Marked Gibbs point processes with unbounded interaction: an existence result

We construct marked Gibbs point processes in $\mathbb{R}^d$ under quite general assumptions. Firstly, we allow for interaction functionals that may be unbounded and whose range is not assumed to be uniformly bounded. Indeed, our typical interaction admits an a.s. finite but random range. Secondly, the random marks -- attached to the locations in $\mathbb{R}^d$ -- belong to a general normed space $\mathcal{S}$. They are not bounded, but their law should admit a super-exponential moment. The approach used here relies on the so-called entropy method and large-deviation tools in order to prove tightness of a family of finite-volume Gibbs point processes. An application to infinite-dimensional interacting diffusions is also presented.


Introduction
In this paper we construct a certain class of continuous marked Gibbs point processes. Recall that a marked point consists of a pair: a location ∈ ℝ , ≥ 1, and a mark belonging to a general normed space S . The interactions we consider here are described by an energy functional , which acts both on locations and on marks. This includes, in particular, the case of -body potentials, but is indeed a more general framework, useful to treat examples coming from the eld of stochastic geometry (as e.g. the area-or the Quermass-interaction model, see Example 1).
The novelty of the results presented in this paper is threefold. Firstly, we do not assume a speci c form of the interaction -like pairwise or -body -but only make assumptions (in Section 2.3) on the resulting energy functional itself. In particular, we do not assume superstability of the interaction, but only rely on the stability assumptions (H ) and (H . ). In the eld of stochastic geometry, in particular, many quite natural energy functionals are stable but not superstable, like the Quermass-interaction model presented in Example 1.
Secondly, the Gibbsian energy functional we consider has an unbounded range: it is nite, but random and not uniformly bounded -as opposed to models treated for example in [1] which deal with a bounded-range interaction; see Assumption (H ). For a very recent existence proof in the case of in nite-range interaction (without marks) see [8]. Moreover, unlike the hyper-edge interactions presented in [7], we treat the case of interactions which are highly non local: the range of the conditional energy on a bounded region of an in nite con guration (see De nition 2.5) requires knowledge of the whole con guration and cannot be determined only by a local restriction of the con guration.
Lastly, we work with a mark reference distribution whose support is a priori unbounded but only ful ls a super-exponential integrability condition (see Assumption (H )).
Let us mention recent works on the existence of marked Gibbs point processes for particular models. In [5] D. Dereudre proves the existence of the Quermassinteraction process as a planar germ-grain model; we draw inspiration from his approach, presenting here an existence result for more general processes, under weaker assumptions. In [3] and [1] the authors treat the case of unbounded marks in ℝ with nite-range energy functional which is induced by a pairwise interaction.
The main thread of our approach is the reduction of the general marked point process to a germ-grain model, where two marked points ( 1 , 1 ), ( 2 , 2 ) ∈ ℝ × S do not interact as soon as the balls with centre and radius ‖ ‖, = 1, 2, do not intersect. The framework we work in requires the introduction of a notion of tempered con gurations (see Section 2.2) in order to better control the support of the Gibbs measure we construct. In this way, the size growth of the marks of far away points is bounded. In Section 3.3 we see that this procedure is justi ed by the fact that the constructed in nite-volume Gibbs measure is actually concentrated on tempered con gurations.
The originality of our method to construct an in nite-volume measure consists in the use of the speci c entropy as a tightness tool. This relies on the fact that the level sets of the speci c-entropy functional are relatively compact in the local convergence topology; see Section 3.2. This powerful topological property was rst shown in the setting of marked point processes by H.-O. Georgii and H. Zessin in [12]. Indeed, we prove in Proposition 3.5, using large-deviation tools, that the entropy of some sequence of nite-volume Gibbs measures is uniformly bounded. This sequence is therefore tight, and admits an accumulation point. Let us remark that the entropy tool relies mainly on stability assumptions of the energy , without the need for superstability. The usual approach (see e.g. [22]), in fact, uses the superstability condition to precisely control the local density of points; in our framework, we do this thanks to an equi-integrability property, which holds on the entropy level sets (see Lemma 3.8). Furthermore, the stability notion we use here is weaker than the classic one of Ruelle, as it includes a term depending on the marks of the con guration. For more details and examples, see Section 2.3.
The last step of the proof consists in showing that this accumulation point satis es the Gibbsian property. Since the interaction is not local and not bounded, this property is not inherited automatically from the nite-volume approximations, but instead requires an accurate analysis, which is done in Section 3.4. In Section 4 we propose an application to in nite-dimensional interacting di usions.

Point-measure formalism
The point con gurations considered here live in the product state space E . . = ℝ × S , ≥ 1, where S , ‖⋅‖ is a general normed space: each point location in ℝ has an associated mark belonging to S . The location space ℝ is endowed with the Euclidean norm |⋅|, and the associated Borel -algebra B(ℝ ); we denote by B (ℝ ) ⊂ B(ℝ ) the set of bounded Borel subsets of ℝ . A set belonging to B (ℝ ) will often be called a nite volume. We denote by B(S ) the Borel -algebra on S .
The set of point measures on E is denoted by M ; it consists of the integer-valued, -nite measures on E : We endow M with the canonical -algebra generated by the family of local counting functions on M , We denote by the zero point measure whose support is the empty set. Since, in the framework developed in this paper, we only consider simple point measures, we identify them with the subset of their atoms: For a point con guration ∈ M and a xed set ⊂ ℝ , we denote by the restriction of the point measure to the set × S : A functional is a measurable ℝ ∪ {+∞}-valued map de ned on M . We introduce speci c notations for some of them: the mass of a point measure is denoted by | |. It corresponds to the number of its atoms if is simple. We also denote by m the supremum of the size of the marks of a con guration: The integral of a xed function ∶ E → ℝ under the measure ∈ M -when it exists -is denoted by For a nite volume , we call local or more precisely -local, any functional satisfying We also de ne the set of nite point measures on E : Moreover, for any bounded subset ⊂ ℝ , M is the subset of M consisting of the point measures whose support is included in × S : Let P(M ) denote the set of probability measures on M . We write ℕ * for the set of non-zero natural numbers ℕ ⧵ {0}. The open ball in ℝ centred in ∈ ℝ with radius ∈ ℝ + is denoted by ( , ).

Mark reference distribution
The mark associated to any point of a con guration is random. We assume that the reference mark distribution on S is such that its image under the map ↦ ‖ ‖ is a probability measure on ℝ + that admits a super-exponential moment, in the following sense: (H ) There exits > 0 such that Throughout Sections 2 and 3 of the paper, the parameter is xed.
Remark. The probability measure is the density of a positive random variable such that 2 + is subgaussian for some > 0 (see e.g. [13], [16]).

Tempered con gurations
We introduce the concept of tempered con guration. For such a con guration , the number of its points in any nite volume , | |, should grow sublinearly w.r.t. the volume, while its marks should grow as a fraction of it. More precisely, we de ne the space M temp of tempered con gurations as the following increasing union where We now prove some properties satis ed by tempered con gurations.
By de nition of 1 (t, 1 2 ), since ( , ) ∈ (0,⌈ ⌉) , The assertion of Lemma 2.2 is illustrated in Figure 1. De ne the germ-grain set Γ of a con guration as usual by where the point is the germ and the ball (0, ‖ ‖) is the grain. Lemma 2.2 then implies that, for tempered con gurations, only a nite number of balls of their germgrain set can intersect a xed bounded subset of ℝ . This remark will be very useful when de ning the range of the interaction in (9).

Energy functionals and nite-volume Gibbs measures
For a xed nite volume ⊂ ℝ , we consider, as reference marked point process, the Poisson point process on E with intensity measure ⊗ ( ). The coe cient is a positive real number, is the Lebesgue measure on , and the probability measure on S was introduced in Section 2.1. In this toy model, since the spatial component of the intensity measure is di use, the con gurations are a.s. simple. Moreover, the random marks of di erent points of the con guration are independent random variables.
To model and quantify a possible interaction between the point locations and the marks of a con guration, one introduces the general notion of energy functional.

De nition 2.3. An energy functional is a translation-invariant measurable functional on the space of nite con gurations
We use the convention ( ) = 0.
Con gurations with in nite energy will be negligible with respect to Gibbs measures.
De nition 2.4. For ∈ B (ℝ ), the nite-volume Gibbs measure with free boundary condition is the probability measure on M de ned by The normalisation constant is called partition function. We will see in Lemma 2.7 why this quantity is well de ned under the assumptions we work with.
Notice how -and therefore -is actually concentrated on M , the nite point con gurations with atoms in .
The measure extends naturally to an in nite-volume measure ; the question we explore in this work is how to do the same for . The rst step in order to de ne an in nite-volume Gibbs measure is to be able to consider the energy of con gurations with in nitely many points. In order to do this, we approximate any (tempered) con guration by a sequence of nite ones ( ) . Using a terminology that goes back to Föllmer [9], we introduce the following De nition 2.5. For ∈ B (ℝ ), the conditional energy of on given its environment is the functional de ned, on the tempered con gurations, as the following limit: where . . = [− , ) is an increasing sequence of centred cubes of volume (2 ) , converging to ℝ .

Remarks.
i. Notice that the conditional energy of nite con gurations con ned in coincides with their energy: ( ) ≡ ( ). In general, however, the conditional energy ( ) of an in nite con guration does not reduce to ( ) because of the possible interaction between (external) points of and (internal) points of . In other words, the conditional energy is possibly not a local functional. In this paper, we are interested in this general framework.
ii. Indeed, we will work with energy functionals for which the limit in (6) is stationary, i.e. reached for a nite (that depends on ). Assumption (H ) below ensures this property. iii. Since only charges con gurations in , can be equivalently de ned as The key property of such conditional energy functionals is the following additivity; the proof of this lemma is analogous to the one in [5], Lemma 2.4, that works in the more speci c setting of Quermass-interaction processes.
Lemma 2.6. The family of conditional energy functionals is additive, i.e. for any ⊂ ∈ B (ℝ ), there exists a measurable function , ∶ M temp → ℝ such that Let us now describe the framework of our study, by considering for the energy functional a global stability assumption (H ), a range assumption (H ) and a locally-uniform stability assumption (H . ): (H ) There exists a constant c ≥ 0 such that the following stability inequality holds where ⊕ (0, r) . . = ∈ ℝ ∶ ∃ ∈ , | − | ≤ r , and where d( ) is the smallest > 0 such that ⊂ (0, ). 1 Equivalently, the limit in (6) is already attained at the smallest ≥ 1 such that ⊃ ⊕ (0, r).
that the following stability of the conditional energy holds, uniformly for all ξ ∈ M t : Remarks. i. Notice how the stability assumption (H ) is weaker than the usual Ruelle stability ( ) ≥ −c| | = −c⟨ , 1⟩, for the presence of the mark-dependent negative term −c⟨ , ‖ ‖ + ⟩. ii. Assumption (9) has the following interpretation: there is no in uence from the points of ( ⊕ (0,r)) on the points of : ( ) = ( ⊕ (0,r) ). The form of the range r models the case where two points x = ( , ), y = ( , ) ∈ E of a con guration are not in interaction whenever ( , ‖ ‖) ∩ ( , ‖ ‖) = ∅. It is easy to see from Lemma 2.2 that ( ) = ( (0,2d( )+2l(t)+2m( )+1) ). Therefore, the range of the energy at the con guration is smaller than r( , ), which is nite but random since it depends on . This range may not be uniformly bounded when varies. Proof.
We provide here examples of energy functionals on marked con gurations, which satisfy the assumptions above. Section 4 provides, in the context of interacting di usions, a further example of a pair interaction that acts on both locations and marks of a con guration, where the mark space is a path space.
Example 1 (Geometric multi-body interaction in ℝ 2 ). Consider the marked-point state space E = ℝ 2 × ℝ + , and recall that one can associate, to any nite con guration Consider, as reference mark measure, a measure on ℝ + satisfying (H ), that is, there exists > 0 such that The Quermass energy functional (see [15]) is de ned as any linear combination of area, perimeter, and Euler-Poincaré characteristic functionals: Notice how this interaction, depending on the values of the parameters , can be attractive or repulsive. It is di cult (and not useful) to decompose this multi-body energy functional as the sum of several -body interactions. The functional satis es assumptions (H ), (H ), and (H . ). Indeed, it even satis es the following stronger conditions: • there exists a constant c such that, for any nite con guration , (two-sided stability) • For any ∈ B (ℝ 2 ) and t ≥ 1, there exists c ′ ( , t) such that, for any ∈ M , ξ ∈ M t , Under these stronger conditions than ours, the existence for the Quermass-interaction model was proved in [5]; notice that is not superstable. For more examples of geometric interactions, see [6]. where is non-negative and null at 0. In both cases, since is a non-negative functional, it satis es (H ) and (H . ). It is also easy to see that, by construction, the range assumption (H ) also holds.

Local topology
We endow the space of point measures with the topology of local convergence (see [11], [12]), de ned as the weak* topology induced by a class of functionals on M which we now introduce.
De nition 2.8. A functional is called tame if there exists a constant > 0 such that We denote by L the set of all tame and local functionals. The topology L of local convergence on P(M ) is then de ned as the weak* topology induced by L , i.e. the smallest topology on P(M ) under which all the mappings ↦ , ∈ L , are continuous.

Construction of an in nite-volume Gibbs measure
Let us rst precise the terminology (see [10]).
De nition 3.1. Let be an energy functional satisfying the three assumptions (H ), (H ), and (H . ). We say that a probability measure on M is an in nitevolume Gibbs measure with energy functional if, for every nite volume ⊂ ℝ and for any measurable, bounded and local functional ∶ M → ℝ, the following identity (called DLR equation after Dobrushin-Lanford-Ruelle) holds under : where , called the Gibbsian probability kernel associated to , is de ned on M by where (ξ) . . = M − ( ξ ) ( ).

Remarks.
1. The probability kernel (ξ, ⋅) is not necessarily well-de ned for any ξ ∈ M . In Lemma 3.9, we will show that this is the case when we restrict it to the subspace M temp . 2. The map ξ ↦ (ξ, ) is a priori not local since ξ ↦ ( ξ ) may depend on the full con guration ξ . 3. The renormalisation factor (ξ) -when it exists -only depends on the external con guration ξ . Therefore (ξ, ⋅) ≡ (ξ , ⋅).
We can now state the main result of this paper: This section will have the following structure. 3.1 We de ne a sequence of stationarised nite-volume Gibbs measures (̄ ) . 3.2 We use uniform bounds on the entropy to show the convergence, up to a subsequence, to an in nite-volume measurē . 3.3 We prove, using an ergodic property, that̄ carries only the space of tempered con gurations. 3.4 Noticing that, for any xed ∈ B (ℝ ),̄ does not satisfy (DLR) , we introduce a new sequence (̂ ) asymptotically equivalent to (̄ ) but satisfying (DLR) . We use appropriate approximations, by localising the interaction, to show that alsō satis es (DLR) .

A stationarised sequence
In this subsection, we extend each nite-volume measure . . = , = [− , ) , de ned on M to a probability measurē on the full space M , invariant under lattice-translations.
Applying Cauchy-Schwarz inequality, we nd: We start by considering the probability measurẽ on M , under which the con gurations in the disjoint blocks . . = + 2 , ∈ ℤ , are independent, with identical distribution . We then build the empirical eld associated to the probability measurẽ , i.e. the sequence of lattice-stationarised probability measures where is the translation on ℝ by the vector ∈ ℤ .

Remarks.
1. As usual we identify the translation on ℝ with the image of a point measure under such translation. 2. So constructed, the probability measurē is invariant under ( ) ∈ℤ .
3. An upper bound similar to (12) holds also under̄ : Moreover, using stationarity and the fact that the covering = ⋃ 1 contains terms, As we will see in the following subsection, in order to prove that (̄ ) admits an accumulation point, it is enough to prove that all elements of the sequence belong to the same entropy level set.

Entropy bounds
Let us now introduce the main tool of our study, the speci c entropy of a (stationary) probability measure on M .
De nition 3.3. Given two probability measures and ′ on M , and any nitevolume ⊂ ℝ , the relative entropy of ′ with respect to on is de ned as where (resp. ′ ) is the image of (resp. ′ ) under the mapping ↦ .

As usual,
De nition 3.4. The speci c entropy of with respect to ′ is de ned by From now on, the reference measure ′ will be the marked Poisson point process with intensity measure ⊗ ( ). In this case, the speci c entropy of a probability measure with respect to is always well de ned if is stationary under the lattice translations ( ) ∈ℤ . Moreover, recall that for any > 0, the -entropy level set P(M ) ≤ . . = ∈ P(M ), stationary under ( ) ∈ℤ ∶ I( | ) ≤ is relatively compact for the topology L , as proved in [12].
From the above proposition we deduce that the sequence (̄ ) ≥1 belongs to the relatively compact set P(M ) ≤ 3 . It then admits at least one converging subsequence which we will still denote by (̄ ) ≥1 for simplicity. The limit measure, here denoted bȳ , is stationary under the translations ( ) ∈ℤ . We will prove in what follows that̄ is the in nite-volume Gibbs measure we are looking for.

Support of the in nite-volume limit measure
We now justify the introduction of a set of tempered con gurations as the right support of each of the probability measures̄ , ≥ 1, as well as of the constructed limit probability measurē .
Proposition 3.6. The measures̄ , ≥ 1, and the limit measurē are all supported on the tempered con gurations, i.e.
In Subsection 3.4, in order to prove Gibbsianity of the limit measure, we need more: a uniform estimate of the support of the measures̄ , ≥ 1. For this reason, we introduce the increasing family (M ) ∈ℕ * of subsets of M temp , de ned by Notice that, thanks to Lemma 2.2, for any t ≥ 1, M t ⊂ M l(t) (see Figure 1).

The limit measure is Gibbsian
We are now ready to prove that the in nite-volumē measure we have constructed satis es the Gibbsian property.
(i) We have to show that, for any ξ ∈ M temp , 0 < (ξ) < +∞. Lemma 2.7 dealt with the free boundary condition case, so this followed from the stability assumption (8). Since ( ξ ) ≠ ( ), this now follows in the same way from (10).
We now state the main result of this subsection: Proposition 3.10. The probability measurē is an in nite-volume Gibbs measure with energy functional .
Proof. Sincē is concentrated on the tempered con gurations, we have to check that, for any nite-volume , the following DLR equation is satis ed under̄ : where is a measurable, bounded and -local functional. Fix ∈ B (ℝ ). We would like to use the fact that its nite-volume approximations (̄ ) satisfy (DLR) ; but since they are lattice-stationary and periodic, this is not true. To overcome this di culty, we use some approximation techniques, articulated in the following three steps: i. An equivalent sequence: We introduce a new sequence (̂ ) and show it is asymptotically equivalent to (̄ ) ii. A cut-o kernel: We introduce a cut o of the Gibbsian kernel by a local functional iii. Gibbsianity of the limit measure: We use estimations via the cut-o kernel since: for to be moved out of , one of the components of should be larger than − 0 ( 0 options for this); there are 2 directions can be moved through ; and the other − 1 components of ∈ ∩ ℤ are left free (2 − 1 options). Calling ′ . . = 0 , we nd Now, for any 4 > 0 (which will be xed later), we split the above integral over the set ∑ ( , )∈ (1 + ‖ ‖ + ) ≥ 4 and its complement. We obtain Fix > 0; for ≥ 2(1+ 4 ) ′ , the rst term is smaller than /2. To control the second term, we apply Lemma 3.8, to the sequence (̄ ) , and the function ( , ) = 1 + ‖ ‖ + ; we nd 4 > 0 such that the second term is smaller than /2, uniformly in , and conclude the proof of this step.
ii. A cut-o kernel: We know that̂ satis es (DLR) , i.e. for any -local and bounded functional If ξ ↦ M ( ) (ξ, ) were a local functional, we would be able to conclude simply by taking the limit in on both sides of the above expression, sincē = lim ̂ for the topology of local convergence. But this is not the case because of the unboundedness of the range of the interaction. We are then obliged to consider some approximation tools.
To that aim, we introduce a ( , 0 )-cut o of the Gibbsian kernels (ξ, ), which takes into account only the points of ξ belonging to a nite volume and having marks smaller than 0 .
converges to 0 as 0 ↑ ∞ and ↑ ℝ uniformly in ξ ∈ M t . Similarly, We denote by the law on ℝ + of the supremum norm of the Langevin di usion starting in 0; since the process . . = sup ∈[0, ] | | 2+ is a submartingale, we can apply Doob's inequality to get Remark. The previous reasoning can be generalised by considering the evolution of the Langevin dynamics in ℝ for any > 2. Assumption (24) should then be reinforced by replacing the (2 + ′ )-exponent with a ( + ′ ) exponent, in order to obtain the niteness of the super-exponential moment (H ).
Let us now describe the kind of interaction we consider between the marked points of a con guration. It is not necessarily superstable, and consists of a pair interaction, concerning separately the (starting) points and their attached di usion paths, and a self interaction.
Note that (27) implies that the potential -de ned on the location space ℝis stable in the sense of Ruelle (see [22] and [17]), with stability constant c , i.e. Under these assumptions, the potential Φ is stable c Φ = c in the following sense: It is then straightforward to prove that such an energy functional satis es the stability assumption (H ) with constant c = c 1 ∨ c Φ , and the range assumption (H ). Moreover, the local uniform-stability assumption (H . ) also holds: Let ∈ M and ξ ∈ M t , t ≥ 1, and denote = ⊕ (0, r). We have We can assume that ξ is of nite energy, and therefore use (27)  Together with the stability of ↦ ( ), this yields the following lower bound for the conditional energy: ( ξ ) ≥ −(c 1 ∨ 2c Φ ) x∈ (1 + ‖ ‖ + ).
Remark. The above is an example of a pair potential with nite but not uniformly bounded range.