ c, then the system is called x-flat (resp., (x, u)-flat). Flatness is closely related to the notion of feedback linearization. It is well known that systems linearizable via invertible static feedback are flat. The control system Ξ is locally linearizable by static feedback if it can be transformed via a local diffeomorphism x = ϕ(x) and a local invertible feedback transformation u = ψ(x, ũ) to a linear controllable system Λ : ẋ = Ax + B ũ (two systems that can be transformed one into each other via transformations of that form are called static feedback equivalent systems). The problem of static feedback linearization was solved by [START_REF] Jakubczyk | On linearization of control systems[END_REF] and [START_REF] Hunt | Linear equivalents of nonlinear time varying systems[END_REF]. In general, a flat system is not linearizable by static feedback, with the exception of the single-input case for which flatness is equivalent to static feedback linearization (thus to the necessary and sufficient conditions of [START_REF] Jakubczyk | On linearization of control systems[END_REF]). Flat systems can be seen as a generalization of static feedback linearizable systems. Namely they are linearizable via dynamic, invertible and endogenous feedback, see [START_REF] Fliess | Flatness and defect of non-linear systems: introductory theory and examples[END_REF], [START_REF] Pomet | A differential geometric setting for dynamic equivalence and dynamic linearization[END_REF] for those notions. In our analysis, we will identify flat excitatory-inhibitory networks that are either static feedback linearizable or become static feedback linearizable after the application of a one-or two-dimensional dynamic precompensator (actually after prolonging one or two suitably chosen controls, which is the simplest dynamic feedback). Given Ξ and an integer p ≥ 1, we construct its p-fold prolongation Ξ p,0,...,0 by, first applying an invertible feedback u = ψ(x, ũ) to get Ξ : ẋ = f (x, ũ), where f = f (x, ψ(x, ũ)), and then prolonging the first (feedback modified) control as ũ(p) 1 = v 1 (which justifies the notation Ξ p,0,...,0 ) and keeping the remaining controls ũℓ = v ℓ , 2 ≤ ℓ ≤ c, unchanged, to obtain

Ξ : ẋ = f (x, u) is flat at (x 0 , ūl 0 ) ∈ X × R c(
Ξ p,0,...,0 ẋ = f (x, y 1 , v 2 , . . . , v c ) ẏj = y j+1 , 1 ≤ j ≤ p -1, ẏp = v 1
, where y 1 = ũ1 . If there exists an invertible feedback transformation u = ψ(x, ũ) leading to a locally static feedback linearizable p-fold prolongation Ξ p,0,...,0 , then the system Ξ is called dynamically linearizable via an invertible p-fold prolongation. In a similar way, we can define the (p 1 , p 2 )fold prolongation Ξ p1,p2,0,...,0 obtained by prolonging the feedback modified controls ũ1 and ũ2 as ũ(p1)

1 = v 1 and ũ(p2) 2 = v 2 ,
resp., as well as dynamically linearizability via an invertible (p 1 , p 2 )-fold prolongation.

B. Problem statement Consider a network of two synaptically coupled excitatory-inhibitory neural modules or subnetworks of form [START_REF] Bressloff | Waves in neural media[END_REF]. Following [START_REF] Wilson | Excitatory and inhibitory interactions in localized populations of model neurons[END_REF], [START_REF] Wilson | A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue[END_REF], we adopt the convention that x n e = 0,

x n i = 0, 1 ≤ n ≤ 2,
is the resting state, and is assumed to reflect low-level background activity and therefore, small negative values of x n e and x n i are allowed, they have physiological significance and represent depression or inhibition of resting activity. For simplicity, from now on, we suppose that all time constants are equal to one, i.e., τ e = τ i = 1, and that the gain functions F e and F i are equal to the shifted logistic sigmoidal function (which is a classical assumption when mathematically modeling excitatory and inhibitory activity in localized populations of neurons [START_REF] Wilson | Excitatory and inhibitory interactions in localized populations of model neurons[END_REF], [START_REF] Wilson | A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue[END_REF]):

F (ξ) = F e (ξ) = F i (ξ) = S (ξ)-S (0), with S (ξ) = 1 1 + e -ξ , (2)
where ξ ∈ R, that is, we work with systems of the form

ẋ1 e = -x 1 e + F (w ee (1, 1)x 1 e + w ee (1, 2)x 2 e -w ei (1, 1)x 1 i -w ei (1, 2)x 2 i ) ẋ1 i = -x 1 i + F (w ie (1, 1)x 1 e + w ie (1, 2)x 2 e -w ii (1, 1)x 1 i -w ii (1, 2)x 2 i ) ẋ2 e = -x 2 e + F (w ee (2, 1)x 1 e + w ee (2, 2)x 2 e -w ei (2, 1)x 1 i -w ei (2, 2)x 2 i ) ẋ2 i = -x 2 i + F (w ie (2, 1)x 1 e + w ie (2, 2)x 2 e -w ii (2, 1)x 1 i -w ii (2, 2)x 2 i ), (3)
with F given by (2). Denote by

f = (f 1 e (x, w), f 1 i (x, w), f 2 e (x, w), f 2 i (x, w)) the right-hand side of (3), i.e., we have ẋ = f (x, w), (4)
where x = (x 1 e , x 1 i , x 2 e , x 2 i ) ∈ R 4 is the state, and w = (w ee (1, 1), . . . , w ii (2, 2)) denotes the connection strengths. The problem that we are studying in this paper can be formalized as follows:

Problem. Consider the EI network given by (3). For a given integer 1 ≤ c ≤ 4, define c controls u n,m ab among w ab (n, m), for a, b ∈ {e, i} and 1 ≤ n, m ≤ 2, leading to a control system ẋ = f (x, u), with rk ∂f ∂u = c, for which the connection strengths not defined as inputs are assumed constant, and study how the subnetworks have to interact -equivalently, which are the conditions that the (constant) configuration strengths have to verify-such that the obtained control system ẋ = f (x, u) is flat.

C. Assumptions From now on, we assume the following:

(A1) All interactions are perfectly symmetric, that is:

w ab (n, m) = w ba (m, n), ∀a, b ∈ {e, i} and 1 ≤ n, m ≤ 2, (5)
meaning that two subnetworks act on each other exactly in the same way and, moreover, that the local interactions between the excitatory and the inhibitory populations within the same network are also identical. In particular, if w ab (n, m) = 0 (meaning that x m b does not directly impact the dynamics of x n a ), then we necessarily have w ba (m, n) = 0, and x n a does not act on x m b either. A network for which (5) holds will be called with maximal symmetry.

(A2) Any connection strength w ab (n, m), for a, b ∈ {e, i} and 1 ≤ n, m ≤ 2, may be modeled as an input for the control EI network and the corresponding input will be denoted by u n,m ab , and since w ab (n, m) = w ba (m, n), the input u n,m ab acts directly on x n a and x m b (it is explicitly involved into the expressions of ẋn a and ẋm b ). (A3) The connection strengths that are not modeled as controls will always be supposed constant (zero or nonzero).

(A4) We work locally, around a nominal point (x 0 , u 0 ) = (x 0 , w 0 ), where u 0 corresponds to nominal values of the connection strengths modeled as inputs, and we always assume that control variables act independently on the system, that is, if c is the number of connection strengths modeled as inputs, then for the obtained control system ẋ = f (x, u), we should have rk ∂f ∂u (x, u) = c, for all (x, u) in a neighborhood of (x 0 , u 0 ), implying, in particular that x 0 ̸ = 0, see ( 2)-(4). Define the set of singularities

S = {(x, u) ∈ R 4 × R c : rk ∂f ∂u (x, u) < c}. (6)
The control network will always be considered around regular points (x 0 , u 0 ) ∈ R 4 × R c \ S. Moreover we suppose that the number c of connection strengths that can be modeled as inputs is strictly less than 4 (if c = 4, which is the state dimension, the fact that the control variables act independently on the system implies that flatness is trivial).

The constructed control system will always be nonlinear with respect to the control (because of the nonlinear sigmoidal function F ), but in the particular case when all inputs correspond to connection strengths of the type w aa (n, n), it can be transformed via a local invertible static feedback transformation into a control-affine one (this was always the case for the control networks studied in [START_REF] Nicolau | Flatness of networks of two synaptically coupled excitatory-inhibitory neural modules[END_REF]).

Flatness will often imply certain relations or conditions involving the connection strengths (that are not modeled as inputs). Conditions of the type w ab (n, m) ̸ = 0 (or more generally, nonzero-type conditions) will be called regularity conditions, they do not restrict the interactions between different subnetworks or between excitatory and inhibitory populations. On the other hand, conditions of the type w ab (n, m) = 0 (or more generally, zero-type conditions) will be called structural conditions because they impose certain constraints on the interactions (and the EI network is flat only for certain configurations of the interactions between subnetworks or between excitatory and inhibitory populations). By interaction configuration between two subnetworks, we will mean all possible values of the associated connection strengths w ab (n, m), a, b ∈ {e, i}, 1 ≤ n, m ≤ 2 (respecting the maximal symmetry assumption). We say that there is no interaction if all w ab (n, m) are zero.

III. MAIN RESULTS

In this section we give our main results: we study flatness of networks consisting of two synaptically coupled EI neural subnetworks of form (3) with maximal symmetry, implying that we have 9 candidates for the inputs. If all controls are chosen among connection strengths of the type w aa (n, n), i.e., for fixed a ∈ {e, i} and 1 ≤ n ≤ 2, the corresponding input is involved in the expression of ẋn a only, and does not appear in the dynamics of any x m b , for (b, m) ̸ = (a, n), then the resulting control system falls into the class of control networks studied in [START_REF] Nicolau | Flatness of networks of two synaptically coupled excitatory-inhibitory neural modules[END_REF], for which we identified and characterized all possible flat configurations. In particular, we showed that the control EI network can always be rendered flat without constraints on the interactions between subnetworks or on the local interactions within the same network, except for the case when only one connection strength is modeled as an input (for which we necessarily need structural conditions). Therefore, from now on, we make the following assumption:

(A5) We suppose that at least one input is defined as w ab (n, m), with a ̸ = b or n ̸ = m.

Assumption (A5) means that we either act on the coupling of the local excitatory-inhibitory populations within the n-th subnetwork (when w ab (n, n), a ̸ = b, plays the role of the input) or on the interaction between the n-th and the m-th subnetworks, when n ̸ = m.

A. Control network with three inputs Assume that we can model three connection strengths as controls. Recall that we work around regular points (x 0 , u 0 ) ∈ R 4 × R 3 \ S, with S given by ( 6), and that control variables act independently on the network, i.e., rk ∂f ∂u (x 0 , u 0

) = c = 3, with f = (f 1 e , f 1 i , f 2 e , f 2
i ) the right hand-side of (3). To make matters more definite, suppose that rk

∂(f 1 e ,f 1 i ,f 2
e ) ∂u (x 0 , u 0 ) = 3 (all other cases can be treated similarly). The control network is: -case 1) either of the form

ẋ1 e = f 1 e (x, u) ẋ2 e = f 2 e (x, u) ẋ1 i = f 1 i (x, u) ẋ2 i = f 2 i (x)
, with u ∈ R 3 , (7) where f 2 i is a function of x only, occurring when no connection strength associated to the inhibitory population of the second subnetwork (i.e., no w ai (n, 2) = w ia (2, n), a ∈ {e, i}, 1 ≤ n ≤ 2) is modeled as an input; -case 2) or of the form

ẋ1 e = f 1 e (x, u) ẋ2 e = f 2 e (x, u) ẋ1 i = f 1 i (x, u) ẋ2 i = f 2 i (x, u) , with u ∈ R 3 , (8)
where all f n a depend explicitly on u; which occurs if connection strengths associated to both subnetworks and both excitatory and inhibitory populations (i.e., w ab (n, m) = w ba (m, n) such that all cases a = e, a = i, n = 1, and n = 2 are present) are necessarily modeled as inputs).

Therefore by applying the invertible (nonlinear) feedback transformation: ũℓ = f n a (x, u), for (a, n) ∈ {(e, 1), (i, 1), (e, 2)}, 1 ≤ ℓ ≤ 3, the three-input network, around (x 0 , u 0 ) ∈ R 4 × R 3 \ S, can always be transformed, in a neighborhood of (x 0 , ũ0 ) with ũ0 such that ũℓ,0 = f n a (x 0 , u 0 ), 1 ≤ ℓ ≤ 3, resp., into:

ẋ1 e = ũ1 ẋ2 e = ũ3 ẋ1 i = ũ2 ẋ2 i = f 2 i (x), (9)
for case 1), or into

ẋ1 e = ũ1 ẋ2 e = ũ3 ẋ1 i = ũ2 ẋ2 i = f 2 i (x, ũ), (10)
for case 2), with the function f 2 i of the feedback modified system depending explicitly (and in a nonlinear way) on at least one control (actually, such that

∂ 2 f 2 i ∂ ũ2 ℓ ̸ = 0, for at least one 1 ≤ ℓ ≤ 3).
It can be easily shown that control network ( 7) is flat at x 0 if and only if (

∂f 2 i ∂x 1 e , ∂f 2 i ∂x 1 i , ∂f 2 i ∂x 2 e
)(x 0 ) ̸ = (0, 0, 0), equivalently (w ie (2, 1), w ii (2, 1), w ie (2, 2)) ̸ = (0, 0, 0), and we actually recover [13, Proposition 2] characterizing flatness for threeinputs EI networks when no relation between the connection strengths is assumed. This is due to the fact that after applying suitable feedback transformations, both network [START_REF] Fritzsche | Nonlinear control of nonobservable non-flat mimo state space systems using flat inputs[END_REF] and that of [START_REF] Nicolau | Flatness of networks of two synaptically coupled excitatory-inhibitory neural modules[END_REF]Proposition 2]) can be transformed into the equivalent control-affine one [START_REF] Hunt | Linear equivalents of nonlinear time varying systems[END_REF]. First of all, notice that there are no structural conditions (i.e., there are no connections strengths that have to be zero or to satisfy a zero-type relation) for flatness of the above three-input EI network and that only a regularity condition is needed (translating into a certain connection strength that has to be nonzero, which is the case for generic systems). The regularity condition means that either the first subnetwork directly affects the second one (more precisely, its inhibitory population) or that there are interactions between the excitatory and the inhibitory populations within the second subnetwork. Finally, observe that when the regularity condition is satisfied, control network ( 7) is actually static feedback linearizable. The next result identifies flat network configurations of form (8) (actually, of the feedback equivalent form [START_REF] Jakubczyk | On linearization of control systems[END_REF], for which the conditions are easier to formulate and use simpler notations).

Proposition 1: Consider the three-input control EI network given by (8) around (x 0 , u 0 ) ∈ R 4 × R 3 \ S and its feedback equivalent network (10) around (x 0 , ũ0 ).

(i) If there exists 1 ≤ ℓ * ≤ 3 such that ∂ f 2 i ∂ ũℓ * ≡ 0 and ∂ f 2 i ∂x n * a * (x 0 , ũ0 ) ̸ = 0, where x n *
a * is such that ẋn * a * = ũℓ * , then system (10) is flat at (x 0 , ũ0 ), implying that control EI network ( 8) is flat at (x 0 , u 0 ) as well. Moreover, system (8) is dynamically linearizable via an invertible either 1-fold or (1,1)-fold prolongation.

(ii) If ∂ f 2 i ∂ ũℓ ̸ ≡ 0, for all 1 ≤ ℓ ≤ 3, then system (10) is not x-flat thus (8) is not x-flat either.
Sketch of proof: (i) Without loss of generality, suppose that ℓ * = 3, i.e.,

∂ f 2 i ∂ ũ3 ≡ 0 and ∂ f 2 i ∂x 2 e (x 0 , u 0 ) ̸ = 0, then it is obvious that (x 1 e , x 1 i , x 2 i
) is a flat output of (10) (and therefore of (8) as well). Moreover, it is easy to see that the control EI network [START_REF] Jakubczyk | On linearization of control systems[END_REF] is never static feedback linearizable (the necessary and sufficient conditions of [START_REF] Jakubczyk | On linearization of control systems[END_REF] are not verified), however its (1,1)-prolongation given by:

ẋ1 e = y 1 1 ẋ2 e = v 3 ẋ1 i = y 1 2 ẋ2 i = f 2 i (x, y 1 1 , y 1 2 ), ẏ1 1 = v 1 ẏ1 2 = v 2 ,
where (x, y 1 1 , y 1 2 ), with y 1 1 = ũ1 , y 1 2 = ũ2 , is the state of the prolonged system and v = (v 1 , v 2 , v 3 ), with v ℓ = uℓ , ℓ = 1, 2, and v 3 = ũ3 , is the new control, is locally static feedback linearizable around (x 0 , ũ0 ). Observe also that in the case when f 2 i depends on one control only (say ũ1 ) only a 1-fold prolongation of ũ1 is needed to dynamically linearize the original system. Suppose now system (10) is flat.

(ii) To show that ( 10) is never x-flat, one can first prove that a system of the form ẋ = f (x, u), x ∈ R 4 , u ∈ R 3 , is x-flat if and only if it can be transformed via a local diffeomorphism x = ϕ(x) and an invertible static feedback transformation u = ψ(x, ũ) into ẋℓ = ũℓ , for 1 ≤ ℓ ≤ 3, and ẋ4 = f 4 (x, ũ1 , ũ2 ). This form actually falls into the class of nonlinear control systems of codimension 1 (with one input less than the number of states) considered in [START_REF] Martin | A geometric sufficient condition for flatness of systems with m inputs and m+1 states[END_REF], where a geometric characterization of that class is provided. It is now enough to show that control network [START_REF] Jakubczyk | On linearization of control systems[END_REF] with f 2 i depending on all three inputs does not verify the necessary and sufficient condition of [START_REF] Martin | A geometric sufficient condition for flatness of systems with m inputs and m+1 states[END_REF]. □ If the function f 2 i of the feedback modified system (10) depends on at most two controls among u ℓ , 1 ≤ ℓ ≤ 3, (equivalently, if each function f n a of (8) involves at most two controls 1 u n,m ab ), then control EI network (10) (equivalently, ( 8)) is flat without structural conditions, only one regularity condition is needed. Contrary to the case of control network [START_REF] Fritzsche | Nonlinear control of nonobservable non-flat mimo state space systems using flat inputs[END_REF] (and, of all results of [START_REF] Nicolau | Flatness of networks of two synaptically coupled excitatory-inhibitory neural modules[END_REF] as well), notice that the nonlinear feedback transformations introduce singularities for flatness involving the state and the control (at points (x, u) such that

∂ f 2 i ∂x n * a *
(x, u) = 0). Notice that Proposition 1(i) gives only sufficient conditions for flatness of a network of form [START_REF] Jakubczyk | On linearization of control systems[END_REF] and does not describe all possible flat configurations. If f 2 i explicitly depends on all controls (see Proposition 1(ii)), then the picture is even more complicated that that of Proposition 1(i). Indeed, even in the case of control systems nonlinear with respect to the control with four states and three controls, the (general) problem of flatness is still open. We can however show that the system is never x-flat. The only possibility for f 2 i to involve all three controls is that the inputs are necessarily defined as w ei (1, 2), w ei (2, 2) and w ii [START_REF] Bressloff | Waves in neural media[END_REF][START_REF] Charlet | Sufficient conditions for dynamic state feedback linearization[END_REF]. So in that case, if the control network is flat, then its flat outputs necessarily 1 Under the assumption that rk

∂(f 1 e ,f 1 i ,f 2 e ) ∂u
= 3, all functions f n a of (8) depend on at most two controls if at least one input is not of the form

w ai (n, 2) = w ia (2, n), with a ∈ {e, i}, 1 ≤ n ≤ 2.
depend on u or its time-derivatives and in general, flatness would imply additional structural conditions.

Conclusion (three inputs):

For both cases 1) and 2) leading, resp., to [START_REF] Fritzsche | Nonlinear control of nonobservable non-flat mimo state space systems using flat inputs[END_REF] and ( 8), if we have enough freedom in choosing which connection strengths are modeled as inputs, then we can always identify EI network configurations that are flat (actually x-flat) without structural conditions (i.e., there are no connections that have to be zero or to satisfy a zero-type relation) and for flatness of the proposed control EI network only a regularity condition is needed.

B. Control network with two inputs

Assume that two connection strengths are modeled as controls leading to a control network ẋ = f (x, u), considered around a nominal point (x 0 , u 0 ) such that rk ∂f ∂u (x 0 , u 0 ) = 2, that is, (x 0 , u 0 ) ∈ R 4 × R 2 \ S, with S given by [START_REF] Fliess | A Lie-Bäcklund approach equivalence and flatness of nonlinear systems[END_REF]. According to Assumption (A5), at least one input is defined as w ab (n, m), with a ̸ = b or n ̸ = m, and let us suppose that rk

∂(f 1 e ,f 1 i ) ∂u (x 0 , u 0 ) = 2
(all remaining cases are similar and we do not treat them here). The control network is: -case 1) either of the form

ẋ1 e = f 1 e (x, u) ẋ2 e = f 2 e (x) ẋ1 i = f 1 i (x, u) ẋ2 i = f 2 i (x) , with u ∈ R 2 , (11)
where f 2 e and f 2 i are functions of x only, occurring when only connection strengths among w ab (1, 1) = w ba (1, 1), a ∈ {e, i} are modeled as inputs; -case 2) or of the form

ẋ1 e = f 1 e (x, u) ẋ2 e = f 2 e (x, u) ẋ1 i = f 1 i (x, u) ẋ2 i = f 2 i (x) , with u ∈ R 2 , (12)
where f 2 i is a function of the state only, occurring when no connection strength among w ai (n, 2) = w ia (2, n), a ∈ {e, i}, 1 ≤ n ≤ 2, is modeled as an input; -case 3) or of the form

ẋ1 e = f 1 e (x, u) ẋ2 e = f 2 e (x, u) ẋ1 i = f 1 i (x, u) ẋ2 i = f 2 i (x, u) , with u ∈ R 2 , (13)
where all f n a depend explicitly on u; this occurs if connection strengths associated to both subnetworks and both excitatory and inhibitory populations (i.e., w ab (n, m) = w ba (m, n) such that all cases a = e, a = i, n = 1, and n = 2 are present) are necessarily modeled as inputs.

By applying the invertible (nonlinear) feedback transformation (recall that rk

∂(f 1 e ,f 1 i ) ∂u (x 0 , u 0 ) = 2): ũℓ = f 1 a (x, u)
, for a = e, i, and 1 ≤ ℓ ≤ 2, the control network, around (x 0 , u 0 ) ∈ R 4 × R 2 \ S, can always be transformed, in a neighborhood of (x 0 , ũ0 ) with ũ0 such that ũℓ,0 = f 1 a (x 0 , u 0 ), a = e, i, and 1 ≤ ℓ ≤ 2, can always be transformed either into:

ẋ1 e = ũ1 ẋ2 e = f 2 e (x) ẋ1 i = ũ2 ẋ2 i = f 2 i (x), (14)
for case 1), or into:

ẋ1 e = ũ1 ẋ2 e = f 2 e (x, ũ) ẋ1 i = ũ2 ẋ2 i = f 2 i (x), (15)
for case 2), with the function f 2 e of the feedback modified system depending explicitly, in a nonlinear way, on at least one control (actually, such that

∂ 2 f 2 e ∂ ũ2 ℓ ̸ ≡ 0, for at least one 1 ≤ ℓ ≤ 2)
and f 2 i depending on x only and remaining unchanged with respect to [START_REF] Martin | A geometric sufficient condition for flatness of systems with m inputs and m+1 states[END_REF], i.e., f 2 i (x) = f 2 i (x), or into:

ẋ1 e = ũ1 ẋ2 e = f 2 e (x, ũ) ẋ1 i = ũ2 ẋ2 i = f 2 i (x, ũ), (16)
for case 3), with each function f 2 e and f 2 i of the feedback modified system depending explicitly on one (and only one) control among ũ1 and ũ2 . More precisely, form ( 16) is such that for 1 ≤ ℓ ′ , ℓ ′′ ≤ 2 with ℓ ′ ̸ = ℓ ′′ , we have f 2 e (x, ũ) = f 2 e (x, ũℓ ′ ) with

∂ 2 f 2 e ∂ ũ2 ℓ ′ ̸ ≡ 0 (but ∂ f 2 e ∂ ũℓ ′′ ≡ 0), and f 2 i (x, ũ) = f 2 i (x, ũℓ ′′ ) with ∂ 2 f 2 i ∂ ũ2 ℓ ′′ ̸ ≡ 0 (but ∂ f 2 i ∂ ũℓ ′ ≡ 0).
Similarly to Section III-A case 1), here as well flatness in case 1) reduces to static feedback linearizability and we actually recover [START_REF] Nicolau | Flatness of networks of two synaptically coupled excitatory-inhibitory neural modules[END_REF]Proposition 3]. Thus all remarks made in the aforementioned paper apply or can be adapted for [START_REF] Kolar | Properties of flat systems with regard to the parameterization of the system variables by the flat output[END_REF], in particular we can render the EI network flat without structural conditions. Let us now turn to case 2). Like for Section III-A case 2), we will identify flat network configurations of form ( 12) when at least one input is not of the form w ae (n, 2) = w ea (2, n), with a ∈ {e, i}, 1 ≤ n ≤ 2, leading to the feedback equivalent form [START_REF] Pomet | A differential geometric setting for dynamic equivalence and dynamic linearization[END_REF] with the function f 2 e depending on one control only (say ũ1 ). Therefore, consider [START_REF] Pomet | A differential geometric setting for dynamic equivalence and dynamic linearization[END_REF] and suppose that

∂ f 2 e ∂ ũ1 ̸ ≡ 0 (implying actually that ∂ 2 f 2 e ∂ ũ2 1 ̸ ≡ 0) and ∂ f 2 e
∂ ũ2 ≡ 0. We will distinguish three mutually exclusive subcases: (A)

∂ f 2 e ∂x 1 i ̸ ≡ 0 but ∂ f 2 i ∂x 1 i ≡ 0, (B) ∂ f 2 e ∂x 1 i ≡ 0 but ∂ f 2 i ∂x 1 i ̸ ≡ 0,
∂ f 2 i ∂x 1 e , ∂ f 2 i ∂x 2 e
)(x 0 ) ̸ = (0, 0), then system (15) is flat at (x 0 , ũ0 ) with either φ = (x 2 e , x 2 i ) or φ = (x 1 e , x 2 i ) a flat output, implying that control EI network (12) is flat at (x 0 , u 0 ) as well. Moreover, system (12) is dynamically linearizable via an invertible 1-fold prolongation. (ii) If f 2 e and f 2 i are in case (B), with

∂ f 2 i ∂x 1 i (x 0 , ũ0 ) ̸ = 0,
and if additionally

∂ f 2 e ∂x 2 i (x 0 , ũ0 ) ̸ = 0, then system (15) is flat at (x 0 , ũ0 ) with φ = (x 1 e , x 2
e ) a flat output implying that control EI network (12) is flat at (x 0 , u 0 ) as well. Moreover, system (12) is dynamically linearizable via an invertible 2fold prolongation. (iii) If f 2 e and f 2 i are in case (C), then (15) is neither x-flat nor (x, u)-flat and thus system (12) is not either.

Sketch of proof: The structural conditions (A) and (B) translate into triangular forms which together with the regularity conditions of (i) and (ii), immediately guarantee that the control network is flat and that it admits the flat outputs given by conditions of (i) and (ii). Moreover, it is clear that it becomes static feedback linearizable via at most 2fold prolongation of ũ1 . If the functions f 2 e and f 2 i are in case (C), then one can show that the system is neither xflat nor (x, u)-flat by applying, for instance, the results of [START_REF] Gstöttner | On the linearization of flat two-input systems by prolongations and applications to control design[END_REF], [START_REF] Kolar | Properties of flat systems with regard to the parameterization of the system variables by the flat output[END_REF] according to which a two-input (x, u)-flat systems can be statically linearized after an at most dim(x)-fold prolongation of a suitably chosen input. □ Notice that for all flat configurations of items (i) and (ii) of the above proposition, we need a structural condition (either condition (A) or condition (B)). If the control EI network satisfies (C) and if it is possible for the system to be flat (so with flat outputs necessarily depending on the derivatives of u), then even if (C) does not contain any structural conditions, flatness will necessarily imply that some interactions are restricted. Let us now consider the case when the connection strengths that are modeled as inputs lead to the control network [START_REF] Nicolau | Flatness of networks of two synaptically coupled excitatory-inhibitory neural modules[END_REF].

Proposition 3: The two-input control EI network given by (13), or equivalently the feedback modified system (16), is not flat.

Sketch of proof: One can use the ruled-manifold criterion [START_REF] Fliess | Flatness and defect of non-linear systems: introductory theory and examples[END_REF], [START_REF] Rouchon | Necessary condition and genericity of dynamic feedback linearization[END_REF] for the feedback modified system, for which f 2 e and f 2 i are such that their second derivatives with respect to ũ are nonzero.

□ Conclusion (two inputs):

If only two connection strengths can be modeled as inputs, the only flat network configuration that we are able to identify without constraining the interactions between the subnetworks (i.e., without structural conditions) are those for which the control network is of form [START_REF] Kolar | Properties of flat systems with regard to the parameterization of the system variables by the flat output[END_REF] and can be transformed into a controlaffine system by feedback transformations. If the connection strengths modeled as inputs yield a network of form [START_REF] Martin | A geometric sufficient condition for flatness of systems with m inputs and m+1 states[END_REF], flat configurations can be identified, they are in a triangular form, and they involve a structural condition. We do not however completely describe flatness of networks of form (12) (i.e., there may be flat control networks of form [START_REF] Martin | A geometric sufficient condition for flatness of systems with m inputs and m+1 states[END_REF], for which neither condition (i) nor condition (ii) of Proposition 2 is satisfied, but if such a flat network exists, then flatness will require structural conditions). If the connection strengths modeled as inputs are such that the network is of form [START_REF] Nicolau | Flatness of networks of two synaptically coupled excitatory-inhibitory neural modules[END_REF], then the control network is never flat. C. Control network with one input We consider next the case when among all connection strengths, only one can be modeled as an input. According to (A5), the input is defined as w ab (n, m), with a ̸ = b or n ̸ = m. Consider the network around a point (x 0 , u 0 ) ∈ R 4 × R \ S, i.e., for which rk ∂f ∂u (x 0 , u 0 ) = 1. Suppose that u = u 1,1 ei = w ei (1, 1) = w ie (1, 1) (all remaining cases are similar and we do not treat them here), implying in particular that x 0 is such that (x 1 e,0 , x 1 i,0 ) ̸ = (0, 0) and assume additionally that x 1 e,0 ̸ = 0. The control network is of the form:

ẋ1 e = f 1 e (x, u) ẋ2 e = f 2 e (x) ẋ1 i = f 1 i (x, u) ẋ2 i = f 2 i (x), (17)
and by applying the invertible (nonlinear) feedback transformation ũ1 = f 1 e (x, u), it can be transformed into:

ẋ1 e = ũ1 ẋ2 e = f 2 e (x) ẋ1 i = f 1 i (x, ũ1 ) ẋ2 i = f 2 i (x), (18)
with f 1 i depending explicitly on ũ1 and such that

∂ 2 f 1 i ∂ ũ2
1 ̸ = 0. Proposition 4: The single-input control EI network given by [START_REF] Waldherr | Flat inputs in the mimo case[END_REF], or equivalently the feedback modified system (18), is not flat.

Sketch of proof: By [START_REF] Charlet | Sufficient conditions for dynamic state feedback linearization[END_REF], the single-input control network is flat if and only if it is static feedback linearizable, that is, satisfies the necessary and sufficient conditions of [START_REF] Jakubczyk | On linearization of control systems[END_REF]. Consider the feedback modified system [START_REF] Wilson | Excitatory and inhibitory interactions in localized populations of model neurons[END_REF]. Since

∂ 2 f 1 i ∂ ũ2 1 ̸ = 0,
it can be easily shown that the structural condition (the involutivity of some distributions) of [START_REF] Jakubczyk | On linearization of control systems[END_REF] is not verified, implying that ( 18) is not static feedback linearizable thus not flat, and consequently [START_REF] Waldherr | Flat inputs in the mimo case[END_REF] is not flat either. □ Conclusion (N = 2, one input): If the input is defined following Assumption (A5), then the single-input control network is never flat. If we drop Assumption (A5), and we define the control by u = u n,n aa = w aa (n, n), with a = e or a = i, then we recover the results of [START_REF] Nicolau | Flatness of networks of two synaptically coupled excitatory-inhibitory neural modules[END_REF] according to which it is possible to render the control network flat but in order to do so, we always need structural conditions for the interactions between the subnetworks and even for the local interactions within the same subnetwork.

2 e∂ ũ1 ̸ ≡ 0 and ∂ f 2 e∂ 2 e ∂x 1 i

 2221 Notice that both cases (A) and (B) necessarily impose structural conditions on the interactions between subnetworks.Proposition 2: Consider the two-input control EI network given by (12) around (x 0 , u 0 ) ∈ R 4 × R 2 \ S, and its feedback equivalent network (15) around (x 0 , ũ0 ). Suppose that f 2 e depends on one and only one control, say ũ1 (i.e.,∂ f ũ2 ≡ 0).The following statements hold: (i) If f 2 e and f 2 i are in case (A), with ∂ f (x 0 , ũ0 ) ̸ = 0, and if additionally (


I Camilier

L Decreusefond
email: laurent.decreusefond@telecom-paristech.fr

QUASI-INVARIANCE AND INTEGRATION BY PARTS FOR DETERMINANTAL AND PERMANENTAL PROCESSES

Keywords: . 1991 Mathematics Subject Classification. 60H07, 60G55 Determinantal point processes, Malliavin calculus, permanental point processes, point processes, integration by parts formula, random thinning

Determinantal and permanental processes are point processes with a correlation function given by a determinant or a permanent. Their atoms exhibit mutual attraction of repulsion, thus these processes are very far from the uncorrelated situation encountered in Poisson models. We establish a quasi-invariance result : we show that if atoms locations are perturbed along a vector field, the resulting process is still a determinantal (respectively permanental) process, the law of which is absolutely continuous with respect to the original distribution. Based on this formula, following Bismut approach of Malliavin calculus, we then give an integration by parts formula.

Motivations

Point processes are widely used to model various phenomena, such as arrival times, arrangement of points in space, etc. It is thus necessary to know into details as large a catalog of point processes as possible. The Poisson process is one example which has been widely studied for a long time. Our motivation is to study point processes that generate a more complex correlation structure, such as a repulsion or attraction between points, but still remain simple enough so that their mathematical properties are analytically tractable. Determinantal and permanental point processes hopefully belong to this category. They were introduced in [START_REF] Macchi | The coincidence approach to stochastic point processes[END_REF] in order to represent configurations of fermions and bosons. Elementary particles belong exclusively to one of these two classes. Fermions are particles like electrons or quarks; they obey the Pauli exclusion principle and hence the Fermi-Dirac statistics. The other sort of particles are particles like photons which obey the Bose-Einstein statistics. The interested reader can find in [START_REF] Tamura | A canonical ensemble approach to the fermion/boson random point processes and its applications[END_REF] an illuminating account of the determinantal (respectively permanental) structure of fermions (respectively bosons) ensemble. A mathematical unified presentation of determinantal/permanental point processes (DPPP for short) was for the first time, introduced in [START_REF] Shirai | Random point fields associated with certain Fredholm determinants. I. Fermion, Poisson and boson point processes[END_REF]. Let χ be the space of locally finite, simple configurations on a Polish space E and K a locally trace-class operator in L 2 (E) with a Radon measure λ. For any positive, compactly supported f and ξ = j δ x j ∈ χ, the α-DPPP is the measure, µ α, K, λ , on χ such that (1)

χ e - R f dξ dµ α,K,λ (ξ) = Det I +α 1 -e -f K 1 -e -f -1 α ,
for the parameters α ∈ A = {2/m; m ∈ N}∪{-1/m, , ∈ N}, where N is the set of positive integers. The values α = -1 and α = 1 correspond to determinantal and permanental point processes respectively. Starting from [START_REF] Aigner | Combinatorial theory[END_REF], existence of α-DPPP is still a challenge as explained in [START_REF] Soshnikov | Determinantal random point fields[END_REF]. Actually, existence is (not easily) proved for α = ±1 and DPPP for other values of α are constructed as superposition of these basic processes. DPPP recently regained interest because they have strong links with the spectral theory of random matrices [START_REF]Random matrices and determinantal processes[END_REF][START_REF] Soshnikov | Determinantal random point fields[END_REF]: for instance, eigenvalues of matrices in the Ginibre ensemble a.s. form a determinantal configuration. DPPP also appear in polynuclear growth [START_REF]Discrete polynuclear growth and determinantal processes[END_REF][START_REF] Johansson | Toeplitz determinants, random growth and determinantal processes[END_REF], non intersecting random walks, spanning trees, zero set of Gaussian analytic functions (see [START_REF] Hough | Determinantal processes and independence[END_REF] and references there in), etc. Mathematically speaking, a few of their properties are known. The most complete references to date are, to the best of our knowledge, [START_REF] Hough | Determinantal processes and independence[END_REF] and [START_REF] Shirai | Random point fields associated with certain Fredholm determinants. I. Fermion, Poisson and boson point processes[END_REF]. Gibbsianness of DPPP, i.e., local absolute continuity of µ α,K,λ with respect to the distribution of a Poisson process, was investigated in several papers by Yoo [START_REF] Yoo | Gibbsianness of fermion random point fields[END_REF][START_REF] Georgii | Conditional intensity and Gibbsianness of determinantal point processes[END_REF]. The conclusion of all these studies seems to be that DPPP are rather hard to describe and analyze, their properties being highly dependent of the kernel and its eigenvalues.

Our aim is to investigate further some of the stochastic properties of α-DPPP. In the spirit of [START_REF] Yoo | Dirichlet forms and diffusion processes for fermion random point fields[END_REF], we are interested in the differential calculus associated to these processes. In [START_REF] Yoo | Dirichlet forms and diffusion processes for fermion random point fields[END_REF], it is shown that a somewhat canonical Dirichlet form associated to DPPP is closed. We here address the problem within the point of view of Malliavin calculus. To date, Malliavin calculus for point processes has been developed namely for Poisson processes ( [START_REF] Bichteler | Calcul de Malliavin pour les diffusions avec sauts : existence d'une densité dans le cas unidimensionnel[END_REF][START_REF] Bass | The Malliavin calculus for pure jump processes and applications to local time[END_REF][START_REF] Bouleau | Energy image density property and the lent particle method for Poisson measures[END_REF][START_REF] Albeverio | Analysis and geometry on configuration spaces[END_REF][START_REF] Privault | Stochastic analysis in discrete and continuous settings with normal martingales[END_REF][START_REF] Decreusefond | Anticipative calculus with respect to filtered Poisson processes[END_REF]) and some of their extensions: Gibbs processes [START_REF]Analysis and geometry on configuration spaces: the Gibbsian case[END_REF], marked processes [START_REF] Albeverio | Analysis and geometry on marked configuration spaces, Infinite dimensional harmonic analysis[END_REF], filtered Poisson processes [START_REF] Decreusefond | Anticipative calculus with respect to filtered Poisson processes[END_REF], cluster processes [START_REF] Bogachev | Analysis on configurations spaces and Gibbs cluster ensembles[END_REF] and Lévy processes [START_REF] Bally | Integration by parts formula for locally smooth laws and applications to sensitivity computations[END_REF][START_REF] Di Nunno | White noise analysis for Lévy processes[END_REF]. There exist three approaches to construct a Malliavin calculus framework for point processes: one based on white noise analysis, one based on a difference operator and chaos decomposition and one which relies on quasi-invariance of the law of Poisson process with respect to some perturbations. This is the last track we follow here since neither the white noise framework nor the chaos decomposition exist so far.

We first show that the action of a diffeomorphism of E into itself onto the atoms of a DPPP yields another DPPP, the law of which is absolutely continuous with the distribution of the original process; a property usually known as quasi-invariance. Then, following the lines of proof of [START_REF] Albeverio | Analysis and geometry on configuration spaces[END_REF][START_REF] Bismut | Calcul des variations stochastiques et processus de sauts[END_REF][START_REF] Bogachev | Analysis on configurations spaces and Gibbs cluster ensembles[END_REF]; we can derive an integration by parts formula for the differential gradient as usually constructed on configuration spaces. This gives another proof of the closability of the Dirichlet form canonically associated to a DPPP as in [START_REF] Yoo | Dirichlet forms and diffusion processes for fermion random point fields[END_REF].

This paper is organized as follows. In part 2, we give definitions concerning point processes and α-determinantal point processes. In part 3, we prove the quasi-invariance for α-DPPP. Then, in Section 4, we compute the integration by parts formula. We begin by determinantal point processes and then extend to α-determinantal point processes. Permanental processes are analyzed on the same basis.

Preliminaries

2.1. Point processes. We remind here some properties of point processes we refer to [START_REF] Daley | An introduction to the theory of point processes[END_REF][START_REF] Kallenberg | Random measures[END_REF] for more details. Let E be a Polish space and λ a Radon measure on (E, B), the Borel σ-algebra on E. By χ we denote the space of all locally finite configurations on E:

χ = {ξ ⊂ E : |ξ ∩ Λ| < ∞ for any compact Λ ⊂ E},
where |A| is the cardinality of a set A. Hereafter we identify a locally finite configuration ξ, defined as a set, and the atomic measure x∈ξ δ x . The space χ is then endowed with the vague topology of measures and B(χ) denotes the corresponding Borel σ-algebra. For any measurable nonnegative function f on E, we denote equivalently:

f, ξ = x∈ξ f (x) = f dξ.
We also denote by χ 0 = {α ∈ χ, | α(E) |< ∞} the set of all finite configurations in χ and χ 0 is equipped with the σ-algebra B(χ 0 ). The restriction of a configuration ξ to a compact Λ ⊂ E, is denoted by ξ Λ . We introduce the set χ Λ = {ξ ∈ χ, ξ(E\Λ) = 0}. Then for any integer n, we denote by χ

(n) Λ = {ξ ∈ χ, ξ(Λ) = n}, the set of all configurations in with n points in Λ. Note that we have χ Λ = ∞ n=0 χ (n)
Λ . Definition 1. A random point process is a triplet (χ, B(χ), µ), where µ is a probability measure on (χ, B(χ)).

Every measure µ on the configuration space χ can be characterized by its Laplace function, that is to say for any measurable non-negative function f on E:

f -→ E µ [e - R f dξ ] = χ e - R f dξ dµ(ξ).
For instance, let π σ denote the Poisson measure on (χ, B(χ)) with intensity measure σ. Then its Laplace transform is, for any measurable non-negative function f :

χ e - R f dξ dπ σ (ξ) = exp E (1 -e -f (x) ) dσ(x)
Another way to describe the distribution of a point process is to give the probabilities P(|ξ

Λ k | = n k , 1 ≤ k ≤ n) for any n and any mutually disjoints Borel subsets of Λ, Λ 1 , • • • , Λ k , 1 ≤ k ≤ n.
For instance, the Poisson measure π σ with intensity measure σ can be defined in this way as:

P(|ξ Λ k | = n k , 1 ≤ k ≤ n) = n k=1 e -σ(Λ k ) σ(Λ k ) n k n k ! .
But in many cases, specifying the joint distribution of the ξ(D)'s is not that simple. It is then easier to describe the distribution of a point process by its correlation functions.

Definition 2. A locally integrable function ρ

n : E n → R + is the n-point cor- relation function of µ if for any disjoint bounded Borel subsets Λ 1 , • • • , Λ m of E and n i ∈ N, m i=1 n i = n: E µ m i=1 |ξ Λ i |! (|ξ Λ i | -n i )! = Λ n 1 1 ×...×Λ nm m ρ n (x 1 , • • • , x n ) dλ(x 1 ) . . . dλ(x n ),
where E µ denotes the expectation relatively to µ.

For example, if m = 1 and n 1 = n, the formula becomes:

E µ |ξ Λ |! (|ξ Λ | -n)! = E µ [|ξ Λ | (|ξ Λ | -1) . . . (|ξ Λ | -n + 1)] = Λ n ρ n (x 1 , • • • , x n ) dλ(x 1 ) . . . dλ(x n ).
We recognize here the n-th factorial moment of |ξ Λ |. In particular:

E µ [|ξ Λ |] = Λ ρ 1 (x) dλ(x),
i.e., ρ 1 is the mean density of particles. More generally, the function ρ n has the following interpretation:

ρ n (x 1 , • • • , x n ) dλ(x 1 ) . . . dλ(x n ) is approxi- mately the probability to find a particle in each one of the [x i , x i + dλ(x i )], i = 1, • • • , n.
A third way to define a point process proceeds via the Janossy densities. Denote by π n,Λ (x 1 , • • • , x n ) the density (assumed to exist) with respect to λ ⊗n of the joint distribution of (x 1 , • • • , x n ) given that there are n points in Λ.

Definition 3. The density distributions or Janossy densities of a random process µ are the measurable functions j n Λ such that:

j n Λ (x 1 , • • • , x n ) = n! µ(ξ(Λ) = n) π n,Λ (x 1 , • • • , x n ) for n ∈ N, j 0 Λ (∅) = µ(ξ(Λ) = 0).
Hence, the Janossy density j n Λ (x 1 , • • • , x n ) appears as the probability density that there are exactly n points in Λ located around x 1 , • • • , x n , and no points anywhere else. For n = 0, j 0 Λ (∅) is the probability that there is no point in Λ. For n ≥ 1, the Janossy densities satisfy the following properties:

• Symmetry:

j n Λ x σ(1) , • • • , x σ(n) = j n,Λ (x 1 , • • • , x n ) , for every permutation σ of {1, • • • , n}. • Normalization constraint. For each compact Λ: +∞ n=0 Λ n 1 n! j n Λ (x 1 , • • • , x n ) dλ(x 1 ) . . . dλ(x n ) = 1.
It is clear that the ρ n 's, j n 's, µ should satisfy some relationships. We will not dwell on that here (see the references cited above), we just mention the relation between µ and j n Λ , which is:

(2)

χ f (ξ) dµ(ξ) = +∞ n=0 1 n! Λ n f (x 1 , • • • , x n ) j n Λ (x 1 , • • • , x n ) dλ(x 1 ) . . . dλ(x n ).
2.2. Fredholm determinants. For details on this part, we refer to [START_REF] Dunford | Linear operators. part ii[END_REF][START_REF] Simon | Trace ideals and their applications[END_REF].

For any compact Λ ⊂ E, we denote by L 2 (Λ, λ) the set of functions square integrable with respect to the restriction of the measure λ to the set Λ. This becomes a Hilbert space when equipped with the usual norm:

f 2 L 2 (λ,Λ) = Λ |f (x)| 2 dλ(x).
For Λ a compact subset of E, P Λ is the projection from L 2 (E) onto L 2 (Λ), i.e., P Λ f = f 1 Λ . The operators we will deal with are special cases of the general category of continuous maps from L 2 (E, λ) into itself.

Definition 4. A map T from L 2 (E) into itself is said to be an integral operator whenever there exists a measurable function, we still denote by T , such that

T f (x) = E T (x, y)f (y) dλ(y).
The function T is called the kernel of T .

Definition 5. Let T be a bounded map from L 2 (E, λ) into itself. The map T is said to be trace-class whenever for one complete orthonormal basis (CONB for short)

(h n , n ≥ 1) of L 2 (E, λ), n≥1 |(T h n , h n ) L 2 | is finite.
Then, the trace of T is defined by

trace(T ) = n≥1 (T h n , h n ) L 2 .
It is easily shown that the notion of trace does not depend on the choice of the CONB. Note that if T is trace-class then T n also is trace-class for any n ≥ 2. Definition 6. Let T be a trace class operator. The Fredholm determinant of (I +T ) is defined by:

Det(I +T ) = exp +∞ n=1 (-1) n-1 n trace(T n ) ,
where I stands for the identity operator.

The practical computations of fractional power of Fredholm determinants involve the so-called α-determinants, which we introduce now. Definition 7. For a square matrix A = (a ij ) i,j=1...n of size n × n, the αdeterminant det α A is defined by:

det α A = σ∈Σn α n-ν(σ) n i=1 a iσ(i) ,
where the summation is taken over the symmetric group Σ n , the set of all permutations of {1, 2, • • • , n} and ν(σ) is the number of cycles in the permutation σ. This is actually a generalization of the well-known determinant of a matrix. Indeed, when α = -1, det -1 A is the usual determinant det A. When α = 1, det 1 A is the so-called permanent of A and for α = 0, det 0 A = i a ii . We can then state the following useful theorem (see [START_REF] Shirai | Random point fields associated with certain Fredholm determinants. I. Fermion, Poisson and boson point processes[END_REF]):

Theorem 1. For a trace class integral operator T , if αT < 1, we have:

Det(I -αT ) -1 α = +∞ n=0 1 n! Λ n det α (T (x i , x j )) 1≤i,j≤n dλ(x 1 ) . . . dλ(x n ). If α ∈ {-1/m; m ∈ N},
Kf (x) = E K(x, y)f (y) dλ(y).
ii) The spectrum of K is included [0, 1[. iii) The map K is locally of trace class, i.e., for all compact Λ ⊂ E, the restriction

K Λ = P Λ KP Λ of K to L 2 (Λ) is of trace class.
For a real α ∈ [-1, 1] and a compact subset Λ ⊂ E, the map J Λ,α is defined by: J Λ,α = (I +αK Λ ) -1 K Λ , so that we have:

(I +αK Λ ) (I -αJ Λ,α ) = I .
For any compact Λ, the operator J Λ,α is also a trace class operator in L 2 (Λ, λ). In the following theorem, we define α-DPPP with the three equivalent characterizations: in terms of their Laplace transforms, Janossy densities and correlation functions. The theorem is also a theorem of existence, a problem which as said above is far from being trivial.

Theorem 2 (See [START_REF] Shirai | Random point fields associated with certain Fredholm determinants. I. Fermion, Poisson and boson point processes[END_REF]). Assume Hypothesis 1 is satisfied. Let α ∈ A. There exists a unique probability measure µ α, K, λ on the configuration space χ such that, for any nonnegative bounded measurable function f on E with compact support, we have:

E µ α, K, λ e - R f dξ = χ e - R f dξ dµ α, K, λ (ξ) = Det I +αK[1 -e -f ] -1 α , (3)
where K[1e -f ] is the bounded operator on L 2 (E) with kernel :

(K[1 -e -f ])(x, y) = 1 -exp(-f (x))K(x, y) 1 -exp(-f (y)).

This means that for any integer n and any

(x 1 , • • • , x n ) ∈ E n ,
the correlation functions of µ α, K, λ are given by:

ρ n, α, K (x 1 , • • • , x n ) = det α (K (x i , x j )) 1≤i,j≤n ,
and for n = 0, ρ 0, α, K (∅) = 1. For any compact Λ ⊂ E, the operator J Λ, α is an Hilbert-Schmidt, trace class operator, whose spectrum is included in [0, +∞[. For any n ∈ N, any compact Λ ⊂ E, and any

(x 1 , • • • , x n ) ∈ Λ n
the n-th Janossy density is given by:

(4) j n Λ,α,K Λ (x 1 , • • • , x n ) = Det (I +αK Λ ) -1/α det α (J Λ, α (x i , x j )) 1≤i,j≤n .
For n = 0, we have j n Λ,α,K Λ (∅) = Det (I + αK Λ ) -1/α . For α = -1, such a process is called a determinantal process since we have, for any n ≥ 1:

ρ n,-1,K (x 1 , • • • , x n ) = det(K(x i , x j )) 1≤i,j≤n .
For α = 1, such a process is called a permanental process, since we have, for any n ≥ 1:

ρ n,1,K (x 1 , • • • , x n ) = πǫΣ n i=1 K(x i , x π(i) ) = per (K(x i , x j )) 1≤i,j≤n .
For any bounded function g : E → R + , and any integral operator T of kernel T (x, y), we denote by T [g] the integral operator of kernel:

T [g](x, y) → g(x)T (x, y) g(y).
For calculations, it will be convenient to use the following lemma: Lemma 1 (see [START_REF] Shirai | Random point fields associated with certain Fredholm determinants. I. Fermion, Poisson and boson point processes[END_REF]). Let Λ be a compact subset of E and f : E → [0, +∞), measurable with supp(f ) ∈ Λ:

Det I +αK Λ [1 -e -f ] -1/α = Det (I +αK Λ ) -1/α Det I -αJ Λ,α [e -f ] -1/α
. By differentiation into the Laplace transform, it is possible to compute moments of f dξ for any deterministic f . We obtain, at the first order: Theorem 3 (see [START_REF] Shirai | Random point fields associated with certain Fredholm determinants. I. Fermion, Poisson and boson point processes[END_REF]). For any non-negative function f defined on E, we have

E Λ f dξ = Λ f (x)K(x, x) dλ(x) = trace(K Λ [f ]).
It is worth mentioning how the existence of α-DPPP is established. For α = -1, there is a non trivial work (see [START_REF] Shirai | Random point fields associated with certain Fredholm determinants. I. Fermion, Poisson and boson point processes[END_REF][START_REF] Soshnikov | Determinantal random point fields[END_REF] and references therein) to show that the Janossy densities satisfy the positivity condition so that a point process with these densities does exist. For α = -1/m, it is sufficient to remark from (3) that the superposition of m independent determinantal point processes of kernel K/m is an α-DPPP fo kernel K. The point is that K/m satisfies Hypothesis 1, in particular that its spectrum is strictly bounded by 1/m < 1, since m > 1. For α = 2, a 2-permanental point process is in fact a Cox process based on a Gaussian random field. We know for sure that there exists X a centered Gaussian random field on E such that:

(5)

E P Λ X 2 (x) dλ(x) = trace(K Λ ),
for any compact Λ ⊂ E and ( 6)

E P [X(x)X(y)] = K(x, y) λ ⊗ λ a.s.,
where P is the probability measure on the probability space supporting X. Then the Cox process of random intensity X 2 (x) dλ(x) has the same distribution as µ 2,K,λ . Indeed, it follows from the formula:

E P exp -(1 -e -f (x) )X 2 (x) dλ(x) = Det(I + 2(1 -e -f )K) -1/2 .
Thus, any 2/m-permanental point process is the superposition of m independent 2-permanental point processes with kernel K/m. Poisson process can be obtained formally as extreme case of 1-permanental process with a kernel K given by K(x, y) = 1 {x=y} . Of course, this kernel is likely to be null almost surely with respect to λ ⊗ λ; nonetheless, it remains that replacing formally this expression in (3) yields the Laplace transform of a Poisson process of intensity λ. Another way to retrieve a Poisson process is to let α go to 0 in [START_REF] Albeverio | Analysis and geometry on configuration spaces[END_REF]. With the above constructions, this means that a Poisson process can be viewed as an infinite superposition of determinantal or permanental point processes.

Theorem 4. When α tends to 0, µ α,K,λ converges narrowly to a Poisson measure of intensity K(x, x) dλ(x).

Proof. For any nonnegative f , for any n ≥ 1,

0 ≤ trace (K Λ [1 -e -f ]) n ≤ trace K Λ [1 -e -f ] , hence, (7) χ exp -f dξ dµ α,K Λ ,λ (ξ)= Det I +αK Λ [1 -e -f ] -1/α = exp - 1 α +∞ n=1 (-1) n-1 n α n trace((K Λ [1 -e -f ]) n ) α→0 ---→ exp -trace(K Λ (1 -e -f )) = E (1 -e -f (x) )K Λ (x, x)dλ(x).
Thus, when α goes to 0, the measure µ α,K Λ ,λ tends towards a measure that we call µ 0,K Λ ,λ . According to [START_REF] Bichteler | Calcul de Malliavin pour les diffusions avec sauts : existence d'une densité dans le cas unidimensionnel[END_REF], µ 0,K Λ ,λ is a Poisson process with intensity K Λ (x, x)dλ(x).

Quasi-invariance

In this part we show the quasi-invariance property for any α-DPPP. Let Diff 0 (E) be the set of all diffeomorphisms from E into itself with compact support, i.e., for any φ ∈ Diff 0 (E), there exists a compact Λ outside which φ is the identity map. For any ξ ∈ χ, we still denote by φ the map:

φ : χ -→ χ x∈ξ δ x -→ x∈ξ δ φ(x) .
For any reference measure λ on E, λ φ denotes the image measure of λ by φ. For φ ∈ Diff 0 (E) whose support is included in Λ, we introduce the isometry Φ,

Φ : L 2 (λ φ , Λ) -→ L 2 (λ, Λ) f -→ f • φ.
Its inverse, which exists since φ is a diffeomorphism, is trivially defined by f • φ -1 and denoted by Φ -1 . Note that Φ and Φ -1 are isometries, i.e.,

Φψ 1 , Φψ 2 L 2 (λ,Λ) = ψ 1 , ψ 2 L 2 (λ φ ,Λ) ,
for any ψ 1 and ψ 2 belonging to L 2 (λ, Λ). We also set:

K φ Λ = Φ -1 K Λ Φ and J φ Λ,α = Φ -1 J Λ,α Φ. Lemma 2.
Let λ be a Radon measure on E and K a map satisfying hypothesis 1. Let α ∈ A. We have the following properties. a) K φ Λ and J φ Λ,α are continuous operators from

L 2 (λ φ , Λ) into L 2 (λ φ , Λ). b) K φ Λ is of trace class and trace(K φ Λ ) = trace(K Λ ). c) Det(I +αK φ Λ ) = Det(I +αK Λ ).
Proof. The first point is immediate according to the definition of an image measure. Since Φ -1 is an isometry, for any (ψ n , n ∈ N) a complete orthonormal basis of L 2 (λ, Λ), the family (Φ

-1 ψ n , , ∈ N) is a CONB of L 2 (λ φ , Λ). Moreover, n≥1 K φ Λ Φ -1 ψ n , Φ -1 ψ n L 2 (λ φ ,Λ) = n≥1 Φ -1 KΦΦ -1 ψ n , Φ -1 ψ n L 2 (λ φ ,Λ) = n≥1 Φ -1 Kψ n , Φ -1 ψ n L 2 (λ φ ,Λ) = n≥1 Kψ n , ψ n L 2 (λ,Λ) .
Hence, K φ Λ is of trace class and trace(K φ Λ ) = trace(K Λ ). Along the same lines, we prove that trace((K φ Λ ) n ) = trace(K n Λ ) for any n ≥ 2. According to Definition 6, the Fredholm determinant of K φ Λ is well defined and point c) follows.

Theorem 5. Assume that K is a kernel operator. Then K φ Λ , as a map from L 2 (λ φ , Λ) into itself is a kernel operator whose kernel is given by ((x, y) → K Λ (φ -1 (x), φ -1 (y))). An analog formula also holds for the operator J Λ,α .

Proof. On the one hand, for any function f , the operator K φ Λ from L 2 (Λ, λ φ ) into L 2 (Λ, λ φ ) is given by :

K φ Λ f (x) = Λ K φ Λ (x, z)f (z)dλ φ (z).
On the other hand, using the definition

K φ Λ = Φ -1 K Λ Φ K φ Λ f (x) = Φ -1 K Λ Φf (x) = Λ K Λ (φ -1 (x), y)f • φ(y)dλ(y) = Λ K Λ (φ -1 (x), φ -1 (z))f (z)dλ φ (z).
The proof is thus complete.

Lemma 3. Let ρ : E → R be non negative and assume that dλ = ρ dm for some other Radon measure on E. Let K satisfy Hypothesis 1. Then, we have the following properties:

(1) The map

K[ρ] is continuous from L 2 (m) into itself. (2) The map K[ρ] is locally trace class and trace(K Λ [ρ]) = trace(K Λ ).
(3) The measure µ α,K,λ is identical to the measure µ α,K[ρ],m . That is to say, in some sense, we can "transfer" a part of the reference measure into the operator and vice-versa.

Proof. Remember that

K[ρ](x, y) = ρ(x) K(x, y) ρ(y). Hence K Λ [ρ]f (x) = ρ(x) Λ K Λ (x, y) ρ(y) dλ(y), thus Λ |K Λ [ρ]f | 2 dm = Λ |K Λ f | 2 dλ,
and the first point follows.

Consider (ψ n , n ∈ N), a CONB of L 2 (λ). Then (ψ n √ ρ, n ∈ N) is a CONB of L 2 (m)
. Furthermore, we have:

n≥1 K Λ [ρ]ψ n , ψ n L 2 (dm) = n≥1 K Λ √ ρψ n , √ ρψ n L 2 (dm) = n≥1 K Λ ψ n , ψ n L 2 (λ) .
Therefore the operator K Λ [ρ] is of trace class and

trace(K Λ [ρ]) = trace(K Λ ).
Similarly we can prove that for any n ≥ 2, we have trace(K n Λ [ρ]) = trace(K n Λ ). Then, using the definition of a Fredholm determinant, we have:

Det(I +αK Λ ) = Det(I +αK Λ [ρ]).
The third point then follows from the characterization of µ α,K[ρ],m by its Laplace transform.

The expression det

α J Λ,α (x i , x j ) 1≤i,j≤n is now denoted det α J Λ,α (x 1 , • • • , x n ).
For any finite random configuration ξ = (x 1 , • • • , x n ), we call J Λ,α (ξ) the matrix with terms (J Λ,α (x i , x j ), 1 ≤ i, j ≤ n). First, remind some results from [START_REF] Albeverio | Analysis and geometry on configuration spaces[END_REF] concerning Poisson measures. For any φ ∈ Diff 0 (E), we define φ * π λ as the image of the Poisson measure π λ with intensity measure λ and λ φ denotes the image measure of λ by φ.

Theorem 6 ([3]

). For any φ ∈ Diff 0 (E), and a Poisson measure π λ with intensity λ:

φ * π λ = π λ φ .
That is to say, for any f nonnegative and compactly supported on E:

(8) E π λ e - R f •φ dξ = exp -1 -e -f dλ φ .
We give the corresponding formula for α-determinantal measures. For any φ ∈ Diff 0 (E), we define φ * µ α,K Λ ,λ as the image of the measure µ α,K Λ ,λ under φ. We prove below that this image measure is an α-DPPP the parameters of which are explicitely known.

Theorem 7. With the notations and hypothesis introduced above. For any φ ∈ Diff 0 (E), for any nonnegative function f on E, for any compact Λ ⊂ E, we have:

E µ α,K Λ ,λ e - R f •φ dξ = E µ α,K φ Λ ,λ φ e - R f dξ (9)
= Det(I + αK φ Λ [1e -f ]) -1/α . That is to say the image measure of µ α,K,λ by φ is an α-determinantal process with operator K φ and reference measure λ φ .

Proof. According to Theorem 2 and Lemma 1, we have for nonnegative f :

E µ α,K Λ ,λ e - R f •φ dξ = Det I +αK Λ [1 -e -f •φ ] -1/α = Det (I +αK Λ ) -1/α Det I -αJ Λ,α [e -f •φ ] -1/α . According to Theorem 1, we get Det I -αJ Λ,α [e -f •φ ] -1/α = +∞ n=0 1 n! Λ n det α J Λ,α (x 1 , • • • , x n )e - P n i=1 f (φ(x i )) dλ(x 1 ) . . . dλ(x n ) = +∞ n=0 1 n! Λ n det α J φ Λ,α (x 1 , • • • , x n )e - P n i=1 f (x i ) dλ φ (x 1 ) . . . dλ φ (x n ) = Det I -αJ φ Λ,α [e -f ] -1/α .
Since Det (I +αK Λ ) = Det I +αK φ Λ , we have:

E µ α,K Λ ,λ e - R f •φ dξ = Det I +αK φ Λ [1 -e -f ] -1/α = E µ α, K φ Λ , λ φ e - R f dξ .
The proof is thus complete.

For α = 2, Theorem 7 says that the image under φ of a Cox process is still a Cox process of parameters K φ Λ and λ φ . Such a process can be constructed as follows: Let X be a centered Gaussian random field satisfying ( 5) and ( 6) and let Y (x) = X(φ -1 (x)). Then, according to Lemma 2, we have: for any compact Λ,

E P Λ Y 2 (x) dλ φ (x) = trace(K φ Λ )
and

E P [Y (x)Y (y)] = K φ (x, y) = K(φ -1 (x), φ -1 (y)), λ φ ⊗ λ φ , a.s..
From Theorem 6, by conditioning with respect to X, we also have:

E µ 2, K, λ e - R f •φ dξ = E P E e - R f •φ dξ X = E P exp -(1 -e -f •φ ) X 2 dλ = E P exp -(1 -e -f ) Y 2 dλ φ .
Thus the two approaches (fortunately) yields the same result.

We now want to prove that µ α, K φ , λ φ is absolutely continuous with respect to µ α, K, λ and compute the corresponding Radon-Nikodym derivative. For technical reasons, we need to assume that there exists a Jacobi formula (or change of variable formula) on the measured space (E, λ). This could be done in full generality for E a manifold; for the sake of simplicity, we assume hereafter that E is a domain of some R d . We denote by ∇ E the usual gradient on R d . We also introduce a new hypothesis.

Hypothesis 2. We suppose that the measure λ is absolutely continuous with respect to the Lebesgue measure m on E. We denote by ρ the Radon-Nikodym derivative of λ with respect to m. We furthermore assume that √ ρ is in H 1,2 loc (K(x, x) dm(x)), i.e., ρ is weakly differentiable and for any compact Λ in E, we have:

∞ > 2 Λ ∇ E ρ(x) 2 K(x, x) dm(x) = Λ ∇ E ρ(x) 2 ρ(x) K(x, x) dm(x) = Λ ∇ E ρ(x) ρ(x) 2 K(x, x) dλ(x).
Then for any φ ∈ Diff 0 (E), λ φ is absolutely continuous with respect to λ and

p λ φ (x) = dλ φ (x) dλ(x) = ρ(φ -1 (x)) ρ(x) Jac(φ)(x),
where Jac(φ)(x) is the Jacobian of φ at point x.

Lemma 4. Assume (E, K, λ) satisfy Hypothesis 1 and 2. Let (u n , n ≥ 0) be a sequence of nonnegative real numbers such that for any x ∈ R,

n≥0 u n n! |x| n < +∞. (10)
For any compact Λ ⊂ E, we have:

(11) E µ α, K Λ , λ u |ξ| det α J Λ,α (ξ) < +∞.
As a consequence, det α J Λ,α (ξ) is µ α, K Λ , λ almost-surely positive.

Proof. According to Theorem 2, we have:

j n Λ,α,K Λ (x 1 , • • • , x n ) = Det(I +αK Λ ) -1/α det α J Λ,α (x 1 , • • • , x n ), hence E u |ξ| det α J Λ,α (ξ) = +∞ n=0 1 n! Λ n u n det α J Λ,α (x 1 , • • • , x n ) j n Λ,α,K Λ (x 1 , • • • , x n ) ⊗ n j=1 dλ(x j ) = Det(I +αK Λ ) -1/α +∞ n=0 u n n! λ(Λ) n < +∞,
because λ is assumed to be a Radon measure and Λ is compact.

Theorem 8. Assume (E, K, λ) satisfy Hypothesis 1 and 2. Then, the measure µ α, K, λ is quasi-invariant with respect to the group Diff 0 (E) and for any φ ∈ Diff 0 (E), we have then:

dφ * µ α, K, λ dµ α, K, λ (ξ) =   x∈ξ p λ φ (x)   det α J φ α (ξ) det α J α (ξ)
•

That is to say that for any measurable nonnegative, compactly supported f on E:

(12) E µ α, K, λ e - R f •φ dξ = E µ α, K, λ e - R f dξ e R ln(p λ φ ) dξ det α J φ α (ξ) det α J α (ξ) .
Proof. Since f is compactly supported and φ belongs to Diff 0 (E), there exists a compact Λ which contains both the support of f and f •φ. According to Theorem 7 and Lemma 5, we have:

E µ α,K Λ ,λ e - R f •φ dξ = E µ α,K φ Λ ,λ φ e - R f dξ = Det I +αK φ Λ -1/α +∞ n=0 1 n! A n = Det (I +αK Λ ) -1/α +∞ n=0 1 n! A n
where for any n ∈ N, the A n are the integrals:

A n = Λ n det α J φ Λ,α (x 1 , • • • , x n )e - P n i=1 f (x i ) dλ φ (x 1 ) . . . dλ φ (x n ) = Λ n det α J φ Λ,α (x 1 , • • • , x n )e - P n i=1 f (x i ) n i=1 p λ φ (x i ) dλ(x 1 ) . . . dλ(x n ) = Λ n det α J Λ,α (x 1 , • • • , x n )α n (x 1 , • • • , x n ) dλ(x 1 ) . . . dλ(x n ),
where

α n (x 1 , • • • , x n ) = det α J φ Λ,α (x 1 , • • • , x n ) det α J Λ,α (x 1 , • • • , x n ) e - P i f (x i ) n i=1 p λ φ (x i ).
Hence according to (4), we can write:

Det (I +αK Λ ) -1/α +∞ n=0 1 n! A n = +∞ n=0 1 n! Λ n j n Λ,α,K Λ (x 1 , • • • , x n )α n (x 1 , • • • , x n ) dλ(x 1 ) . . . dλ(x n ).
Thus, we have [START_REF] Daley | An introduction to the theory of point processes[END_REF].

Should we consider Poisson process either as a 0-DPPP or as an α-DPPP with the singular kernel mentioned above, we see that the last fraction in [START_REF] Daley | An introduction to the theory of point processes[END_REF] reduces to 1 and we find the well known formula of quasi-invariance for Poisson processes (see [START_REF] Albeverio | Analysis and geometry on configuration spaces[END_REF]). In the following, we define:

L φ µ α, K, λ (ξ) =   x∈ξ p λ φ (x)   det α J φ α (ξ) det α J α (ξ) •
Then formula [START_REF] Daley | An introduction to the theory of point processes[END_REF] can be rewritten as:

E µ α, K, λ e - R f •φ dξ = E µ α, K, λ e - R f dξ L φ µ α, K, λ (ξ) .

Integration by parts formula

In this section, we prove the integration by parts formula. The proof relies on a differentiation within [START_REF] Daley | An introduction to the theory of point processes[END_REF]. We thus need to put a manifold structure on χ. The tangent space T ξ χ at some ξ ∈ χ is given as L 2 (dξ), i.e., the set of all maps V from E to R such that:

|V (x)| 2 dξ(x) < ∞.
Note that if ξ ∈ χ 0 then T ξ χ can be identified as R |ξ| with the euclidean scalar product.

We consider V 0 (E) the set of all C ∞ -vector fields on E with compact support. For any v ∈ V 0 (E), we construct: φ v t : E → E, t ∈ R, where the curve, for any x ∈ E t ∈ R → φ v t (x) is defined as the solution to:

d dt φ v t (x) = v(φ v t (x)) and φ v 0 (x) = x.
Because v ∈ V 0 (E), there is no explosion and φ v t is well-defined for each t ∈ R. The mappings {φ v t , t ∈ R} form a one-parameter subgroup of diffeomorphisms with compact support, that is to say:

• ∀t ∈ R, φ v t ∈ Diff 0 (E). • ∀t, s ∈ R, φ v t • φ v s = φ v t+s . In particular, (φ v t ) -1 = φ v -t .
• For any T > 0, there exists a compact K such that φ v t (x) = x for any x ∈ K c , for any |t| ≤ T.

In the following, we fix v ∈ V 0 (E). For any ξ ∈ χ, we still denote by φ v t the map:

φ v t : χ -→ χ ξ = x∈ξ δ x i -→ x∈ξ δ φ v t (x) ∈ χ.
Definition 8. A function F : χ → R is said to be differentiable at ξ ∈ χ whenever for any vector field v ∈ V 0 (E), the directional derivative along the vector field v

∇ v F (ξ) = d dt F (φ v t (ξ))
t=0 is well defined.

Since φ v t does not change the number of atoms of ξ, if ξ belongs to χ 0 , this notion of differentiability coincides with the usual one in R |ξ| and

∇ v F (x 1 , • • • , x n ) = n i=1 ∂ i F (x 1 , • • • , x n )v(x i ), if ξ = {x 1 , • • • , x n }.
In the general case, a set of test functions is defined as is : Following the notations from [START_REF] Albeverio | Analysis and geometry on configuration spaces[END_REF], for a function F : χ → R we say that F ∈ FC ∞ b (D, χ) if:

F (ξ) = f h 1 dξ, • • • , h N dξ , for some N ∈ N, h 1 , • • • , h N ∈ D = C ∞ (E), f ∈ C ∞ b (R N ). Then for any F ∈ FC ∞ b (D, χ), given v ∈ V 0 (E)
, we have:

F (φ v t (ξ)) = f h 1 • φ v t dξ, • • • , h N • φ v t dξ .
It is then clear that the directional derivative of such F exists and that:

∇ v F (ξ) = N i=1 ∂ i f h 1 dξ, • • • , h N dξ ∇ E v h i dξ.
The gradient ∇F of a differentiable function F is defined as a map from χ into T χ such that, for any v ∈ V 0 (E),

∇ x F (ξ)v(x) dξ(x) = ∇ v F (ξ).
If ξ ∈ χ 0 and F is differentiable at χ, then In this part, we assume α = -1 and that Hypothesis 1 and 2 hold. We denote by β λ (x) the logarithmic derivative of λ, given by: for any x in E,

∇ x F (ξ) = |ξ| i=1 ∂ i F ({x 1 , • • • , x |ξ| })1 {x=x i } . If ξ belongs to χ, for any F ∈ FC ∞ b (D, χ), ∇ x F (ξ) = n i=1 ∂ i f h 1 dξ, • • • , h N dξ ∇ E h i (x).
β λ (x) = ∇ρ(x) ρ(x) on {ρ(x) > 0},
and β λ (x) = 0 on {ρ(x) = 0}. Then, for any vector field v on E with compact support, we denote by B λ v the following function on χ:

B λ v : χ -→ R ξ -→ B λ v (ξ) = E β λ (x).v(x) + div(v(x)) dξ(x),
where x.y is the euclidean scalar product of x and y in E.

If λ = m, B m v (ξ) = E div(v(x)) dξ(x)
and according to Theorem 3,

E [|B m v (ξ)|] ≤ E | div(v(x))|K(x, x) dλ(x) ≤ v ∞ trace(K Λ ) < ∞,
where Λ is a compact containing the support of v. As in [START_REF] Yoo | Dirichlet forms and diffusion processes for fermion random point fields[END_REF], we now define the potential energy of a finite configuration by

U : χ 0 -→ R ξ -→ -log det J(ξ).
Hypothesis 3. The functional U is differentiable at every configuration ξ ∈ χ 0 . Moreover, for any v ∈ V 0 (E), there exists c > 0 such that for any ξ ∈ χ 0 , we have

(13) ∇U (ξ), v L 2 (dξ) ≤ u |ξ| det J(ξ)
,

where (u n = cn n/2 , n ≥ 1) satisfy [START_REF] Bouleau | Energy image density property and the lent particle method for Poisson measures[END_REF].

Theorem 9. Assume that the kernel J is once differentiable with continuous derivative. Then, Hypothesis 3 is satisfied.

Proof. Let ξ = {x 1 , • • • , x n } ∈ χ 0 and let Λ be a compact subset of E whose interior contains ξ. Since J(., .) is differentiable (y 1 , • • • , y n ) -→ -log det (J(y i , y k ), 1 ≤ i, k ≤ n) is differentiable. The chain rule formula implies that t -→ log det (J(φ v t (x i ), φ v t (x k )), 1 ≤ i, k ≤ n) is differentiable and its differential is equal to 1 det J(φ v t (ξ)) trace Adj(J(φ v t (x i ), φ v t (x k ))) E v t ( ∂J(ξ) ∂x ) t + ( ∂J(ξ) ∂y ) t E v t ,
where ( ∂J(ξ) ∂x ) t is the matrix with terms ∂J Λ ∂x (φ v t (x i ), φ v t (x j ))

x i , x j ∈ξ , ( ∂J(ξ) ∂y ) t is the matrix with terms ∂J Λ ∂y (φ v t (x i ), φ v t (x j ))

x i , x j ∈ξ

, and E v t is the diagonal matrix with terms (v(φ v t (x i ))) x i ∈ξ . For t = 0, this reduces to

∇U (ξ), v L 2 (dξ) = 1 det J(ξ) trace Adj(J(ξ)) E v 0 ( ∂J(ξ) ∂x ) 0 + ( ∂J(ξ) ∂y ) 0 E v 0 . Since J is continuous and Λ is compact, ∂J ∂y (ξ) HS ≤ |ξ| J ∞ and E v 0 (ξ) HS ≤ |ξ| 1/2 v ∞ .
Hence, there exists c independent of ξ such that

∇U (ξ), v L 2 (dξ) ≤ c |ξ| 2 1 det J(ξ) | trace(Adj(J(ξ)))|.
From [15, page 1021], we know that for any n × n matrix A, for any x and y in R n , we have

|(Adj A)x.y| ≤ y A n-1 HS (n -1) -(n-1)/2 . It follows that | trace(Adj A)| = | n j=1 (Adj A)e j .e j | ≤ n A n-1 HS (n -1) -(n-1)/2 ,
where (e j , j = 1, • • • , n) is the canonical basis of R n . Since J is bounded, J(ξ) HS ≤ |ξ| J ∞ , hence there exists c independent of ξ such that

∇U (ξ), v L 2 (dξ) ≤ c det J(ξ) |ξ| |ξ|/2 .
The proof is thus complete.

Corollary 1. Assume that hypothesis 3 holds. For any v ∈ V 0 (E), for any ξ ∈ χ 0 , the function

t -→ H t (ξ) = det J(φ v t (ξ)) det J(ξ) is differentiable and sup |t|≤T dH t (ξ) dt ≤ u |ξ| det J(ξ)
,

where (u n , n ≥ 0) satisfy [START_REF] Bouleau | Energy image density property and the lent particle method for Poisson measures[END_REF].

Proof. According to Hypothesis 3, the function (t → U (φ v t (ξ))) is differentiable and ( 14)

dU (φ v t (ξ)) dt = ∇U (φ v t (ξ)), v L 2 (dφ v t (ξ)) . For any t, φ v t is a diffeomorphism hence, Theorem 8 applied to φ v t and φ v -t implies that µ -1, K φ v t ,λ φ v t
and µ -1, K, λ are equivalent measure. According to Lemma 4, for any t, det

J φ v t (ξ) is µ -1, K φ v t ,λ φ v t -a.s. positive hence it is also µ -1, K, λ -a.s. positive. Since for any ξ ∈ χ 0 , t → det J φ v t (ξ) = exp(-U (φ v t (ξ))
) is continuous, it follows that there exists a set of full µ -1, K, λ measure on which det J φ v t (ξ) > 0 for any |t| ≤ T , for any ξ. Furthermore,

dH t (ξ) dt = - det J(φ v t (ξ)) det J(ξ) dU (φ v t (ξ)) dt .
In view of ( 14) and of Hypothesis 3, this means that sup

|t|≤T dH t (ξ) dt ≤ det J(φ v t (ξ)) det J(ξ) u |ξ| det J(φ v t (ξ)) = u |ξ| det J(ξ) ,
since φ v t (ξ) has the number of atoms as ξ. Lemma 5. Assume that λ = m and set

P t (ξ) = x∈ξ p φ v t (x) = x∈ξ Jac φ v t (x).
For any v ∈ Diff 0 (E), for any configuration ξ ∈ χ, P is differentiable with respect to t and we have

d log P t dt (ξ) = div v - t 0 ∇ E div v • η r,t . v(η r,t ) dr dξ,
where for any r ≤ t, x → η r,t (x) is the diffeomorphism of E which satisfies:

η r,t (x) = x - t r v(η s,t (x)) ds.
In particular for t = 0, we have:

(15) d dt   x∈ξ p λ φ v t (x)   t=0 = B m v (ξ).
Moreover, there exists c > 0 and κ > 0 such that for any ξ ∈ χ 0 , [START_REF] Georgii | Conditional intensity and Gibbsianness of determinantal point processes[END_REF] sup

t≤T dP t dt (ξ) ≤ ce κ|ξ| .
Proof. Introduce, for any s ≤ t, x -→ η s,t (x), the diffeomorphism of E which satisfies:

η s,t (x) = x - t s v(η r,t (x)) dr.
As a comparison, we remind that

φ v t (x) = x + t 0 v(φ v s (x)) ds. It is well- known that the diffeomorphism x -→ η 0,t (x) is the inverse of x -→ φ v t (x).
Then using [START_REF] Üstünel | Transformations of measure on Wiener spaces[END_REF], we have: [START_REF] Hough | Determinantal processes and independence[END_REF] Jac

φ v t (x) = d(φ v t ) * m(x) dm(x) = exp t 0 div v • η r,t (x) dr , and:
x∈ξ

Jac φ v t (x) = exp   x∈ξ t 0 div v • η r,t (x) dr   .
Hence, we have:

x∈ξ d dt log Jac φ v t (x) = x∈ξ d dt t 0 div v • η r,t (x) dr
= x∈ξ div v(x) - t 0 ∇ E div v • η r,t (x).v(η r,t (x)) dr.
The first and second point follow easily. Now, v is assumed to have bounded derivatives of any order, hence for any ξ ∈ χ 0 ,

d log P t dt (ξ) ≤ c|ξ|, (18)
where c does not depend neither from t nor ξ. According to [START_REF] Hough | Determinantal processes and independence[END_REF], there exists κ > 0 such that for any ξ ∈ χ 0 , we have:

(19) |P t (ξ)| ≤ exp(κ|ξ|).
Thus, combining ( 18) and ( 19), we get [START_REF] Georgii | Conditional intensity and Gibbsianness of determinantal point processes[END_REF].

We are now in position to prove the main result of this section.

Theorem 10. Assume (E, K, λ) satisfy Hypothesis 1, 2 and 3, let α = -1.

Let F and G belong to FC ∞ b . For any compact Λ, we have:

(20) χ Λ ∇ v F (ξ)G(ξ) dµ -1,K Λ ,λ (ξ) = - χ Λ F (ξ)∇ v G(ξ) dµ -1,K Λ ,λ (ξ) + χ Λ F (ξ)G(ξ) B λ v (ξ) + ∇ v U (ξ) dµ -1,K Λ ,λ (ξ).
Proof. In view of Lemma 3, we can replace J by J[ρ] and assume λ = m, i.e., λ is the Lebesgue measure. Note that

B m v (ξ) = div v(x) dξ(x).
Let Λ be a fixed compact set in E, remember that χ Λ ⊂ χ 0 . Let M be an integer and χ M = {ξ ∈ χ 0 , |ξ| ≤ M }. It is crucial to note that χ M is invariant by any φ ∈ Diff 0 (E). On the one hand, by dominated convergence, we have:

d dt χ M F (φ v t (ξ))G(ξ) dµ -1,K Λ [ρ],m (ξ) t=0 = χ M d dt (F (φ v t (ξ))) t=0 G(ξ) dµ -1,K Λ [ρ],m (ξ) = χ M ∇ v F (ξ)G(ξ) dµ -1,K Λ [ρ],m (ξ).
On the other hand, we know from ( 12) that ( 21)

χ M F (φ v t (ξ))G(ξ) dµ -1,K Λ [ρ],m (ξ) = χ Λ F (φ v t (ξ))G(ξ)1 {|ξ|≤M } dµ -1,K Λ [ρ],m (ξ) = χ Λ F (ξ)G(φ v -t (ξ))1 {|φ v -t (ξ)|≤M } dµ -1,K φ v t Λ [ρ],m φ v t (ξ) = χ Λ F (ξ)G(φ v -t (ξ))1 {|ξ|≤M } L φ v t -1, K[ρ], λ (ξ) dµ -1,K Λ [ρ],m (ξ).
According to Corollary 1 and Lemma 5, the function (t → L

φ v t -1, K[ρ], λ (ξ)
) is differentiable and there exists c such that:

sup t≤T dL φ v t -1, K[ρ], λ dt (ξ) ≤ u |ξ| det J(ξ) ,
where (u n , n ≥ 0) satisfy [START_REF] Bouleau | Energy image density property and the lent particle method for Poisson measures[END_REF]. Lemma 4 implies that the right-hand-side of the last inequality is integrable with respect to µ -1, K Λ , λ , thus, we can differentiate inside the expectations in [START_REF] Kallenberg | Random measures[END_REF] and we obtain:

χ Λ ∇ v F (ξ)G(ξ)1 {|ξ|≤M } dµ -1,K Λ ,m (ξ) = χ Λ F (ξ) (-∇ v G(ξ) + G(ξ) (B m v (ξ) + ∇ v U (ξ))) 1 {|ξ|≤M } dµ -1,K Λ ,m (ξ).
According to Hypothesis 3 and Lemma 4, by dominated convergence, we have:

χ Λ ∇ v F (ξ)G(ξ) dµ -1,K Λ ,m (ξ) = χ Λ F (ξ) (-∇ v G(ξ) + G(ξ) (B m v (ξ) + ∇ v U (ξ))) dµ -1,K Λ ,m (ξ).
Now, we remark that

∇ v U [ρ](ξ) = ∇ v log detJ[ρ](ξ) = ∇ v log   x∈ξ ρ(x)detJ(ξ)   = ∇ v log ρ(x) dξ(x) + ∇ v U (ξ) = ∇ E ρ(x) ρ(x) .v(x) dξ(x) + ∇ v U (ξ).
Moreover, we have

B m v (ξ) + Λ ∇ E ρ(x) ρ(x) .v(x) dξ(x) = B λ v (ξ),
and in view of Theorem 3,

E Λ ∇ E ρ(x) ρ(x) .v(x) dξ(x) 2 ≤ E Λ ∇ E ρ(x) ρ(x) 2 dξ(x) E Λ |v(x)| 2 dξ(x) ≤ v 2 ∞ trace(K Λ ) Λ ∇ E ρ(x) ρ(x) 2 K(x, x)ρ(x) dm(x).
Then, Hypothesis 2 implies that B λ v is integrable and we get [START_REF]Random matrices and determinantal processes[END_REF] in the general case. 4.2. α-determinantal point processes. We now prove the integration by parts formula for α-determinantal point processes where α = -1/s for s integer greater than 2. In principle, we could follow the previous lines of proof modifying the definition of U as

U (ξ) = -log det α J α (ξ)
and assuming that Hypothesis 3 is still valid. Unfortunately, there is no (simple) analog of Theorem 9 since there is no rule to differentiate an αdeterminant and control its derivative.

We already saw that such an α-DPPP can be obtained as the superposition of s determinantal processes of kernel K/s.

Let (E 1 , λ 1 , K 1 ), • • • , (E s , λ s , K s ) be s Polish spaces each equipped with a Radon measure and s linear operators satisfying Hypothesis 1 on their respective space. We set E = ∪ s i=1 {i} × E i , that is to say E is the disjoint union of the E i 's, often denoted as ⊔ s i=1 E i . An element of E is thus a couple (i, x) where x belongs to E i for any i ∈ {1, • • • , s}. On the Polish space E, we put the measure λ defined by

E f (i, x) dλ(i, x) = E f (i, x) dλ i (x).
We also define K as

Kf (i, x) = E i K i (x, y)f (y) dλ i (y). A compact set in E is of the form Λ = ∪ s i=1 {i} × Λ i where Λ i is a compact set of E i hence K Λ f (i, x) = Λ i K i (x, y)f (y) dλ i (y).
This means that K is a kernel operator the kernel of which is given by: [START_REF] Macchi | The coincidence approach to stochastic point processes[END_REF] K((i, x), (j, y)) = K i (x, y)1 {i=j} .

In particular, for ξ = ((i l , x l ), l = 1,

• • • , n), we have det K(ξ) = s j=1 det K(ξ j )
where ξ j = {x, (j, x) ∈ ξ}.

It is straightforward that K is symmetric and locally of trace class. Moreover, its spectrum is equal to the union of the spectra of the K i 's. For, if ψ is such that Kψ = αψ then ψ(i, .) is an eigenvector of K i and thus α belongs to the spectrum of K i . In the reverse direction, if ψ is an eigenvector of K i associated to the eigenvalue α then the function f (j, x) = ψ(x)1 {i=j} is square integrable with respect to λ and is an eigenvector of K for the eigenvalue α. If we assume furthermore that each of the E i 's is a subset of R d , we can define the gradient on E as

∇ E f (i, x) = ∇ E i f (i, x).
Now χ E is the set of locally finite point measures of the form

ξ = j δ (i j , x j ) .
With these notations, it is clear that Hypothesis 1, 2 and 3 are satisfied provided they are satisfied for each index i. Thus [START_REF]Random matrices and determinantal processes[END_REF] is satisfied.

Now take E 1 = . . . = E s , λ 1 = . . . = λ s and K 1 = . . . = K s . We introduce the map Θ defined as:

Θ : E -→ E 1 (i, x) -→ x.
Consistently with earlier defined notations, we still denote by Θ the map

Θ : χ E -→ χ E 1 ξ -→ (j, x)∈ξ δ x .
Then, according to what has been said above, µ -1/s, sK 1 , λ 1 is the image measure of µ -1, K, λ by the map Θ. Set

ξ n = (i, x)∈ξ δ x 1 {i=n} .
The reciprocal problem, interesting in its own sake and useful for the sequel, is to determine the conditional distribution of ξ n given Θξ.

Theorem 11. Let s be an integer strictly greater than 1, for F non-negative or bounded, for any Λ compact subset of E,

(23) E [F (ξ 1 ) | Θξ] = η⊂Θξ F (η) × |Θξ| |η| j β, (s-1)K 1,Λ (Θξ\η) j -1, K 1,Λ (η) j α,sK 1,Λ (Θξ)
,

where β = -1/(s -1).

Note that ( 23) also holds for s = 1 with the convention that j β, 0 (η) = 0 for η = ∅ and j β, 0 (∅) = 1, which is analog to the usual convention 0 0 = 1.

Proof. Let ζ = ξ 2 ∪ . . . ∪ ξ s , we known that ζ is distributed as µ -β, -K 1 /β, λ 1 . Consider Ξ, the map Ξ : χ E 1 × χ E 1 -→ χ E 1 × χ E 1 (η 1 , η 2 ) -→ (η 1 , η 1 ∪ η 2 ).
By construction, the joint distribution of Ξ(ξ 1 , ζ) is the same as the distribution of (ξ 1 , Θξ). For any η ⊂ Θξ ∈ χ 0 , we set:

R(η, Θξ) = |Θξ| |η| j β, (s-1)K 1,Λ (Θξ\η) j -1, K 1,Λ (η) j α,sK 1,Λ (Θξ) • Hence, we have E [F (ξ 1 )G(Θξ)] = E [(F ⊗ G) • Ξ(ξ 1 , ζ)] = ∞ j=0 ∞ k=0 1 j! 1 k! Λ j ×Λ k F ({x 1 , • • • , x j })G({x 1 , • • • , x j } ∪ {y 1 , • • • , y k }) × j -1, K 1,Λ (x 1 , • • • , x j ) j β, (s-1)K 1,Λ (y 1 , • • • , y k ) dλ(x 1 ) . . . dλ(y k ) = ∞ j=0 ∞ k=0 1 (k + j)! Λ j ×Λ k F ({x 1 , • • • , x j })(GR)({x 1 , • • • , x j } ∪ {y 1 , • • • , y k }) × j α,sK 1,Λ (x 1 , • • • , x j , y 1 , • • • , y k ) dλ(x 1 ) . . . dλ(y k ) = ∞ m=0 1 m! Λ m   j≤m F ({x 1 , • • • , x j })R({x 1 , • • • , x j }, {x 1 , • • • , x m })   × G({x 1 , • • • , x m }) j α,sK 1,Λ (x 1 , • • • , x m ) dλ(x 1 ) . . . dλ(x m ) = χ E 1 η⊂ω F (η)R(η, ω) G(ω) dµ α, sK 1 , λ 1 (ω).
The proof is thus complete.

This formula can be understood by looking at the extreme case of Poisson process. Assume that Θξ is distributed according to a Poisson process of intensity λ dm. Then, ξ 1 is a Poisson process of intensity s -1 λ dm and ζ also is a Poisson process of intensity (1s -1 )λ dm. The couple (ξ 1 , Θξ) can then be constructed by random thinning of Θξ: Keep each point of Θξ independently of the others, with probability 1/s; the remaining points will be distributed as ξ 1 . The conditional expectation of a functional F (ξ 1 ) given Θξ is then the sum of the values of F taken for each realization of a thinning multiplied by the probability of each thinned configuration. Since |Θξ| is assumed to be known, the atoms of Θξ are independent and identically dispatched along E, hence the probability to obtain a specific configuration is binomially distributed of parameters |Θξ| and 1/s. This means that

E [F (ξ 1 ) | Θξ] = η⊂Θξ F (η) × |Θξ| |η| 1 s |η| 1 - 1 s |Θξ|-|η| .
This corresponds to [START_REF] Privault | Stochastic analysis in discrete and continuous settings with normal martingales[END_REF] for α = 0. As a consequence, ( 23) can be read as a generalization of this procedure where the points cannot be drawn independently and with equal probability because of the correlation structure.

For h any map from E 1 into E 1 , we define h ⊔ by

h ⊔ : E -→ E (i, x) -→ (i, h(x)).
With this notation at hand, for v in V 0 (E 1 ), (φ v t ) ⊔ is the solution of the equations:

d(φ v t ) ⊔ (i, x) = v ⊔ ((φ v t ) ⊔ (i, x)), 1 ≤ i ≤ m.
Note that we only consider a restricted set of perturbations of configurations in the sense that we move atoms on each "layers" without "crossing": By the action of (φ v t ) ⊔ , an atom of the form (i, x) is moved into an atom of the form (i, y), leaving its first coordinate untouched.

Theorem 12. Assume that (E 1 , K 1 , λ 1 ) satisfy Hypothesis 1, 2 and 3. Let s = -1/α be an integer greater than 1. For F and G cylindrical functions, for v ∈ V 0 (E 1 ), we have:

χ Λ ∇ v F (ω)G(ω) dµ α, sK 1,Λ ,λ 1 (ω) = - χ Λ F (ω)∇ v G(ω) dµ α, K 1, Λ ,λ 1 (ω) + 1 |α| χ Λ F (ω)G(ω) η⊂ω (B λ 1 v (η) + ∇ v U (η))R(η, ω) dµ α, sK 1,Λ ,λ 1 (ω).
Proof. We first apply [START_REF]Random matrices and determinantal processes[END_REF] to the process ξ = (ξ 1 , • • • , ξ s ). Remember that Θξ is equal to ξ 1 ∪ . . . ∪ ξ s . A cylindrical function of Θξ is a function of the form:

H(Θξ) = f ( h 1 dΘξ, • • • , h N dΘξ) where h 1 , • • • , h N ∈ D = C ∞ (E 1 ), f ∈ C ∞ b (R N ). Such a functional can be written as F • Θ(ξ) where F is a cylindrical function of ξ. Moreover, for v ∈ V 0 (E 1 ), ∇ v H(Θξ) = lim t→0 1 t (H(φ v t (Θξ) -H(Θξ)) = lim t→0 1 t F (Θ(φ v t ) ⊔ (ξ) -F (Θξ) = ∇ v ⊔ F (Θξ). ( 24
)
In view of ( 22), ( 25)

U (ξ) = -log det J(ξ 1 , • • • , ξ s ) = s j=1 U (ξ j ).
Analyzing the proof of (20), we see that the intrinsic definition of B λ v is

B λ v (ξ) = div λ (v) dξ where div λ (v)(x) = d dt d(φ v t ) * λ dλ (x) t=0 .
In view of ( 24), we only need to consider flows on E associated to vector fields of the form v ⊔ for v ∈ V 0 (E 1 ). Hence,

B λ v ⊔ (ξ) = s j=1 B λ j v (ξ j ). (26)
It follows from the previous considerations that:

χ Λ ⊔ ∇ v ⊔ F (Θξ)G(Θξ) dµ -1,K Λ ,λ (ξ) = - χ Λ ⊔ F (Θξ)∇ v ⊔ G(Θξ) dµ -1,K Λ ,λ (ξ)
+ χ Λ ⊔ F (Θξ)G(Θξ) B λ v ⊔ (ξ) + ∇ v ⊔ U (ξ) dµ -1,K Λ ,λ (ξ)
where Λ ⊔ = ∪ s j=1 {i} × Λ. Since the ξ j 's are independent and identically distributed, according to [START_REF] Simon | Trace ideals and their applications[END_REF] and [START_REF] Soshnikov | Determinantal random point fields[END_REF], we have

E B λ v ⊔ (ξ) + ∇ v ⊔ U (ξ) Θξ = sE B λ 1 v (ξ 1 ) + ∇ v U (ξ 1 ) Θξ = - 1 α η⊂Θξ (B λ 1 v (η) + ∇ v U (η))R(η, Θξ).
Thus, we obtain:

χ Λ ∇ v F (ω)G(ω) dµ α, sK 1,Λ ,λ 1 (ω) = - χ Λ F (ω)∇ v G(ω) dµ α, K 1, Λ ,λ 1 (ω) - 1 α χ Λ F (ω)G(ω) η⊂ω (B λ 1 v (η) + ∇ v U (η))R(η, ω) dµ α, sK 1,Λ ,λ 1 (ω).
The proof is thus complete.

4.3. α-permanental point processes. For permanental point processes, we begin with the situation where α = 1. In this case, j 1, K Λ , λ ({x 1 , • • • , x n }) = Det(I + K Λ ) -1 per(J(x i , x j ), 1 ≤ i, j ≤ n).

We aim to follow the lines of proof of Theorem 10, for, we need some preliminary considerations. For any integer n, let D[n] be the set of partitions of {1, • • • , n}. The cardinal of D[n] is known to be the n-th Bell number (see [START_REF] Aigner | Combinatorial theory[END_REF]), denoted by We slightly change the definition of the potential energy of a finite configuration as

U : χ 0 -→ R ξ -→ -log per J(ξ).
A new hypothesis then arises:

Hypothesis 4. The functional U is differentiable at every configuration ξ ∈ χ 0 . Moreover, for any v ∈ V 0 (E), there exists (u n , n ≥ 1) a sequence of nonnegative real as in Lemma 4 such that for any ξ ∈ χ 0 , we have Hence the result.

Remark 1. The finite rank condition is rather restrictive but the sequence (B n n n/2 , n ≥ 1) has not a finite exponential generating function thus we can't avoid it. In order to circumvent this difficulty one would have to improve known upper-bounds on permanents.

We can then state the main result for this subsection.

Theorem 14. Assume that (E, K, λ) satisfy Hypothesis 1, 2 and 4. Let F and G belong to FC ∞ b . For any compact Λ, we have:

χ Λ ∇ v F (ξ)G(ξ) dµ 1,K Λ ,λ (ξ) = - χ Λ F (ξ)∇ v G(ξ) dµ 1,K Λ ,λ (ξ) + χ Λ F (ξ)G(ξ) B λ v (ξ) + ∇ v U (ξ) dµ 1,K Λ ,λ (ξ).
Proof. Same as the proof of Theorem 10.

Now then, we can work as in Subsection 4.2 and we obtain the integration by parts formula for α-permanental point processes.

Corollary 2. Assume that (E 1 , K 1 , λ 1 ) satisfy Hypothesis 1, 2 and 4. Let s = 1/α be an integer greater than 1. For F and G cylindrical functions, for v ∈ V 0 (E 1 ), we have: A trivial consequence of the previous results is that, for any α-DPPP known to exist, the operator ∇ is closable and can thus be extended to D 2,1 . With the same lines of proof we retrieve the result of ( [START_REF] Yoo | Gibbsianness of fermion random point fields[END_REF]), which says that the Dirichlet form: E(F, F ) = E [ ∇F, ∇F ] is closable.

χ Λ ∇ v F (ω)G(ω) dµ α, sK 1,Λ ,λ 1 (ω) = - χ Λ F (ω)∇ v G(ω) dµ α, K 1, Λ ,λ 1 (ω) + 1 α χ Λ F (ω)G(ω) η⊂ω (B λ 1 v (η) + ∇ v U (η))R(η, ω) dµ α, sK 1,Λ ,λ 1 (ω).

4. 1 .

 1 Determinantal point processes. In what follows, c and κ are positive constant which may vary from line to line.

B= e e x - 1 .

 1 n and which can be computed by their exponential generating function: for any real x,For an n × n matrix A = (a ij , 1 ≤ i, j ≤ n) and for τ a subset of {1, • • • , n}, we denote by A[τ ] the matrix (a ij , i ∈ τ, j ∈ τ ). For a partition σ of {1, • • • , n}, ι(σ) is the number of non-empty parts of σ. This means that σ = (τ 1 , • • • , τ ι(σ) ), where the τ i 's are disjoint subsets of {1, • • • , n} whose union is exactly {1, • • • , n}. Then, we set det A[σ] = ι(σ) j=1 det J[τ j ].It is proved in[START_REF] Chu | Determinant, permanent, and MacMahon's master theorem[END_REF] Corollary 1.7] that[START_REF] Üstünel | Transformations of measure on Wiener spaces[END_REF] perA = σ∈D[n](-1) n+ι(σ) det A[σ].

( 29 )

 29 ∇U (ξ), v L 2 (dξ) ≤ u |ξ| per J(ξ)•An analog of Theorem 9 now becomes Theorem 13. Assume that K is of finite rank N and that the kernel J is once differentiable with continuous derivative. Then, Hypothesis 4 is satisfied.Proof. Since K is of finite rank N there are at most N points in any configuration. It is clear from[START_REF] Üstünel | Transformations of measure on Wiener spaces[END_REF] that (t → U (φ v t (ξ))) is differentiable. Since |detJ(ξ)[τ ]| ≤ c|τ | |τ /2| where |τ | is the cardinal of τ ∈ D[|ξ|], we get ∇U (ξ), v L 2 (dξ) ≤ c B |ξ| |ξ| |ξ|/2 per J(ξ) 1 {|ξ|≤N } .

4. 4 . 2 , 1 = 2 =

 4212 Consequences. We define the norm ||.|| 2,1 on FC ∞ b (D, χ) by: ||F || 2 ||F || 2 L 2 (µ) + E ||∇F || E F 2 + E |∇ x F | 2 dξ(x) .and we call D 2,1 the closure of FC ∞ b (D, χ) for the norm ||.|| 2,1 .

  this is true without the condition αT < 1.

	2.3. Determinantal-permanental point processes. The following set
	of hypothesis is of constant use.
	Hypothesis 1. The Polish space E is equipped with a Radon measure λ.
	The map K is an Hilbert-Schmidt operator from L 2 (E, λ) into L 2 (E, λ)
	which satisfies the following conditions:
	i) K is a bounded symmetric integral operator on L 2 (E, λ), with kernel
	K(., .), i.e., for any x ∈ E,