Dr Michelle C White

Clare Hommers

Peter Sarah Parry

White Mc

Pain management in 100 episodes of severe mucositis in children

Keywords: pain, analgesia, ketamine, hematology, oncology, opioid pain, analgesia, ketamine, hematology, oncology, opioid

INTRODUCTION

Oral and gastrointestinal (GI) mucositis is a common, debilitating inflammatory condition affecting the mucosal tissues of the alimentary tract following many anticancer treatments. From a patients perspective, mucositis is one of the most debilitating side effects of their treatment [START_REF] Bellm | Patient reports of complications of bone marrow transplantation[END_REF] . It occurs in up to 100% of patients undergoing high dose chemotherapy with haemopoietic stem cell (HSCT) or bone marrow transplantation (BMT); over 50% of patients undergoing head and neck radiotherapy; and can occur in 5 -15% of patients receiving standard chemotherapy.

The severity of mucositis has been evaluated using a number of different scales. Two of the most commonly used scales are the World Health Organisation (WHO) Oral Toxicity Score [START_REF]Handbook for reporting results of cancer treatment[END_REF] and the National Cancer Institute Common Toxicity Criteria (NCI-CTC) for Oral Mucositis [3;3;3]. The NCI-CTC score grades severity from 0 to 4 with severe mucositis defined as Grade 3 or 4. Severe mucositis is associated with significant pain that may necessitate dose reductions, delays or premature termination of therapy causing excess morbidity and mortality. Symptoms last from a few days to several weeks. Basic oral care and systemic analgesics are the mainstay of treatment. Intravenous morphine is recommended as the analgesia of choice for children [START_REF] Mouth | Mouth Care for Children and Young People with Cancer: Evidencebased Guideline[END_REF].

It is recognised that in some children with cancer, pain may become 'morphine -resistant' and the use of ketamine has proved effective [START_REF] Klepstad | Long-term treatment with ketamine in a 12-year-old girl with severe neuropathic pain caused by a cervical spinal tumor[END_REF][START_REF] Tsui | Intravenous ketamine infusion as an adjuvant to morphine in a 2year-old with severe cancer pain from metastatic neuroblastoma[END_REF][START_REF] Conway | Use of continuous intravenous ketamine for endstage cancer pain in children[END_REF][START_REF] Ugur | Oral ketamine for pain relief in a child with abdominal malignancy[END_REF]. Ketamine is a NMDA receptor antagonist and can be used as both an analgesic and anaesthetic. Traditionally its use in routine acute pain management has been limited by fear of psychomimetic side effects. However, recently it has regained popularity at low doses in many types of acute and chronic pain [START_REF] White | Pain management in fulminating ulcerative colitis[END_REF][START_REF] Macpherson | Ketamine and midazolam delivered by patient-controlled analgesia in relieving pain associated with burns dressings[END_REF][START_REF] Bredmose | Pre-hospital use of ketamine in paediatric trauma[END_REF].

We have used ketamine as an adjunct to morphine in children with mucositis whose pain is difficult to manage with morphine alone. We conducted a retrospective evaluation of the pain management of 100 episodes of severe mucositis in children and present our findings. We collected data from 103 consecutive episodes of severe mucositis in 85 children (some patients had more than one episode of mucositis). Severe mucositis was defined as Grade 3 or 4 according to the NCI-

CTC scales (3).

A standard data collection form was used and the following data recorded: patient demographics, primary diagnosis, 24 hourly morphine consumption, 24 hourly ketamine consumption, pain scores and side effects. Pain was assessed hourly with an appropriate validated tool, most children used a self-reporting tool, Wong and Baker Faces with a 0-10 numerical rating scale [START_REF] Wong | Pain in children: comparison of assessment scales[END_REF] or a visual analogue score, but for children unable to self-report the child's nurse used the FLACC observational tool [START_REF] Merkel | The FLACC: a behavioral scale for scoring postoperative pain in young children[END_REF].

Pain was charted as is the standard clinical practice in Bristol Royal Hospital for Children on a categorical scale: 0 = no pain (pain score = 0); 1 = mild pain (pain scores 1 -3); 2 = moderate pain (pain scores 4 -7); 3 = severe pain (pain scores 8 -10). The following side effects were recorded: over sedation (somnolent and difficult to rouse), nausea, vomiting, pruritus, urinary retention and psychomimetic.

Pain management protocols are shown in Table 1. Morphine and ketamine are known to be stable when mixed in a saline solution [START_REF] Schmid | The stability of a ketamine-morphine solution[END_REF]. Patient controlled analgesia (PCA) or nurse controlled analgesia (NCA) was used as appropriate for each child. After initial dosing, the amount of opioid was rapidly titrated to effect under the direction of the 24 hour Acute Pain Service. Ketamine therapy was initiated at the discretion of the pain service or at the request of the child. In circumstances where a child had previously received ketamine adjuvant analgesia with good effect, it is our routine practice to allow these children to choose their preferred analgesia (morphine or morphine and ketamine PCA / NCA) on subsequent occasions. All children received regular intravenous paracetamol as per the standard practice in our hospital.

Table 1 Pain Management Protocols

Morphine PCA Morphine 1 mg.kg -1 made up to 50 mL with 0.9% saline Bolus dose: 1-2mL (=20-40mcg.kg -1 )

Background infusion: 0-2mL.h -1 (=0-40mcg.kg.h -1 )
Lockout period: 5 min The initial 4 hour maximum dose was changed as the child's analgesic requirements changed

RESULTS

Of the 103 consecutive mucositis episodes in 85 children we excluded 3 mucositis episodes. Two were excluded because the child's analgesia was changed from morphine to fentanyl. In one child this was because of significant renal impairment secondary to the chemotherapy; and the other child was changed to fentanyl because of sepsis induced hypotension and fentanyl was considered to provide greater cardiovascular stability in this situation. The third child was excluded because the records were incomplete. This left a total of 100 episodes of mucositis in 82 children for analysis. Of the 82 children, The median daily morphine usage for Group 1 and Group 2 are shown in Figure 1. The morphine consumption of children in Group 2 (i.e. those requiring the addition of ketamine to supplement morphine analgesia) was more than double the consumption of children in Group 1 (i.e. those whose pain management required morphine alone). Children in Group 2 were also more likely to be female and older in age (median age 12 years versus 7 years) as shown in Table 2. The average duration of symptoms was 7 days in Group 1 (morphine only), and 12 days in Group 2 (adjuvant ketamine). On average, adjuvant ketamine was started after 5 days which corresponded with a reduction in the average daily morphine consumption as can be seen in Figure 1.

In total, we collected data for 627 days of morphine only therapy and 233 days of morphine and ketamine therapy. The average pain scores and incidences of side effects for the total number of days of morphine (n = 627) and morphine and ketamine (n = 233) therapy are shown in Tables 3 and4

DISCUSSION

We have described the analgesic management of 100 episodes of severe mucositis in children. Our data show that (24/92) 26% of episodes of mucositis required the addition of ketamine to the standard morphine PCA/NCA in order to control pain. Prior to the addition of ketamine, this group of children had rapidly escalating opioid requirements needing approximately twice as much morphine as those requiring morphine alone (1000 versus 500 mcg.kg.day -1 , see Figure 1). Adjuvant ketamine therapy was started an average of 5 days after the onset of symptoms and was associated with morphine consumption of over 1000 mcg.kg.day -1 (see Figure 1). In 8/100 mucositis episodes, a morphine and ketamine PCA was preferred and requested by 4 children who were undergoing repeat chemotherapy regimes associated with mucositis.

Ketamine has been used in the treatment of adult cancer pain and was the subject of a Cochrane review [START_REF] Bell | Ketamine as an adjuvant to opioids for cancer pain[END_REF]. However there are very few reported uses in children with cancer. Four case reports describe a total of five children [START_REF] Klepstad | Long-term treatment with ketamine in a 12-year-old girl with severe neuropathic pain caused by a cervical spinal tumor[END_REF][START_REF] Tsui | Intravenous ketamine infusion as an adjuvant to morphine in a 2year-old with severe cancer pain from metastatic neuroblastoma[END_REF][START_REF] Conway | Use of continuous intravenous ketamine for endstage cancer pain in children[END_REF][START_REF] Ugur | Oral ketamine for pain relief in a child with abdominal malignancy[END_REF], who received ketamine during end of life care. Finkel [START_REF] Finkel | Ketamine as an adjuvant for treatment of cancer pain in children and adolescents[END_REF], describes the use of ketamine in 11 adolescents with uncontrolled pain; in 8 out of 11 ketamine improved pain control and modulated the need for escalating opioid analgesia. In our case series, 24/92 (26%) of episodes of severe mucositis required adjuvant ketamine because of escalating morphine requirements and suboptimal pain control; and in 8 out of 100 mucositis episodes, a ketamine and morphine PCA was requested by children who had experienced this form of analgesia previously and expressed a preference for it. Our data show adjuvant ketamine treatment appears to be associated with reduced morphine consumption and improved pain scores (see Group 2: Figure 1 and Table 3). Ketamine was used in almost one third (28/100) of mucositis episodes (totalling 233 days of treatment) and was well tolerated with minimal side effects (Table 4). Psychomimetic side effects, such as hallucinations, are reported to occur in up to 10% of children [START_REF] Sussman | A comparative evaluation of ketamine anesthesia in children and adults[END_REF] and traditionally have limited the role of ketamine in acute pain management, but none of our children experienced hallucinations.

Mucositis is a painful condition affecting the mouth and alimentary tract characterised by painful erythema and ulceration which, when severe, prevents swallowing thereby necessitating intravenous hydration and parenteral nutrition. Children even find it impossible to swallow saliva. Intravenous opioids are the analgesia of choice [START_REF] Mouth | Mouth Care for Children and Young People with Cancer: Evidencebased Guideline[END_REF]. However, in our series morphine PCA / NCA provided suboptimal analgesia in 24 / 92 episodes (26%).

Ketamine is a N-methyl-D-aspartate (NMDA) receptor antagonist and can be used as both an analgesic and an anaesthetic. Painful stimuli activate NMDA receptors and cause spinal cord hyperexcitability.

This leads to complex changes causing central sensitization, wind-up phenomena and severe pain [START_REF] Woolf | The induction and maintenance of central sensitization is dependent on Nmethyl-D-aspartic acid receptor activation; implications for the treatment of post-injury pain hypersensitivity states[END_REF].

Laboratory data suggests blockage of the NMDA receptor can prevent central sensitisation thereby alleviating pain [19;20]. In this study of 100 mucositis episodes, we found that in 24 episodes (19 children) where morphine requirements were high and continuing to escalate, the addition of ketamine appeared to be beneficial. The quality of analgesia improved, and the morphine requirements reduced.

The reason for the apparent 'morphine resistance' in some children is unclear and may relate to pharmacogenetic reasons. We found a trend towards greater opioid requirements in older females, a group known to experience more pain and higher morphine consumption in other settings [START_REF] Dahan | Sex-specific responses to opiates: animal and human studies[END_REF]. An alternative explanation could be opioid tolerance although no child was receiving prior maintenance opioids. The apparent 'morphine resistant' pain was not related to neuropathic pain. Some children did have neuropathic pain, but this was clearly distinguishable from the mucositis pain and was treated with gabapentin administerd nasogastrically to good effect. Different chemotherapeutic regimes may induce hepatic enzymes and accelerate morphine clearance, thereby necessitating higher opioid requirements.

However, we did not record this data and the heterogenous nature of our population would preclude analysis. Even in a homogenous population this would be difficult to study given the variety of anticancer protocols which vary by tumour type and stage of disease, the inter-patient variability of toxic effects on hepatic metabolism and renal clearance, and the vast numbers of chemotherapy agents with overlapping toxicity profiles [START_REF] Latham | Anesthetic considerations for the pediatric oncology patient--part 1: a review of antitumor therapy[END_REF]. This study did not aim to evaluate the pharmacokinetics but our results indicate that further investigation in this area is warranted.

Treatment of mucositis hinges on good oral care and analgesia. A study of the perspectives of children and their parents found that although children were aware of the significance of oral care, the discomfort associated with oral care forced some children to neglect this aspect of treatment. Parents also found oral care a stressful event for them and their children. Furthermore, mucositis presented children with a difficult dilemma whereby it was too painful to chew and swallow food yet they felt very hungry [START_REF] Cheng | Oral mucositis: a phenomenological study of pediatric patients' and their parents' perspectives and experiences[END_REF].

Adult patients rate pain from mucositis as one of the most debilitating side effects of their cancer treatment [START_REF] Bellm | Patient reports of complications of bone marrow transplantation[END_REF]. Optimal pain management guidelines are therefore critically important. Our data suggests that children with escalating morphine requirements who appear to benefit from adjuvant ketamine, tended to be older (median age 12 versus 7 years), were more likely to be female, and had a longer average duration of symptoms (12 v 7 days).

This study has several limitations, namely it is a retrospective analysis, we did not compare analgesia regimens, and we did not attempt to determine risk factors for 'opioid resistant' mucositis pain.

Benzodiazepines are known to reduce the incidence of hallucinations [START_REF] Lilburn | Ketamine sequelae. Evaluation of the ability of various premedicants to attenuate its psychic actions[END_REF] but are not routinely prescribed by our Pain Management Team when using ketamine analgesia. However, in our hospital diazepam is sometimes prescribed for anxiolysis in children by their oncologists. We did not record benzodiazepine use, which is another limitation of this study. However, we believe this descriptive evaluation of 100 episodes of mucositis in children (totalling 627 days of morphine only therapy and 233 days of combined morphine and ketamine therapy) adds evidence to the literature regarding the treatment of mucositis and the use of ketamine in children with cancer-related pain.

In summary, this study shows that morphine PCA provides inadequate analgesia in 26% of children with severe mucositis (Grade 3 or 4) despite these children receiving higher than average amounts of morphine (1000 mcg/kg/day compared with 500 mcg/kg/day, see Figure 1). Addition of ketamine to the morphine See separate file uploaded.

METHODS

  The clinical records of children with severe mucositis requiring intravenous opioid analgesia between May 2006 and April 2009 were reviewed. The project was registered with the Audit and Service Evaluation Department. Individual parental (or patient) consent was deemed unnecessary for this retrospective study.

Initial 4

 4 hour maximum dose * : 400mcg.kg-1 Morphine and Ketamine PCA Morphine 1 mg.kg -1 + Ketamine 1mg.kg -1 made up to 50 mL with 0.9% saline Bolus dose: 1-2 mL(=20-40mcg.kg -1 morphine and ketamine) Background infusion: 0-2mL.h -1 (=0-40mcg.kg.h -1 morphine and ketamine) lockout period 5 min, Initial 4 hour maximum dose * : 1000mcg.kg-1 Morphine NCA Morphine 1 mg.kg -1 made up to 50 mL with 0.9% saline Bolus dose: 1-2mL (=20-40mcg.kg -1 ) Background infusion: 0-2mL.h -1 (=0-40mcg.kg.h -1 ) maximum dose * : 400mcg.kg-1 Morphine and Ketamine NCA Morphine 1 mg.kg -1 + Ketamine 1mg.kg -1 made up to 50 mL with 0.9% saline Bolus dose: 1-2 mL(=20-40mcg.kg -1 morphine and ketamine) Background infusion: 0-2mL.h -1 (=0-40mcg.kg.h -1 morphine and ketamine) Lockout period: 20 min, Initial 4 hour maximum dose * : 1000mcg.kg -1 *

  stem cell transplant (HSCT); acute lymphoblastic leukaemia (ALL); acute myeloid leukaemia (AML); non-Hodgkins lymphoma (NHL); primitive neuroectodermal tumour (PNET)

  Figure 1: Daily morphine usage -Group 1 and 2 (data shown as median and interquartile

  children were aged 12 -14 years (Table2) and all chose to have ketamine and morphine from the onset of their symptoms because they had found it beneficial and preferable to morphine only on a previous admission. The addition of ketamine, for inadequate pain control or morphine related side effects, was

	needed in 24 / 92 (26%) of mucositis episodes. Analgesia was administered by PCA in 77 episodes and
	by NCA in 23 episodes (numbers of PCA and NCA per group: Group 1 = 53 : 15; Group 2 = 18 : 6;
	Group 3 = 6 : 2 respectively).
	F o r
	P
	e
	e r
	R
	e
	v i e
	w
	10 had more than one patient episode of mucositis (six children each had 2 episodes of mucositis during
	the study period; one child had 3 episodes; two children each had 4 episodes; and one child had 5
	episodes). Patient demographics and diagnosis are shown in Table 2.
	68 mucositis episodes (59 children) were treated with morphine only (Group 1); 24 mucositis episodes

(19 children)

required the addition of ketamine to supplement the morphine analgesia (Group 2); and 8 patient episodes (in 4 children) requested ketamine and morphine from the outset (Group 3). These

Table 2 Patient demographics and diagnosis

 2

		Morphine only	Morphine with	Morphine with
			ketamine added after	ketamine from
			initiation of treatment	initiation of treatment
	Number of mucositis	68 (59)	24 (19)	8 (4)
	episodes (number of
	children)
	Age (years) median [IQR]	7 [3 -14]	12 [6 -12]	13 [12 -14]
	Sex M : F	43 : 25	8 : 16	4 : 4
	(number of mucositis

Table 4 Incidence of side effects per total treatment days. Values given as absolute number per total treatment days (%)

 4

	Table 3	Median pain scores (0 = no pain, 1 = mild pain, 2 = moderate pain, 3 = severe
	pain)
			Pain Score
	Group 1:	r o F Morphine only	1
	Group 2a: Group 2b:	Morphine + ketamine (before ketamine was added) 2 P e Morphine + ketamine (after ketamine was added) 1
	Group 3:	e r Morphine + ketamine from the outset	1
		R
		e
		v i e
			w
		Morphine only days (n = 627)	Morphine + Ketamine days (n = 233)
		(Group 1 + Group 2a)	(Group 2b + Group 3)
	Over sedation	19 / 627 (3.0)	3 / 233 (1.3)
	Nausea	23 / 627 (3.7)	3 / 233 (1.3)
	Vomiting	25 / 627 (4.0)	9 / 233 (4.0)
	Pruritus	59 / 627 (9.5)	13 / 233 (5.7)

respectively. No episodes of psychomimetic side effects including hallucinations were reported.

ACKNOWLEDGEMENTS

The authors wish to thank Dr Tim Hooper for help with data collection.


Cédric Bernardin
email: cedric.bernardin@umpa.ens-lyon.fr

Valeria Ricci
email: valeria.ricci@unipa.it

A SIMPLE PARTICLE MODEL FOR A SYSTEM OF COUPLED EQUATIONS WITH ABSORBING COLLISION TERM

Keywords: 1991 Mathematics Subject Classification. 82C22, 82C21, 60F05, 60K35 Interacting particle systems, large numbers limit, absorption

We study a particle model for a simple system of partial differential equations describing, in dimension d ≥ 2, a two component mixture where light particles move in a medium of absorbing, fixed obstacles; the system consists in a transport and a reaction equation coupled through pure absorption collision terms. We consider a particle system where the obstacles, of radius ε, become inactive at a rate related to the number of light particles travelling in their range of influence at a given time and the light particles are instantaneously absorbed at the first time they meet the physical boundary of an obstacle; elements belonging to the same species do not interact among themselves. We prove the convergence (a.s. w.r.t. the product measure associated to the initial datum for the light particle component) of the densities describing the particle system to the solution of the system of partial differential equations in the asymptotics a d n n -κ → 0 and a d n ε ζ → 0, for κ ∈ (0, 1 2 ) and ζ ∈ (0, 1 2 -1 2d ), where a -1 n is the effective range of the obstacles and n is the total number of light particles.

Introduction

In this paper we propose a microscopic model for a system of partial differential equations consisting in a transport equation, having pure loss collision term, and a pure loss reaction equation; the equations are self-consistently coupled and they are meant to describe the evolution in time of the phase space densities in a binary mixture where the two species interact in such a way that each species inhibits the activity of the other, with an interaction proportional to their macroscopic (position space) densities. In particular, we want to rigorously derive the system of equations in a suitable asymptotics for the particle system.

The system of equations we shall deal with is a sort of very simple reactive transport system, though without conservation of masses; more complex reactive transport systems arise in many different contexts, like for instance in the modeling of biological systems, of porous media, of radiative transfer or, more in general, of various systems in the presence of chemical reactions (see e.g. [SDL, MM]). We are interested here in the analysis of the simplest reactive self-consistent coupling (i.e. the absorption coupling), and we shall therefore not include other terms in the equations.

From the point of view of particle modeling, an analysis of the absorption selfconsistent coupling for one species nonlinear equations can be found in models for reaction-diffusion equations, such as the ones proposed in [NOR] and [Szn]; in both papers, the particle system evolves according to a Brownian motion (to get the diffusion) and destruction of particles occurs with different mechanism: in the first paper the destruction is stochastic, with a death rate for the Brownian particles which is a function of stochastic exponential times, in the second one it is deterministic, and it happens as soon as collisions between particles occur.

We shall consider a binary semi-deterministic system where both kinds of interactions occur. More precisely, the particle system we shall consider consists in point like (light) particles moving uniformly among fixed, spherical particles (obstacles).

Particles not belonging to the same species do not interact among themselves and particles of different species interact in the following way: a light particle becomes inactive (or is absorbed) at the first time it meets an obstacle and an obstacle becomes inactive (or disappears) in a stochastic interval of time whose size is connected, through a local mean-field type interaction, to the number of light particles traveling within the area of detection (range) of the obstacle.

The main difficulty in the derivation of the (otherwise simple) system of partial differential equations originates from the self consistent structure of the problem: in order to overcome the mathematical troubles introduced from the self-consistent terms, we shall adopt a natural scheme for facing self-consistent problems in partial differential equations and particle systems and we shall prove the convergence using two levels of approximation of the original particle system, the first one eliminating the self-consistent structure and the second one reducing the correlations between the two species with respect to the original many-particle system. Our formalism is quite explicit and the strategy we shall adopt is similar to the procedure used in [NOR]. The adjustment of this procedure to our particle system is not trivial, since, at variance with the situation treated in [NOR], we deal with a two species system interacting in a non symmetric way and the instantaneous absorption of the light particles is in some sense a singular interaction with respect to the local mean-field type interaction which govern the obstacles lifetime, i.e. the interaction type considered in [NOR]. The presence of the deterministic component (the light particles) imposes a sufficiently careful analysis of the trajectories, in the style of the analysis performed for particle models of linear equations based on similar deterministic components (see e.g. [BGW], [DR1,DR2]). Nevertheless, the structure of the stochastic component (the obstacles) simplifies by a considerable amount the mathematics with respect to a two component, totally deterministic system.

Our final theorem establishes a weak law of large numbers for the empirical measures (associated to the two species of particles in the mixture) to the solution of the system of partial differential equations, almost surely with respect to the initial distribution of positions in the phase space of the light particles and in probability with respect to the distributions of positions and life time of the obstacles. This weak law of large numbers is valid in an asymptotics where the radius and the effective range of the obstacles vanish, keeping a finite action in the limit, and the number of particles grows up to infinity, these quantities being related in a way that we shall determine while proving the theorem. As a part of this relationship, we shall prove a simple condition (relating the diminishing rate of the effective range of the obstacles to the number of light particles) which guarantees the weak convergence of the product of an empirical measure times a sufficiently regular approximant of a Dirac delta distribution toward the product of their weak limits (under suitable regularity assumptions on the weak limit of the empirical measure and on the choice of the delta's approximant). This condition entails very useful bounds for our estimates and allows to get easily the required asymptotic result.

The equations, the particles model and the main result

Throughout the paper we shall use the following notations: in dimension d ≥ 2, we denote by x ∈ R d the position of a light particle and by v ∈ R d its velocity; t ∈ R + is the time variable. In general, configurations of M variables in R d will be denoted by boldface letters with subscript M (y M = (y 1 , . . . , y M )) and sequences of variables by capital letters with the subscript ∞ (Y ∞ = {y i } ∞ i=1 ). When needed (e.g., in the functional spaces) we shall use R d

x and R d v resp. for the position and the velocity spaces. For p ∈ R d and r ∈ R + we shall denote by B r (p) = {y ∈ R d : |p -y| ≤ r} the ball of radius r centered in p. For a given z = (x, v) ∈ R d × R d , the free flow associated to the light particle with initial position (in the phase space) z is T t (z) = (T t 1 (z), T t 2 (z)) = (x + vt, v), t ∈ R + , while x(t) = T t 1 (z), t ∈ R + , is its trajectory, and for ε > 0 (1)

T ε (t, z) = y ∈ R d ; ∃s ∈ [0, t), |y -x(s)| ≤ ε
denotes the flow tube of radius ε associated to the trajectory up to the time t. Unless differently stated, for a given random variable η, we denote by P η its associated probability distribution, and for a measure π (random variable η), we denote by E π (E η ) the expectation w.r.t. π (η). In order to simplify the notation, in many stochastic variables depending on configurations y M , we shall label this dependence by a simple subscript M , instead of rewriting each time the whole configuration.

We shall moreover denote ⇀ the weak convergence (convergence in law) in the space of finite measures and * ⇀ the *-weak (vague) convergence on the space of Radon measures, and by K (with some subscript) any generic constant whose value needs not to be specified. We shall sometimes shorten the notation for sums and difference of functions with the same argument as (f ± g)(w) = f (w) ± g(w).

For T > 0, we consider the following system of partial differential equations for the two densities

f = f (t, x, v), σ = σ(t, x), t ∈ [0, T ], x ∈ R d , v ∈ R d :        ∂ t f + v • ∇ x f = -C d |v|σf ∂ t σ = -Θ( R d dvf )σ f (0, x, v) = f 0 (x, v) σ(0, x) = σ 0 (x), (2)
where Θ and C d = {ω∈R d :|ω|=1} |n•ω|dω, |n| = 1, are positive constants. In order to have a convenient existence and uniqueness theorem for the solution (cf. Appendix 6.1), we assume

f 0 ∈ L 1 (R d v ; W 1,∞ (R d x )), vf 0 ∈ L 1 (R d v ; W 1,∞ (R d x )) ∩ L ∞ (R d x × R d v ), v 2 f 0 ∈ L 1 (R d v ; L ∞ (R d x )) and σ 0 ∈ W 1,∞ (R d x ).
We shall show that the system (2) can be derived from a semi-deterministic particle system of the kind we specified in the Introduction. The particle system will be described in next paragraphs.

Obviously, once established the correct asymptotics to get equation (2) from the particle system we have chosen, we shall simply write lim n→∞ to denote a limit in this asymptotics (n and ε being related). This peculiar use of the notation will be consequent to the hypothesis in the environments where the notation is used.

2.1. The particle system: initial data statistics. We consider configurations of spherical fixed obstacles of radius ε with stochastically distributed positions at time t = 0 and we denote by c M = (c 1 , . . . , c M ) the coordinates of their centers. Obstacles may overlap (i.e., configurations such that for some i, k, |c ic k | ≤ 2ε are allowed) and M = 0 corresponds to absence of obstacles. In this paper, we shall assume that obstacles positions follow a Poisson distribution with parameter µ ε , i.e. that the probability distribution of finding M obstacles in a bounded measurable set Λ ⊂ R d is given by:

(3)

P (dc M ) = e -µε[Λ]L µ M ε M ! dc 1 . . . dc M ,
where (here and in what follows) [Λ] L denotes the Lebesgue measure of the set Λ. We shall adapt the initial datum σ 0 (x) to this choice for the statistics of the obstacles.

We consider then n point-like particles, located initially at positions x 1 , . . . , x n and moving uniformly among the obstacles with velocities v 1 , . . . , v n ; we shall denote the phase space coordinate of the i-th light particles as z i = (x i , v i ). A point in the n-particles phase space is denoted as z n = (x 1 , v 1 , . . . , x n , v n ) and a sequence of initial data is denoted as

Z ∞ = {z i } ∞ i=1 ∈ (R d × R d ) ∞ .
We describe both species of particles by means of their empirical measure (see e.g. [G1]).

Given a 1-particle probability density f 0 , we denote by P the (infinite product) probability measure defined on the space of infinite sequences (R d × R d ) ∞ by f 0 , i.e. such that, for n = 1, 2, . . . and

A (i) x × A (i) v ⊂ R d × R d , (4) P(Z ∞ : z i ∈ A (i) x × A (i) v , 1 ≤ i ≤ n) = n k=1 A (k) x ×A (k) v f 0 (x, v)dxdv.
The sequence of initial empirical measures for the light particles {µ 0 n } ∞ n=1 is then such that:

(5)

µ 0 n (t, x, v; z n ) = 1 n n i=1 δ zi (x, v) ⇀ f 0 (x, v) P -a.e. w.r.t.Z ∞ .
We shall require moreover some regularity condition on f 0 .

2.2. The particle system: dynamics. We define now the dynamics of the system.

We consider a sequence τ ∞ = {τ 1 , . . . , τ k , . . .} of independently distributed exponential variables and we define the following stochastic functions:

• The risk function at a given position c,

(6) V n,ε,M (t, c) = 1 n n i=1 t 0 dsq n (x i (s) -c)ξ n,ε,M (s, z i ),
where q n (x) = a d n q(a n x), is an (at least continuous) approximant, up to a multiplicative constant, of the Dirac delta distribution, with q a non negative, radial (q(x) = q(|x|)) function such that R d q(y)dy = Θ > 0 and ξ n,ε,M is the stochastic function defined below.

• The life functions 1 , resp. ξ n,ε,M (t, z) for a light particle with initial position and velocity (x, v) and η n,ε,M (t, c k ) for an obstacle centered in

c k ∈ {c 1 , . . . , c M }, (7)
ξ n,ε,0 (t, z) = 1 M ≥ 1 ξ n,ε,M (t, z) = I {x(s) / ∈ M h=1 Bε(c h )ηn,ε,M (s,c h ) ∀s∈[0,t)} η n,ε,M (t, c k ) = I {Vn,ε,M (t,c k )<τ k } .
• The maximal collision time between a light particle with initial phase space position z = (x, v) and an obstacle located in c,

(8) T z,c = inf s∈R+ {s : |x(s) -c| ≤ ε},
with no reference to the activity of both particles. Because no life functions are involved in this definition, T z,c can be infinite.

In the particle system defined through ( 6) and ( 7), the obstacles become inactive at stochastically distributed times defined at a given position c k through the risk function (6) as the time

t k = t k (c k , τ k ) s.t. V (t k , c k ) = τ k ,
while the light particles are absorbed at the first time they meet an active obstacle (i.e. as soon as |x i (T coll )c k | = ε for some k and T coll < t k ; notice that T coll is defined w.r.t. active particles, and it is therefore a different variable w.r.t. T z,c ).

Unless we require q to have compact support, a light particle located at x can affect the life function of a far obstacle (it suffices that q(x-c) = 0); on the contrary, a given light particle interacts only with obstacles which are met by its trajectory in space. Since, for a given n, the volume including all light particle trajectories up to time T , V n , is such that

(9) V n ⊂ B max i=1,...,n |xi|+ max j=1,...,n |vj |T +1 (0) = B n ,
and, in studying the interacting system, we do not need to consider quantities related to obstacles which can not be met from any light particles, we may (for a given n) restrict the expectation value with respect to (3) to the volume Λ = Λ n , for

1 We use, here and later, the notation

M k=1 Bε(c k )η n,ε,M (s, c k ) to mean k∈{1,...,M }: η n,ε,M (s,c k )=1 Bε(c k ).
Although not formally correct, this notation allows us to write less cumbersome formulas.

a growing sequence of bounded Lebesgue measurable sets {Λ n } n s.t. B n ⊂ Λ n and lim n→∞ Λ n = R d . We denote by E n c the expectation value with respect to the centers distribution (3) with such a choice for Λ. We shall denote by

E n = E n c E τ = E τ E n c ,
where E τ is the expectation value w.r.t. to the exponential times sequence τ ∞ , and by P n the corresponding probability distributions.

2.3. Scaling laws: heuristics. In order to derive the system (2) from the particle system defined above in the macroscopic limit where the radius ε of the obstacles vanishes, we have to choose the scaling for densities of both species of particles in a suitable way.

As a first requirement, we want the mean free path of the light particles to be finite at the scale at which the system is observed, in such a way to keep track, in the chosen asymptotics, of the interaction between the light particles and the obstacles. To this purpose, we fix the rate µ ε of the Poisson process (3) to be such that:

(10) µ ε ε d-1 = µ > 0
and, since we consider a uniform initial macroscopic distribution of obstacles, in (2) we assume σ 0 (x) = µ. When V n,ε,M ≡ 0 (i.e. q ≡ 0), formula (10) defines the so called Boltzmann-Grad scaling, which has been analysed for linear particle systems in particular in connection with the asymptotics of the Lorentz gas and its variants (see e.g. [Ga, Sp, BBS], [DR1] for the stochastic case, [BGW, G2, RW] for the periodic case and the beautiful review [G3] -focused on the periodic case-). When V n,ε,M ≡ 0, (10) guarantees the finiteness of the mean free path for a given light particle as soon as V n,ε,M , in flow tubes and for a non-negligible set of obstacles configurations, is bounded uniformly in n. This is true, under the conditions on f 0 specified at the beginning of Section 2 and the conditions on q given in Section 2.2 (i.e. q ∈ L 1 (R d ) ∩ C(R d )), whenever a d n n -1 2 ≤ const. A second requirement is to have, in the ε → 0 asymptotics, vanishing correlations between light particles and obstacles, so as to obtain a Markovian limit (i.e. without memory effects), coherently with the structure of (2). Since the dominant part of correlations is associated to grazing crossings of light particle trajectories, and in particular to the mean volume V ε g spanned from grazing trajectories crossings within the effective range of an obstacle, this requirement connects the scaling in the effective range a -1 n to the scaling in the mean density of the obstacles. In order to have negligible correlations, the mean number of multiple collisions per unit volume has to vanish in the limit. This leads to the condition

µ ε a d n V ε g → 0. Notice that possibly V ε g = O(ε d-1
), and having a better asymptotic behavior depends essentially on the regularity of the limit measure of the light particle component.

We have then a third, technical, requirement due to the form of ( 6): since eventually we want to obtain a mean-field limit, and therefore we want to be able to perform limits of mean values with respect to µ 0 n of sequences of functions converging to singular limits (such as Dirac delta distributions), we need a condition assuring the convergence (in a suitable sense) of the product of such sequences times the initial empirical measure: this is achieved if the empirical measure converges faster to its (sufficiently regular) limit than the chosen approximant concentrates, in such a way that the empirical measure is in practice equivalent to its limit density well before than the delta's approximant concentrates in its center. We can guess roughly that this happens if the fraction of particles fluctuating around mean values in a volume corresponding to the effective range of an obstacle vanishes in the chosen asymptotics, and therefore 1 √ n = o(a -d n ). We shall prove a more precise asymptotics in Appendix 6.2 Given (10) and the just described scaling laws, for a given configuration of obstacles, the empirical measure at time t for the light particle component (representing its mesoscopic density in phase space) is:

(11) µ n (t, x, v; z n , c M ) = 1 n n i=1 δ T t (zi) (x, v)ξ n,ε,M (t, z i )
and the (macroscopic) density of the obstacle component (i.e. the number of obstacles per unit volume) is expressed as:

(12) σ n (t, x; z n , c M ) = ε d-1 M k=1 δ c k (x)η n,ε,M (t, c k ).
2.4. The limit process. Under the conditions we assumed on (f 0 , σ 0 ), as shown in paragraph 6.1 in the Appendix, the problem (2) admits a unique solution

(f, σ) ∈ L ∞ ([0, T ] × R d × R d ) × L ∞ ([0, T ] × R d ), with dvf ∈ L ∞ ([0, T ] × R d ) (and actually dvf (t, •, v) ∈ C b (R d ))
. This solution can be expressed in semi-explicit form as:

f (t, x, v) = f 0 (x -vt, v)e -C d |v| t 0 dsσ(s,x-v(t-s)) σ(t, x) = σ 0 (x)e -Θ t 0 ds R d dvf (s,x,v) . (13)
In order to be able to compare solutions of (2) with the stochastic measures ( 11) and ( 12), we define the risk functions and the life functions associated to the limit process described by (2) as:

V f L (t, c) = Θ t 0 ds R d dvf (s, c, v) η f L (t, c) = I {VL(t,c)<τ } (14) U σ L (t, z) = C d |T t 2 (z)| t 0 dsσ(s, T s 1 (z)) ξ σ L (t, z) = I {UL(t,z)<τp} , (15)
where τ and τ p are independently distributed exponential variables and τ = τ k when c = c k . In this way, the semi-explicit form of the solution (13) can be re-expressed as

f (t, x, v) = f 0 (x -vt, v)E τp ξ σ L (t, T -t (x, v)) σ(t, x) = σ 0 (x)E τ η f L (t, x) . (16)
In order to simplify the notation, we shall omit in what follows the dependence on the density functions (

V L = V f L , U L = U σ L ).
We want to establish a (weak) law of large numbers for the measures (11) and (12). More precisely, we shall prove the following theorem (we recall that S (R n ) denotes the space of C ∞ (R n ) functions of rapid decay at infinity [Sch, DL]):

Theorem 1. Consider the non-negative functions f 0 and q and assume

• f 0 ∈ S (R d × R d ) L 1 (R d v ; W 1,∞ (R d x )
) is a probability density such that

(17) R d dvf 0 ∈ S (R d ) with R d dvf 0 (0, v) > 0,
and

vf 0 ∈ L 1 (R d v ; W 1,∞ (R d x )), v 2 f 0 ∈ L 1 (R d v ; L ∞ (R d x )); • q is a radial function s.t. q ∈ S (R d ) and R d dxq(x) = Θ > 0; • {a n } ∞
n=1 is such that a n > 0, lim n→∞ a n = ∞ and there exists some κ ∈ (0, 1 2 ) such that

(18) lim n→∞ a d n n κ = 0; • {ε} ∞ n=1 = {ε n } ∞ n=1 s.t. ε n > 0 and (19) lim n→∞ a d n ε ζ = 0 for some ζ ∈ (0, 1 2 -1 2d
). Then, P-almost everywhere w.r.t. sequences of initial data Z ∞ and in probability w.r.t. P and P τ , when n → ∞

(20) µ n (t, x, v; z n , c M ) = 1 n n i=1 δ T t (zi) (x, v)ξ n,ε,M (t, z i ) ⇀ f (t, x, v) (21) σ n (t, x; z n , c M ) = ε d-1 M i=1 δ ci (x)η n,ε,M (t, c i ) * ⇀ σ(t, x),
where ξ n,ε,M , η n,ε,M are defined in paragraph 2.2 and (f, σ) is the unique solution of

       ∂ t f + v • ∇ x f = -C d |v|σf ∂ t σ = -Θ( R d dvf )σ f (0, x, v) = f 0 (x, v) σ(0, x) = µ. (22)
We shall see in paragraph 6.2 in the appendix that hypothesis ( 18), together with suitable regularity assumptions on f 0 (all included in our theorem), guarantees the following P-a.s. convergence

|v| j µ 0 n (x, v)q n (x) ⇀ Θ|v| j f 0 (x, v)δ 0 (x) (23) |v| j µ 0 n ⇀ |v| j f 0 ⊗ k i=1 |v| j µ 0 n ⇀ ⊗ k i=1 |v| j f 0 for j = 0, 1, 2, . . . , P , k = 1, . . . Q, with given positive integers P, Q.

Definitions of the approximating systems and proof of the main theorem

The main difficulty in studying (2) and its associated particle system ( 6),( 7) is that we have to deal with a self-consistent problem. The first step is therefore to find, both for the limit system (2) and for the particle system defined from ( 6),( 7), suitable approximating systems which do not share this self-consistent structure.

We first recall here without proof, since it will be useful for the sequel and we shall use it largely in various steps of the convergence proof, lemma 3.2 in [NOR], concerning bounds of distances of stochastic variables of the form η(t) = I {S(t)<τ } in terms of distances between their associated risk function S.

Lemma 1. Let τ > 0 be an exponentially distributed real variable and define, for

i = 1, 2, t ∈ [0, T ], η i (t) = I {Si(t)<τ }
, where S i , S are non negative, non decreasing, right continuous random functions. Then, for any δ > 0

(24) |η 1 (t) -η 2 (t)| ≤ 1 δ |S 1 (t) -S(t)| + 1 δ |S 2 (t) -S(t)| + I {|S(t)-τ |<δ} .
Remark 1. Whenever at least one or both risk functions S i , i = 1, 2 are independent of the exponential time τ , the bound simplifies in the first case as

E τ |η 1 (t) -η 2 (t)| ≤ 1 δ E τ |S 1 (t) -S 2 (t)| + 2δ
, and in the second case as

E τ |η 1 (t) - η 2 (t)| ≤ |S 1 (t) -S 2 (t)| .
3.1. Approximation of the limit system. As shown in Appendix 6.1, under suitable hypothesis on f 0 , the solution of the system of equations ( 2) can be obtained as the k → ∞ limit of the sequence of solutions of the sequence of linear systems defined as:

f (0) (t, x, v) = f 0 (x -vt, v), σ (0) (t, x) = µ (25)        ∂ t f (k) + v • ∇ x f (k) = -C d |v|σ (k-1) f (k) ∂ t σ (k) = -(Θ R d dvf (k-1) )σ (k) f (0, x, v) = f 0 (x, v) σ(0, x) = µ k = 1 . . . .
More precisely, the sequence of semiexplicit solutions of ( 25) is, for k = 1, 2, . . .,

f (k) (t, x, v) = f 0 (x -vt, v)e -C d |v| t 0 dsσ (k-1) (s,x-v(t-s)) σ (k) (t, x) = µe -Θ t 0 ds R d dvf (k-1) (s,x,v) , (26)
and we have, when k → ∞ (cf. Appendix 6.1)

(27) f -f (k) L ∞ ([0,T ]×R d ×R d ) → 0 , R d dv(f -f (k) ) L ∞ ([0,T ]×R d ) → 0, σ -σ (k) L ∞ ([0,T ]×R d ) → 0.
The risk functions associated to the k-th system (25) are defined as:

V (k) (t, c) = Θ t 0 ds R d dvf (k-1) (s, c, v) (28) Ū (k) (t, z) = C d |T t 2 (z)| t 0 dsσ (k-1) (s, T s 1 (z)), (29)
and, because we shall need it later (cf. the definition of the system (33)), we define V (0) (t, c) = 0. Their associated life functions are:

η(k) (t, c) = I { V (k) (t,c)<τ } ξ(k) (t, z) = I { Ū(k) (t,z)<τp} , (30)
with τ and τ p exponentially distributed times. Of course, (26) can be expressed in terms of V (k) and Ū (k) in a form analogous to (16).

3.2. Approximation of the particle system. In the same spirit, we may also approximate, for each n, the particle system described by ( 6), ( 7) by a suitable sequence of systems. For an initial datum z n for the n particles phase space position and configuration of obstacles c M (M = 0 in the absence of obstacles), we define this system, for integers M, k ≥ 1, j = 1, . . . , n and ℓ = 1, . . . , M , as:

(31) ξ (0) n,ε,0 (t, z j ) = ξ (k) n,ε,0 (t, z j ) = 1 V (0) n,ε,M (t, c ℓ ) = 0, η (0) n,ε,M (t, c ℓ ) = 1, ξ (0) n,ε,M (t, z j ) = 1 V (k) n,ε,M (t, c ℓ ) = 1 n n i=1 t 0 dsq n (x i (s) -c ℓ )ξ (k-1) n,ε,M (s, z i ) η (k) n,ε,M (t, c ℓ ) = I {V (k) n,ε,M (t,c ℓ )<τ ℓ } ξ (k) n,ε,M (t, z j ) = I {xj (s) / ∈ M h=1 Bε(c h )η (k-1) n,ε,M (s,c h ) ∀∈[0,t)} .
This sequence is a linearization of the original particle system (which is its formal limit when k → ∞) in the same way as ( 25) is a linearization of (2) and it satisfies what is called in [NOR] sandwiching property (see ( 48)). This property implies, in a given asymptotics for n and ε:

(32)

E n |V (k) n,ε,M -V (k) | → 0 =⇒ E n |V n,ε,M -V L | → 0 E n |ξ (k) n,ε,M -ξ(k) | → 0 =⇒ E n E τp |ξ n,ε,M -ξ L | → 0 E n E τ |η (k) n,ε,M -η(k) | → 0 =⇒ E n E τ |η n,ε,M -η L | → 0 E n | dzµ n φ(ξ (k) n,ε,M -ξ(k) )|→ 0 =⇒ E n E τp | dzµ n φ(ξ n,ε,M -ξ L )
| → 0 and using these implications we shall be able to bypass the direct evaluation of quantities related to the particle system (6),( 7).

The advantage in dealing with the two approximating sequences of systems defined by ( 25) and (31) instead of the original systems is that, for each k, the two components evolve in a given field, associated to the functions defined at the previous step k -1 in the sequence, and the original self-consistent structure is lost.

The next step would be then to show that ( 25) and ( 31) are, for a given k, asymptotically equivalent. Unfortunately, the system defined from (31), when ε is positive, still keeps strong correlations between all light particles and obstacles positions (in the phase space), correlations which are absent in (25); this makes hard the direct comparison of the two systems. Therefore, we need to define an intermediate system in which correlations among light particles and obstacles are further reduced. We shall prove then that this system is equivalent in the limit n → ∞ both to (25) and to (31).

3.3.

A system asymptotically equivalent to both approximating systems. For an initial datum z n for the n particles phase space position, and for configurations of obstacles c M (M = 0 in the absence of obstacles), we define, for integers M, k ≥ 1, j = 1, . . . , n and ℓ = 1, . . . , M , the intermediate system in the following way ( 33)

ξ(0) n,ε,0 (t, z j ) = ξ(k) n,ε,0 (t, z j ) = 1 ξ(0) n,ε,M (t, z j ) = 1, ξ(k) n,ε,M (t, z j ) = I {xj (s) / ∈ M h=1 Bε(c h )I { V (k-1) (s,c h )<τ h } ∀s∈[0,t)} Â(k) n,ε,M (t, c ℓ ) = 1 n n i=1 t 0 ds q n (x i (s) -c ℓ ) ξ(k-1) n,ε,M (s, z i ) η(k) n,ε,M (t, c ℓ ) = I { Â(k) n,ε,M (t,c ℓ )<τ ℓ }
where τ k , 1 ≤ k ≤ M , are independent exponentially distributed times. In (33), the life function of a light particle at level k is defined through fictitious obstacles life functions (I { V (k-1) (t,c)<τc} ), corresponding to the obstacles life functions at level k -1 associated to (25); the life function of an obstacle at level k, I { Â(k) n,ε,M <τc} , is defined through the light particles life functions at level k -1. In this way, the correlation between light particles and obstacles is weaker, compared to the same correlation in system (31), and this allows us to prove, when n and ε verify conditions ( 18) and ( 19), the convergence to both systems ( 25) and (31) in quadratic mean w.r.t. the expectation value E n .

3.4. Proof of the main theorem. We proceed now with the proof of our main theorem. To this purpose, we assume we have already proved the sandwiching property for the system (31) and the asymptotic equivalence of this last system to (25): we shall postpone to next sections the proof of lemmas and propositions concerning these two facts, since they are the core of the derivation; as pointed out in section 3.2, the convergence of ( 31) to (25) will be obtained passing through the equivalence in this asymptotics of (33) to both systems (31) and ( 25).

Proof of Theorem 1. Given the definitions in the previous paragraphs, we write

ξ n,ε,M = (ξ n,ε,M -ξ L ) + (ξ L -ξ(k) ) + ξ(k) and f = (f -f (k) ) + f (k) .
Then, using formula (13) for f and (26) for f (k) , we can write, for all φ ∈ C b (R d ×R d ) and for all k ≥ 1:

E n 1 n n i=1 φ(T t (z i ))ξ n,ε,M (t, z i ) - R d ×R d dxdvφ(x, v)f (t, x, v) ≤ (34) E n E τp 1 n n i=1 φ(T t (z i ))(ξ n,ε,M -ξ L )(t, z i ) + (35) φ ∞    E n E τp   1 n n j=1 |ξ L -ξ(k) |(t, z j )   + R d ×R d dxdvC d |v|f 0 (x, v) t 0 ds|σ -σ (k-1) |(s, x(s)) + (36) E n E τp 1 n n i=1 φ(T t (z i )) ξ(k) (t, z i ) - R d ×R d dxdvf (k) (t, x, v)φ(x, v) .
The term (34) vanishes on a set of full measure w.r.t. P as a consequence of the sandwiching property valid for (31) (cf. Corollary 1, Sec. 4.3) and of the convergence of ( 31) to (25) in the asymptotics specified in ( 18), (19) (cf. Proposition 4, Sec. 5).

Thanks to (23), and in particular to the convergence |v|µ 0 n ⇀ |v|f 0 , the term in curly brackets ( 35) is bounded (excepted on a set of zero measure with respect to P) by K a σσ (k-1) L ∞ ([0,T ]×R d ) , so that, by choosing a suitable k, it can be made arbitrarily small because of ( 27).

Using Cauchy-Schwarz's inequality, we get then for (36)

E n E τp 1 n n i=1 φ(T t (z i )) ξ(k) (t, z i ) - R d ×R d dxdvf (k) (t, x, v)φ(x, v) 2 ≤ E n 1 n n i=1 φ(T t (z i ))E τp [ ξ(k) (t, z i )] 2 + R d ×R d dxdvf (k) (t, x, v)φ(x, v)× × R d ×R d dxdvf (k) (t, x, v)φ(x, v) - 2 n n i=1 φ(T t (z i ))E τp [ ξ(k) (t, z i )] .
Because of the hypothesis on

f 0 , φ(T t (•))E τp [ ξ(k) (t, •)] ∈ C b (R d × R d )
, so that, thanks to (23) (and in particular to µ 0 n ⇀ f 0 ), on a set of full measure w.r.t. P we have 1

n n i=1 φ(T t (z i ))E τp [ ξ(k) (t, z i )] → R d ×R d dxdvf (k) (t, x, v)φ(x, v),
and (36) vanishes on this set.

In the same way, by writing

η n,ε,M = (η n,ε,M -η L ) + (η L -η(k) ) + η(k) and σ = (σ -σ (k) ) + σ (k) , we get, for all ψ ∈ C K (R d ) and for all k ≥ 1, E n ε d-1 M i=1 ψ(c i )η n,ε,M (t, c i ) - R d dxψ(x)σ(t, x) ≤ (37) ψ ∞ E n ε d-1 M i=1 I suppψ (c i )|η n,ε,M -η L |(t, c i ) + (38) E n ε d-1 M i=1 I suppψ (c i )|η (k) -η L |(t, c i ) + suppψ dx|σ (k) -σ|(t, x) (39) +E n ε d-1 M i=1 ψ(c i )η (k) (t, c i ) - R d dxψ(x)σ (k) (t, x) ,
where the term in round brackets ( 38) is (everywhere) bounded by

K b dv(f - f (k-1) ) L ∞ ([0,T ]×R d )
and can be made arbitrarily small because of ( 27).

The term (37) vanishes on a set of full measure w.r.t. P for the same reasons as (34), i.e. as a consequence of the sandwiching property valid for (31) and of the convergence of ( 31) to (25) in the asymptotics specified in ( 18), ( 19).

As for (39), we can write

E n ε d-1 M i=1 ψ(c i )η (k) (t, c i ) - R d dxψ(x)σ (k) (t, x) 2 ≤ Λn dcψ(c)σ (k) (t, c) - R d dcψ(c)σ (k) (t, c) 2 + ε d-1 Λn dcψ 2 (c)σ (k) (t, c). Since ψ ∈ C K (R d ), we have ψ h (•)σ (k) (t, •) ∈ L 1 (R d
)) for h = 1, 2 and when n grows to infinity we have both Λ n → R d and ε → 0, so that this term vanishes, in the chosen asymptotics. Collecting all the assertions about the different terms, the theorem is proved.

4. Asymptotic equivalence of ( 31) and ( 25) and convergence of the particle system to the limit system

We collect in the present section the Lemmas and Propositions which will help us to build up the last step of the proof of our main theorem.

4.1.

A few useful bounds. Let us list a few bounds and formulas which we shall use often in our calculations.

It will be useful to adopt in next sections, for stochastic variables λ Fn,M of the form λ Fn,M (t,

z j ) = I {xj (s) / ∈ M h=1 Bε(c h )I {F n,M (s, c h )<τ h } ∀s∈[0,t)} , (with F n,M (s, c h ) = F cM ,τM ,zn (s, c h ) a given random function) the following rep- resentation: (40) λ Fn,M (t, z j ) = M h=1 1 -I {c h ∈Tε(t,zj)} I {Fn,M (Tz j ,c h , c h )<τc h } .
Given any couple of stochastic variables λ F (1) n,M , λ F (2) n,M of the form (40), using the trivial inequality

| h I A h - h I B h | ≤ h |I A h -I B h |, for M ≥ 1, we get the bound (41) λ F (1) n,M 1 -λ F (2) n,M 2 (t, z j ) ≤ M h=1 I c h ∈Tε(t,zj ) |η F (1) n,M -η F (2) n,M |(T zj ,c h , c h ), where η F (i) n,M (T zj ,c h , c h ) = I {F (i) n,M (Tz j ,c h ,c h )<τc h } , i = 1, 2.
The bound (41) will be largely used in the last paragraph of this section.

Moreover, because of the hypothesis on f 0 (assigning initially uniformly bounded number and kinetic energy limit densities for the light particle component, cf. Remark 9 in Appendix 6.2), P-a.e. with respect to Z ∞ , we may use the bounds

1 n n j=1 q n (x j (s) -c) ∞ ≤ K 1 (42) 1 n n j=1 q n (x j (s) -c)[T ε (s, z j )] L ∞ ≤ ε d-1 C d T K 2 (43)
where K 1 , K 2 > 0 are constant independent of n.

4.2.

Quantities related to flow tubes. When bounding correlations in our particle system, we shall need to evaluate expectation values with respect to the Poisson distribution (3) and to µ 0 n on volumes corresponding to the intersections of flow tubes.

Defining, for two vectors v, w ∈ R d , cos α(v, w) = v•w |v||w| (and α ij = α(v i , v j )), and recalling the notation B r (p) introduced at the beginning of Section 2 to denote the ball of radius r centered in p, we have, for the intersection of two flow tubes associated to the particles with initial phase space positions z i and z j , the following trivial bound:

Lemma 2. For any β ∈ (0, 1 d ): (44) [T ε (t, z j ) ∩ T ε (t, z i )] L ≤ [B 2 (0)] L ε d(1-β) I | sin αij |≥ε β + C d ε d-1 min(|v i |, |v j |)T I | sin αij |<ε β .
Proof. We have always

[T ε (t, z j ) ∩ T ε (t, z i )] L ≤ min([T ε (t, z j )] L , [T ε (t, z i )] L ) = C d ε d-1 min(|v i |, |v j |)t
and, whenever sin α ij = 0, denoting as y the (unique) crossing point for the trajectories x i (s), x j (s) (i.e. y = x i + v i s 1 = x j + v j s 2 for some s 1 , s 2 ∈ R),

T ε (t, z j ) ∩ T ε (t, z i ) ⊂ B 2ε sin α ij (y).
We can therefore write:

(45) |T ε (t, z j ) ∩ T ε (t, z i )| ≤ |B 2 (0)| ε sin α ij d ∧ C d ε d-1 min(|v i |, |v j |)t
and (44) follows.

Thanks to the weak convergence of the initial empirical measure toward a regular density, we can estimate the measure (w.r.t. µ 0 n ) of the set corresponding, for a given velocity w ∈ R d , to the grazing collisions. We can prove in fact the following Lemma:

Lemma 3. Let µ 0 n ⇀ f 0 ∈ S (R d × R d ), β > 0 and w ∈ R d . Assume (18). Denoting cos α i = vi•w
|vi||w| , the following bound is verified

1 n n i=1 I | sin αi|<ε β ≤ Kε ζ + o(a -d n ) for any ζ ∈ (0, d-1 2 β).
Proof. We observe first that we have lim

n→∞ #{j:|vj •w|=|w||vj|} n = 0, since f 0 ∈S (R d ×R d ),
so that, uniformly in ε, the contribution of the corresponding term vanishes when n → ∞ and it is actually o(a -d n ) in the prescribed asymptotics 2 . We now evaluate the contribution of grazing crossings of particle trajectories. We choose a suitable (standard) C ∞ regularization Rδ of I |v•w| =|v||w| , for instance Rδ is s.t.:

I |v•w| =|v||w| -Rδ ≤ e - sin 2 α(v,w) δ 2 (δ 2 -sin 2 α(v,w)) I | sin α(v,w)|≤δ .
Here the parameter δ denotes the radius of the set where the regularization differs from the original characteristic functions (i.e. the set in R d × R d s.t. | sin α(v, w)| ≤ δ), and, for all δ > 0 and 0 < ι < d-1 2 , we write:

1 n n i=1 I |vi•w| =|vi||w| I | sin αi|<ε β ≤ 1 n n i=1 ε ιβ (1 -|vi•w| 2 |vi| 2 |w| 2 ) ι 2 Rδ (v i ) + o(δ) ≤ ε ιβ R d ×R d dxdv f 0 (x, v) (1 -|v•w| 2 |v| 2 |w| 2 ) ι 2 + K R ε ιβ δ 3+ι + o(δ).
The error coming from the regularization of the characteristic function

I |v•w| =|v||w| is o(δ) because of the weak convergence of µ 0 n toward f 0 ∈ S (R d × R d
) and, for the same reason, all constants (here and in the following estimates) are uniform in w 2 In fact, since by Toeplitz's lemma, for all a ∈ (0, 1],

1 n n k=1 ( #{j:|v j •w|=|w||v j |} k ) a vanishes
when n → ∞, we have at least

#{j:|v j •w|=|w||v j |} n = o( 1 √ n ) ).
and n and depend only on the dimension d and on few L p norms of f 0 . We obtain then, for suitable choices of δ,

1 n n i=1 I |vi•w| =|vi||w| I | sin αi|<ε β ≤ Kε ζ ,
for any ζ ∈ (0, ιβ) and therefore for any ζ ∈ (0, d-1 2 β). Collecting the two estimates, the lemma is proved.

We shall use Lemma 3 for free choices of β ∈ (0,

1 d ), so that ζ ∈ (0, d-1 2 β) ⊂ (0, 1 2 -1 2d ).
Remark 2. The estimate obtained in Lemma 3 is valid for any given sequence of empirical measures

{µ 0 n } such that µ 0 n ⇀ f 0 ∈ S (R d × R d )
and it is actually a stronger assertion with respect to what we need to prove Theorem 1 (which is valid Pa.e. w.r.t. Z ∞ ). In order to prove Lemma 3, where, as stated in the introductory sentence to the lemma, we bound the measure of a set with respect to the empirical measure µ 0 n , we need to evaluate the measure of the set {(x, v) : |v • w| = |w||v|} with respect to the measure µ 0 n because the measure of this set vanishes only asymptotically (i.e. with respect to the limit measure with density f 0 ). We choose to evaluate this measure with respect to the parameter a d n (i.e. as o(a -d n )) for further convenience.

A similar, simpler statement can be proved P-a.e. w.r.t. Z ∞ : in this case we obtain the bound 1 n n i=1 I | sin αi|<ε β ≤ Kε ζ (P-a.e. w.r.t. Z ∞ ). This alternative statement could be used instead of Lemma 3 to prove our main theorem. We prefer nevertheless to use Lemma 3, getting in this way bounds which are (as much as possible) valid on the whole subset of initial sequences Z ∞ such that the associated sequence of empirical measures {µ 0 n } ∞ n=1 converges weakly to

f 0 ∈ S (R d × R d ).
Remark 3. Notice that, since 1

n 2 i,j:|vi•vj |=|vi||vj | 1 = o(a -d n
) and all bounds are uniform in the velocity w, we get also

(46) 1 n 2 n i,j=1 I | sin αij |≤ε β ≤ 1 n n j=1 ( 1 n n i=1 I | sin αij |≤ε β ) ≤ Kε ζ + o(a -d n ).
A last very useful tool will be the following parametrization of the points c ∈ Tε(t,z).

Let u = Tz,c be the maximal collision time defined in (8); then we may define the change of variables

c = x(u) + εω, u ∈ [0, t), ω ∈ ∂B1(0).
Using this parametrization, for each non negative function

V ∈ L ∞ ([0, T ]; W 1,∞ (R d )) we can write V (Tz,c, c) = V (u, x(u) + εω), getting then (47) Tε (t,z) dce -V (Tz,c,c) = ε d-1 C d |v| t 0 du[e -V (u,x(u)) + φε(u, x(u))]
where we may bound the remainder through the trivial inequality |e -ye -x | ≤ |x -y|, for x, y ≥ 0, and the function φε is such that sup

[0,T ] φε L ∞ (R d ) = O(ε) → 0 when ε → 0.
4.3. The sandwiching property. We prove now the sandwiching property for the system defined by ( 31) and its implication on the convergence of (7).

Lemma 4. (sandwiching property) Consider Vn,ε,M , ξn,ε,M , ηn,ε,M defined by ( 6), ( 7) and ξ

(k) n,ε,M , V (k) n,ε,M , η (k)
n,ε,M defined by (31). Then, for k = 1, 2 . . .,

ξ (2k-1) n,ε,M ≤ ξ (2k+1) n,ε,M ≤ ξn,ε,M ≤ ξ (2k) n,ε,M ≤ ξ (2k-2) n,ε,M V (2k-2) n,ε,M ≤ V (2k) n,ε,M ≤ Vn,ε,M ≤ V (2k+1) n,ε,M ≤ V (2k-1) n,ε,M η (2k-1) n,ε,M ≤ η (2k+1) n,ε,M ≤ ηn,ε,M ≤ η (2k) n,ε,M ≤ η (2k-2) n,ε,M . (48)
Proof. From definitions ( 6), ( 7) and ( 31), we have 0 =

V (0) n,ε,M ≤ Vn,ε,M ≤ V (1) n,ε,M , which implies η (0) n,ε,M ≥ ηn,ε,M ≥ η (1) n,ε,M , which in turn implies ξ (1) n,ε,M ≤ ξn,ε,M ≤ ξ (2) n,ε,M ≤ ξ (0) n,ε,M = 1.
We have therefore, using again these definitions:

ξ (1) n,ε,M ≤ ξ (3) n,ε,M ≤ ξn,ε,M ≤ ξ (2) n,ε,M ≤ ξ (0) n,ε,M V (0) n,ε,M ≤ V (2) n,ε,M ≤ Vn,ε,M ≤ V (3) n,ε,M ≤ V (1) n,ε,M η (1) n,ε,M ≤ η (3) n,ε,M ≤ ηn,ε,M ≤ η (2) n,ε,M ≤ η (0) n,ε,M
and ( 48) is valid for k = 1.

Since the following chain is also valid:

V (2k-2) n,ε,M ≤ V (2k) n,ε,M ≤ Vn,ε,M ≤ V (2k+1) n,ε,M ≤ V (2k-1) n,ε,M =⇒ η (2k-2) n,ε,M ≥ η (2k) n,ε,M ≥ ηn,ε,M ≥ η (2k+1) n,ε,M ≥ η (2k-1) n,ε,M =⇒ ξ (2k-1) n,ε,M ≤ ξ (2k+1) n,ε,M ≤ ξn,ε,M ≤ ξ (2k+2) n,ε,M ≤ ξ (2k) n,ε,M =⇒ V (2k) n,ε,M ≤ V (2k+2) n,ε,M ≤ Vn,ε,M ≤ V (2k+3) n,ε,M ≤ V (2k+1) n,ε,M
the statement of the proposition follows by induction.

We may then prove the following corollary to Lemma 4:

Corollary 1. Consider Vn,ε,M , ξn,ε,M , ηn,ε,M and V (k) n,ε,M , ξ (k) n,ε,M , η (k)
n,ε,M defined as in Lemma 4 and VL and ξL given by ( 14) and ( 15), with (f, σ) unique solution of (2). Then, when n → ∞, ε → 0, for any sub-linear operators

L on L ∞ ([0, T ] × R d × R d ) L|V (k) n,ε,M -V (k) | → 0 =⇒ L|Vn,ε,M -VL| → 0 L|ξ (k) n,ε,M -ξ(k) | → 0 =⇒ LEτ p |ξn,ε,M -ξL| → 0 LEτ |η (k) n,ε,M -η(k) | → 0 =⇒ LEτ |ηn,ε,M -ηL| → 0 L| R d ×R d dzµ 0 n φ(ξ (k) n,ε,M -ξ(k) )| → 0 =⇒ LEτ p | R d ×R d dzµ 0 n φ(ξn,ε,M -ξL)| → 0.
Proof. We denote here by φ+ = φ ∧ 0, φ-= -φ ∨ 0 resp. the positive and the negative part of φ.

Let Fn,ε,M be one among the non negative functions Vn,ε,M , 1ξn,ε,M , 1ηn,ε,M , dzµ 0 n φ+ ξn,ε,M , dzµ 0 n φ-ξn,ε,M and F (k) n,ε,M its approximant of order k (defined through (31)). Let FL be its associate limit function (defined through ( 14) and ( 15)) and F (k) the approximant of FL (defined through (28),( 29) and ( 30)).

Because of (48), we have, for a given k ≥ 1:

F (2k) n,ε,M -FL ≤ Fn,ε,M -FL ≤ F (2k+1)
n,ε,M -FL, and therefore,

|Fn,ε,M -FL| ≤ |FL -F (2k) | + |FL -F (2k+1) |+ |F (2k+1) -F (2k+1) n,ε,M | + |F (2k) -F (2k) n,ε,M |. Since VL -V (k) L ∞ ([0,T ]×R d ) and Eτ p [ξL -ξ(k) ] L ∞ ([0,T ]×R d ×R d ) vanish in the k → ∞ limit (cf. (27)), and since Eτ p [|ηL -η(k) |] ≤ |VL -V (k) | (cf.
Remark 1), the assertion is proved.

As already pointed out, Corollary 1 is valid in particular when L is an expectation value operator.

Equivalence between (33) and (25).

4.4.1. Motion of light particles in a decaying medium: equivalence of the light particle components of ( 33) and ( 25). We prove here a lemma which describes the behavior of the light particle component of the system (33) in the n → ∞ limit. In this lemma, we consider a particle system such that the rate of death of the obstacles is given and independent from the light particles component, while the light particles are instantaneously absorbed each time they collide with an obstacle. The correspondent limit system consists of equations which are coupled only in a very weak way and the parameters determining the asymptotics are independent.

Let g(t, c) ∈ L 1 ([0, T ]; W 1,∞ (R d )) be a non negative function. The risk function V (t, c) defined, for t ≤ T , by

V (t, c) = t 0 g(s, c)ds
is non decreasing as a function of t.

As in the previous sections, we define:

(49)

η V (t, c h ) = I {V (t,c h )<τ h } ξ V n,ε,M (t, zi) = I {x i (s) / ∈ M k=1 Bε (c k )η V (s,c k ) ∀∈[0,t)}
.

We have then:

Lemma 5. Let g(t, c) ∈ L 1 ([0, T ]; W 1,∞ (R d )) and ξ V n,ε,M be defined by (49). Consider the sequence Z∞ ∈ (R d × R d ) ∞ and its associated sequence of empirical measures {µ 0 n } ∞ n=1 . Assume µ 0 n ⇀ f0 ∈ S and |v|µ 0 n ⇀ |v|f0 ∈ S . Then ∀φ ∈ C b (R d × R d ), lim n→∞ ε→0 E n 1 n n i=1 ξ V n,ε,M (t, zi)φ(T t (zi)) - R d ×R d dxdvf (t, x, v)φ(x, v) 2 = 0
(independently of the order of the limits), where (f, σ) is the (unique) solution of the following system of partial differential equations:

       ∂tf + v • ∇xf = -|v|C d σf ∂tσ = -gσ f (0, x, v) = f0(x, v) σ(0, x) = µ.
Remark 4. Of course, when V = V (k-1) , we have ξ V n,ε,M = ξ(k) n,ε,M , so that obviously (by the triangular inequality) Lemma 5 implies, when ε → 0, n → ∞, the limit

E n [| dxdvµ 0 n ( ξ(k) n,ε,M - ξ(k) )φ|] → 0.
Proof. According to the definition of maximal collision time (8), we have

ξ V n,ε,M (t, zj) = 1 ⇐⇒ ∀h : 1 ≤ h ≤ M, Tz j ,c h ∧ t ≥ inf s∈R + {η V (s, c h ) = 0} ∧ t. The function F (•, c) = inf{z : z = V -1 (•, c)} is well defined and inf{s ∈ R+; η V (s, c h ) = 0} = F (τc h , c h ).
Moreover, the maximal collision time Tz j ,c is greater than t if and only if c does not belong to the tube Tε(t, zj). It follows that

P n ξ V n,ε,M (t, zj) = 1 = P n {1 ≤ h ≤ M : Tz j ,c h ∧ t ≥ F (τc h , c h ) ∧ t} = E n M h=1 I {c h / ∈Tε (t,z j )} + I {c h ∈Tε(t,z j ); Tz j ,c h ≥V -1 (τc h ,c h )∧t} = exp -µε Tε(t,z j ) dce -V (Tz j ,c,c) .
Using (47), because of the weak convergence of µ 0 n and |v|µ 0 n , we get

(50) E n 1 n n i=1 ξ V n,ε,M (t, zi)φ(T t (zi)) n→∞,ε→0 → R d ×R d dxdvf (t, x, v)φ(x, v),
the result being independent of the ordering of the limits. From (50), we prove Lemma 5 as follows. We call

E1 n = E n     1 n 2 n i,j=1 i =j ξ V n,ε,M (t, zi)ξ V n,ε,M (t, zj)φ(T t (zi))φ(T t (zj))     E2 n = R d ×R d dxdvf (t, x, v)φ(x, v)-E n 2 n n i=1 ξ V n,ε,M (t, zi)φ(T t (zi)) .
We write then:

E n 1 n n i=1 ξ V n,ε,M (t, zi)φ(T t (zi)) - R d ×R d dxdvf (t, x, v)φ(x, v) 2 = E n 1 n 2 n i=1 ξ V n,ε,M (t, zi)φ 2 (T t (zi)) + E1 n + E2 n R d ×R d dxdvf (t,x,v)φ(x,v)
where

E n 1 n 2 n i=1 ξ V n,ε,M (t, zi)φ 2 (T t (zi)
) , when n → ∞, obviously vanishes and, thanks to (50),

E2 n R d ×R d dxdvf (t, x, v)φ(x, v) → - R d ×R d dxdvf (t, x, v)φ(x, v) 2 .
As for E1 n we observe that, for i = j:

E n ξ V n,ε,M (t, zi)ξ V n,ε,M (t, zj) = exp -µε Tε (t,z i ) dc e -V (Tz i ,c,c) -µε Tε (t,z j ) dc e -V (Tz j ,c,c) × exp µε
Tε (t,z j ) Tε(t,z i ) dc I {c:Tz i ,c≤Tz j ,c} e -V (Tz j ,c,c) + I {c:Tz i ,c>Tz j ,c} e -V (Tz i ,c,c) .

We write 1 =

I | sin α ij |≥ε β + I | sin α ij |<ε β , for β < 1 d .
Then, using ( 46) and [Tε(t, zj ) ∩ Tε(t, zi))]L ≤ Kµε d-1 (always valid for intersections of flow tubes, as shown also in the proof of Lemma 2) we have:

1 n 2 n i,j=1 i =j I | sin α ij |<ε β E n ξ V n,ε,M (t, zi)ξ V n,ε,M (t, zj) ≤ 1 n 2 n i,j=1 i =j e µε[Tε(t,z j ) Tε (t,z i )] L I | sin α ij |<ε β ≤ Kµε ζ + o(a -d n ).
As for the remaining part (since 46) and Lemma 2 we get:

I | sin α ij |≥ε β = 1 -I | sin α ij |<ε β ), from (
1 n 2 n i,j=1 i =j φ(T t (zi))φ(T t (zj))I | sin α ij |>ε β E n ξ V n,ε,M (t, zi)ξ V n,ε,M (t, zj) = o(a -d n ) + O(ε ζ ) + 1 n 2 n i,j=1 φ(T t (zi))φ(T t (zj))× × exp -µε Tε(t,z i ) dce -V (Tz i ,c,c) + Tε (t,z j ) dce -V (Tz j ,c,c) +Ktε 1-dβ
since in this case µε[Tε(t, zj) Tε(t, zi))]L = O(ε 1-dβ ) → 0, so that (using (47) once more)

E1 n → R d ×R d dxdvf (s, x, v)φ(x, v) 2
and Lemma 5 is proved.

We emphasize again that all results in this Lemma are independent of the ordering of the limits n → ∞, ε → 0. 4.4.2. Equivalence of the obstacle components of ( 33) and (25). Lemma 5 describes the asymptotic average behavior of the light particle component of the system (33). The asymptotic average behavior of its risk function, for an obstacle located in c, should be equivalent to the behavior associated to the risk (28). This fact is described by the proposition proved in this section and its corollary (notice that the risk function of a given obstacle with position c h , differently from the generic risk function associated to a given position, is defined only if there exists an obstacle in c h ). The second proposition will be useful in the proof of equivalence of ( 33) to (31), in the next section.

It will be convenient to define the following stochastic functions. For z

= (x, v) ∈ R d × R d : (51) ρ (k) ε (t, c, z) = 1 -I {c∈Tε(t,z)} I { V (k) (Tz,c,c)<τc} ρ (k) (t, z) = t 0 du ∂B 1 (0) |v • ω|dω exp -V (k) (u, x(u)) Q ij n (s, t, c) = qn(xi(s) -c)qn(xj(t) -c) R ij (k) ε (s, t, c) = Eτ c [ρ (k) ε (s, c, zi)ρ (k) ε (t, c, zj)].
Given these definitions, a fundamental tool in all estimates in this section is the following lemma: Lemma 6. Take a sequence Z∞ ∈ (R d × R d ) ∞ and its associated sequence of empirical measures {µ 0 n } ∞ n=1 . Take a sequence {Λn} ∞ n=1 of bounded, Lebesgue measurable sets such that Bn ⊂ Λn (with Bn defined in ( 9)) and assume µ 0 n ⇀ f0 ∈ S and |v|µ n 0 ⇀ |v|f0 ∈ S . For 0 ≤ s ≤ t, we have

(52) 1 [Λn]L Λn dc R ij (k) ε (t, s, c) = 1 - ε d-1 [Λn]L [ρ (k) (t, zi) + ρ (k) (s, zj)(1 -δij )] + R (2) ε (zi, zj , t, s) [Λn]L where, for each ζ ∈ (0, 1 2 -1 2d ), (53) 1 n n i=1 R (2) ε (zi, zj) L ∞ ([0,T ]×[0,T ]) ≤ Kr1ε d-1+ζ |vj | + ε d-1 [o(a -d n )]
and

(54) lim ε→0 ε 1-d 1 n 2 n i,j=1 R (2) ε (zi, zj ) L ∞ ([0,T ]×[0,T ]) = 0.
Proof. Formula (52) follows, through simple calculations, from (47), expanding the expectation value in the definition of

R ij (k) ε
, given in ( 51), and writing

Tε (t,z) dce -V (k) (Tz,c,c) = ε d-1 ρ (k) (t, z) + C d |v| t 0 duφε(u, x(u)) ;
doing this, we obtain for the remainder

|R (2) ε (zi, zj, t, s)| ≤ ε d-1 C d (|vi| + |vj |) ϕε L ∞ ([0,T ]×R d ) +[Tε(s, zj) ∩ Tε(t, zi)]L (55)
where

ϕε L ∞ ([0,T ]×R d ) = O(ε) → 0.
We use then the bound (55) to obtain ( 53) and ( 54). In the single sum in (53), so as in the double sum in (54), thanks to the weak convergence of |v|µ 0 n , the contribution of the first term on the right-hand side of ( 55) is O(ε d ). As for the contribution of [Tε(s, zj)∩Tε(t, zi)]L, we bound the Lebesgue measure of the intersection of the tubes using Lemma 2; then, thanks to the weak convergence of µ 0 n toward the (sufficiently) regular function f0, we can use Lemma 3 and ( 46), with β < 1 d , to estimate the contribution coming from the grazing collisions (being the remaining part in both sums o(ε d-1 )). So, from Lemma 3 we get straightforwardly (53) and, from (46), we get the bound:

(56) 1 n 2 n i,j=1 R (2) ε (zi, zj ) L ∞ ([0,T ]×[0,T ]) ≤ KRε d-1 [ε ζ + o(a -d n )],
and, when ε → 0, we obtain (54).

We may now prove the main proposition of this section:

Proposition 1. Consider the stochastic variables defined in ( 33) and ( 28) and the sequences of positive real numbers {an} ∞ n=1 and {εn} ∞ n=1 satisfying conditions ( 18) and ( 19), i.e. lim n→∞ a d n n -κ = 0, for some κ ∈ (0, 1 2 ), and

lim n→∞ a d n ε ζ = 0, for some ζ ∈ (0, 1 2 -1 2d
). Then, P-almost everywhere w.r.t. sequences of initial data Z∞,

lim n→∞ [Λn]L sup t∈[0,T ] E n Â(k) n,ε,M (t, c) -V (k) (t, c) 2 I M ≥1 = 0
where {Λn} is an increasing sequence of bounded Lebesgue measurable sets such that Bn ⊂ Λn (with Bn defined in ( 9)) and

lim n→∞ Λn = R n .
Proof. We expand the square power in the expectation value and we calculate the value of the three resulting terms. Because of definition (51), using (40), we can write:

ξ(k-1) n,ε,M (s, zj ) = M h=1 ρ (k-2) ε (s, c h , zj ).
We substitute this expression in the definition of Â(k) n,ε,M , in system (33), and we obtain (resumming on the Poissonian variables):

[Λn]LE n Â(k) n,ε,M (t, c) 2 I {M ≥1} = 1 n 2 n i,j=1 [0,t] 2 ds1 ds2 exp µε[Λn]L Λn dc R ij (k-2) ε (s1, s2, c) [Λn]L -1 -e -µε[Λn] L × × Λn dc Q ij n (s1, s2, c)R ij (k-2) ε (s1, s2, c) Λn dc R ij (k-2) ε (s1, s2, c) [Λn]L .
Then, using Lemma 6, we have:

(57) [Λn]LE n Â(k) n,ε,M (t, c) 2 I {M ≥1} = 1 n 2 n i,j=1 [0,t] 2 ds1 ds2 e -µ[ρ (k-2) (s 1 ,z i )+ρ (k-2) (s 2 ,z j )(1-δ ij )] ×(1 + ε 1-d R ij(3) ε ) Λn dc Q ij n (s1, s2, c)R ij (k-2) ε (s1, s2, c) where R ij(3) ε = R (3) ε (s1, s2, zi, zj) ≤ KtR (2) ε (s1, s2, zi, zj) is such that ε 1-d n 2 n i,j=1 R ij(3) ε L ∞ [0,T ]×[0,T ] ≤ KR 3 (ε ζ + o(a -d n )).
We recall that, using Lemma 6 and since q ∈ S (R d ), we have, P-almost everywhere w.r.t. sequences of initial data Z∞ and for any D ⊂ R d , the bounds

(58) ε 1-d n 2 n i,j=1 R (3) ε L ∞ [0,T ]×[0,T ] D dc Q ij n (s, t, c) ≤ KqΘa d n q ∞(ε ζ + o(a -d n )), 1 n 2 n i,j=1 Tε (t,z i ) dc Q ij n (s, t, c) ≤ C d tε d-1 (a d n q ∞ ) K1 n n i=1 |vi|.
Therefore, using the definition of R

ij (k-2) ε
, given in ( 51), we get the equality

[Λn]LE n Â(k) n,ε,M (t, c) 2 I {M ≥1} = R (4) ε (t)+ 1 n 2 n i,j=1 [0,t] 2 ds1ds2e -µ[(ρ (k-2) (s 1 ,z i )+ρ (k-2) (s 2 ,z j )(1-δ ij )] Λn dc Q ij n (s1, s2, c)
where

sup t∈[0,T ] R (4) ε (t) ≤ 4KqΘT 2 q ∞ a d n (ε ζ + o(a -d n )) + O(ε d-1 a d n ) → 0, since lim n→∞ a d n ε ζ = 0 with ζ ∈ (0, 1 2 -1 2d ). Let z = (x, v) and z = (y, w). Define the sequence of measurable sets Λ s n [z] = {a ∈ R d : ∃b ∈ Λn s.t. a = b -x(s)} and the sequence of functions Q * Λ s n (z, x) = (I Λ s n [z] qn) * qn(x).
Since q is a radial function, we have the following equality:

1 n 2 n i,j=1 e -µ[(ρ (k-2) (s 1 ,z i )+ρ (k-2) (s 2 ,z j )(1-δ ij )] Λn dc Q ij n (s1, s2, c) = (R d ×R d ) 2 dzdz[µ 0 n ⊗ µ 0 n ](z, z)e -µ[ρ (k-2) (s 1 ,z)+ρ (k-2) (s 2 ,z)] Q * Λ s 2 n (z, x(s1) -y(s2)) + 1 n R d dzµ 0 n (z)e -µρ (k-2) (s 1 ,z) [1 -e -µρ (k-2) (s 2 ,z) ]Q * Λ s 2 n (z, x(s1) -y(s2)),
where the last term is bounded by

2 n dzµ 0 n (qn * qn)(x(s1) -y(s2)) ≤ 2 a d n n q ∞ Θ.
In the first term we can use Fubini's theorem, rewriting

(R d ×R d ) 2 dzdz[µ 0 n ⊗ µ 0 n ](z, z)e -µ[ρ (k-2) (s 1 ,z)+ρ (k-2) (s 2 ,z)] Q * Λ s 2 n (z, x(s1) -y(s2)) = R d da (R d ×R d ) 2 dzdz[µ 0 n ⊗ µ 0 n ](z, z) e -µρ (k-2) (s 1 ,z) qn(a -x(s1)) × × e -µρ (k-2) (s 2 ,z) qn(a -y(s2))I Λ s 2 n [z] (a -y(s2)
) and, because of (42), we can use Lebesgue's dominated convergence theorem to pass to the limit into the integral.

{Λn} ∞

n=1 is an increasing sequence of Lebesgue measurable set and

IΛ n → 1 in L ∞ (R d ). From the P-a.e. weak convergence ⊗ Q k=1 qnµ 0 n ⇀ ⊗ Q k=1 δ0f0 and the L ∞ convergence of I Λ s n [y] , we have [0,t] 2 ×(R d ×R d ) 2 ds1ds2dzdz[µ 0 n ⊗ µ 0 n ](z, z)× × e -µ[ρ (k-2) (s 1 ,z)+ρ (k-2) (s 2 ,z)] Q * Λ s 2 n (z, x(s1)-y(s2)) → R d dc V (k) (t, c) 2
, so that we obtain finally

lim n→∞ 1 n 2 n i,j=1 [0,t] 2 ×Λn ds1 ds2 dc Q ij n e -µ[ρ (k-2) (s 2 ,z i )+ρ (k-2) (s 1 ,z j )(1-δ ij )] = R d dc V (k) (t, c) 2 .
We compute now the double product and, because of the definitions of ρ

(k)
ε , Λn and of the boundedness of V (k) , we get :

[Λn]LE n Â(k) n,ε,M (t, c) V (k) (t, c)I {M ≥1} = O(a d n ε d-1 )+ 1 n n j=1 t 0 ds1 exp -µε Tε (s 1 ,z j ) dc e -V (k-2) (Tz j ,c,c) × Λn dc V (k) (t, c)qn(xj(s1) -c) .
Using (47), P-a.e. with respect to Z∞, we get:

[Λn]LE n Â(k) n,ε,M (t, c) V (k) (t, c)I {M ≥1} = 1 n n i=1 t 0 ds1e -µρ (k-2) (s 1 ,z i ) Λn dc V (k) (t, c)qn(xi(s1) -c) + o(ε)
so that, because of the P-a.e. weak convergence of qnµ 0 n and the hypothesis on f0, we obtain:

-2 lim n→∞ [Λn]LE n Â(k) n,ε,M (t, c) V (k) (t, c)I {M ≥1} = -2 R d dc V (k) (t, c) 2 .
For the last term we have:

[Λn]LE n | V (k) (t, c)| 2 I {M ≥1} = (1 -e -µε [Λn] L ) Λn dc V (k) (t, c) 2 , whose limit is R d dc V (k) (t, c) 2
because of the L ∞ convergence of IΛ n . Hence, we have established Proposition 1.

In addition to Proposition 1, we shall need its corollary Corollary 2. Under the same hypothesis as Proposition 1, for the obstacles life functions defined in (30) and in (33) and for all φ ∈ CK (R d ), P-almost everywhere w.r.t. sequences of initial data Z∞:

lim n→∞ E n [ε d-1 M i=1 I suppφ (ci)|ηn,ε,M (t, ci) -η(t, ci)|] = 0.
Proof. Since for all δ > 0 (cf. Remark 1)

E n [ε d-1 M i=1 I suppφ (ci)|ηn,ε,M (t, ci) -η(t, ci)|] ≤ 1 δ E n [ε d-1 M I suppφ (c)| V (k) (t, c) - Â(k) n,ε,M (t, c)|] + 2δµ |suppφ ∩
Λn| , the proof follows easily bounding the first term on the right-hand side by Cauchy-Schwarz's inequality and using the identity

E n [(ε d-1 M ) 2 I suppφ (c)] = µ 2 [Λn]L[suppφ ∩ Λn]L
and Proposition 1.

We shall need moreover a modified form of the proposition, whose proof will be only sketched, being essentially the same as the one of Proposition 1 Proposition 2. Under the same hypothesis as Proposition 1,P-almost everywhere w.r.t. sequences of initial data Z∞:

lim n→∞ sup 1≤h≤n 1 |v h | E n | Â(k) n,ε,M -V (k) | 2 (Tz h ,c 1 , c1)M I c 1 ∈Tε(T,z h ) = 0.
Proof. The proof is obtained along the same line as the proof of Prop. 1, since Tz h ,c 1 is independent of the stochastic times τ ∞, is bounded by T and all quantities involved depend on Tz h ,c 1 in a simple way, so to allow to get estimates uniform in z h . Using, instead of (58), the estimates

ε 1-d n 2 n i,j=1 R ij(3) ε L ∞ [0,T ]×[0,T ] Q ij n (s, t, c) ≤ Kra d n (ε ζ + o(a -d n )), 1 n 2 n i,j=1 Tε (t,z h )∩Tε(t,z i ) dc Q ij n (s, t, c) ≤ KQ|v h |ε d-1 a d n (ε ζ + o(a -d n ))
which are a consequence of Lemma 6 and of the bounds (42), (43), we perform the change of variables (47) and we get:

1 |v h | E n M Â(k) n,ε,M (Tz h ,c 1 , c1) 2 I c 1 ∈Tε(T,z h ) = O(a d n (ε ζ + o(a -d n )))+ µC d n 2 n i,j=1 T 0 ds [0,s] 2 ds1ds2Q ij n (s1, s2, x h (s))e -µ[ρ (k-2) (s 1 ,z i )+ρ (k-2) (s 2 ,z j )] ,
while for the double product, we have

1 |v h | E n M I {c 1 ∈Tε(T,z h )} Â(k) n,ε,M (t, c1) V (k) (t, c1) = O(a d+1 n ε)+ µC d n n j=1 T 0 ds V (k) (s, x h (s)) s 0 ds1qn(xj(s1) -x h (s)) exp[-µρ (k-2) (s1, zj)]. Of course 1 |v h | E n M V (k) (t, c1) 2 I c 1 ∈Tε(T,z h ) = µC d T 0 ds V (k) (s, x h (s)) 2 + O(ε).
Then, observing that, thanks to the P-a.e. weak convergence of qnµ 0 n , the quantity

1 |v h | E n | Â(k) n,ε,M (Tz h ,c 1 , c1) -V (k) (Tz h ,c 1 , c1)| 2 M I c 1 ∈Tε(T,z h )
vanishes for all fixed h ∈ N, uniformly in z h because of the choice of the initial data (cf. Remark 9 in Appendix 6.2) and since all error estimates in this section are themselves uniform in z h , the Proposition is proved.

4.5. Equivalence between ( 33) and (31): the O, P-frozen system. The last step to complete the proof of our main theorem is to estimate the distance between the particle system (33) and the particle system (31). The difficulty, here, originates from the complicate dependence of the life functions, defining both systems, on the stochastic configuration: in particular, this dependence compels us to use (24) for estimating differences of life functions for both species, preventing the use of any of its simplified forms, mentioned in Remark 1.

Since the system (33) is equivalent in the large particles number limit to the system (25), a mean-field system, what should happen for systems (33) and ( 31) to be equivalent in the limit is that (31) is somehow stable (in the large particles number limit) w.r.t. deletion (or addition) of a finite number of particles of the two species. This is in fact the case, as we shall show in the first Lemma in this section. Then, deleting light particles and obstacles in a suitable way from system (31), we shall be able to simplify the dependence on the stochastic configuration in the estimates and finally to prove the asymptotic equivalence between ( 33) and (31).

Let O and P be finite subsets of N and k the level of approximation. In this section, we introduce a sequence of particle systems where the obstacles with labels in O (resp. light particles with labels in P) have no interaction with the light particles (resp. with the obstacles). These systems will give us a tool to estimate the required distance.

For a given configuration of obstacles, cM = (c1, . . . , cM ) and a particle initial datum zn = (z1, . . . , zn), fix the sets of integers O ⊆ {1, . . . , M }, P ⊆ {1, . . . , n} and define, for integers M, k ≥ 1:

(59) γ (0,∅,P) n,ε,0 (t, zj) = γ (k,∅,P) n,ε,0 (t, zj) = I {j / ∈P} , γ (0,O,P) n,ε,M (t, zj ) = I {j / ∈P} , δ (0,O,P) n,ε,M (t, c ℓ ) = I {ℓ / ∈O} , γ (k,O,P) n,ε,M (t, zj) = I {x j (s) / ∈ M h=1 Bε(c h )δ (k-1,O,P) (c h ,s) ∀s∈[0,t)} I {j / ∈P} , δ (k,O,P) n,ε,M (t, c ℓ ) = I {A (k,O,P) n,ε,M (t,c ℓ )≤τc ℓ } I {ℓ / ∈O} , A (k,O,P) n,ε,M (t, c ℓ ) = 1 n n i=1 t 0 qn(xi(s) -c ℓ )γ (k-1,O,P) n,ε,M (s, zi)ds.
Remark that A (k,O,P) n,ε,M (t, c ℓ ) does not depend on τc h , xj, vj for any h ∈ O and any j ∈ P, and it does not depend on c h , h ∈ O, whenever ℓ = h. Moreover, for (O, P) = (∅, ∅), the system defined above is the k-th level approximation system defined in (31), Section 3.2, i.e.

A (k,∅,∅) n,ε,M = V (k) n,ε,M , γ (k,∅,∅) n,ε,M = ξ (k) n,ε,M , δ (k,∅,∅) n,ε,M = η (k)
n,ε,M .

4.5.1. Proof of the equivalence between ( 33) and (31) using the O, P-frozen system. Using the bound (41), we can use the same strategy as in [NOR].

To shorten the notation, we define in this paragraph, for sets O, O1 ⊆ {1, . . . , M } and P, P1 ⊆ {1, . . . , n}:

A nM (k) O,P (O1, P1) = A (k,O∪O 1 ,P∪P 1 ) n,ε,M -A (k,O,P) n,ε,M , for M = 1, . . ., I O,P,0 ε,t,z i (c ℓ ) = 0 I O,P,M ε,t,z i (c ℓ ) = M I {c ℓ ∈Tε(t,z i )} I O⊂{1,.
..,M } I P⊂{1,...,n} , and I M ε,t,z i = I ∅,∅,M ε,t,z i . We prove the following lemmas: Lemma 7. Consider the stochastic variables defined in (59). Then, under the same hypothesis as in Proposition 1, for all integers k ≥ 1, p ≥ 1 and for all O1, P1 ⊂ N s.t. 0 < #O1, #P1 < ∞, P-almost everywhere w.r.t. sequences of initial data Z∞, we have the limits

(60) lim n→∞ 1 n p 1≤j 1 ≤n ,..., 1≤jp ≤n sup z j 0 ∈R d ×R d 1 |vj 0 | E n [0,T ] p p h=1 dsj h qn(xj h (sj h ) -ci h )I {c i h ∈Tε(s j h-1 ,z j h-1 )} ×I O∪O 1 ,P∪P 1 ,M ε,s jp ,z jp (c ℓ )M #{i 1 ,...,ip,ℓ}-1 A nM (k) O,P (O1, P1)(Tz jp ,c ℓ , c ℓ ) = 0 lim n→∞ sup z j 0 ∈R d ×R d 1 |vj 0 | E n I O∪O 1 ,P∪P 1 ,M ε,s j 0 ,z j 0 (c ℓ )A nM (k) O,P (O1, P1)(Tz j 0 ,c ℓ , c ℓ ) = 0
for all i1, . . . , ip ∈ O ∪ O1 and sj 0 < T , zj 0 = (xj 0 , vj 0 ). Limits ( 60) are valid also in the case where T replaces Tz j ,c ℓ .

Remark 5. This lemma shows that particle systems obtained from (31) by deleting a finite number of light particles and/or obstacles are equivalent, in the prescribed asymptotics.

Proof. The proof is obtained by induction.

For k = 1 and p ≥ 0, we have

A nM (1) O,P (O1, P1)(Tz jp ,c ℓ , c ℓ ) ≤ 1 n h∈P 1
Tz jp ,c ℓ 0 dsqn(x h (s)c ℓ ).

and the second limit in (60) follows straightforwardly, since

I O∪O 1 ,P∪P 1 ,M ε,s jp ,z jp (c ℓ )A nM (1) O,P (O1, P1)(Tz jp ,c ℓ , c ℓ ) ≤ I O∪O 1 ,P∪P 1 ,M ε,s jp ,z jp (c ℓ ) a d n n #P1 q ∞ T and E n [I O∪O 1 ,P∪P 1 ,M ε,s jp ,z jp (c ℓ )] ≤ µC d T |vj p |, for p ≥ 0.
Denoting N = #{i1, . . . , ip, ℓ}, we build a partition of labels by grouping them in the following way.

We start from the label ℓ and we denote as

i k (1) 1 , . . . , i k (1) w 1 , with k (1) 1 < k (1) 2 < . . . < k (1)
w 1 , the w1 labels among the p + 1 labels i1, . . . , ip, ℓ having common value ℓ; notice that k

(1) w 1 = p + 1, being ci p+1 = c ℓ the obstacle associated to the light particle label jp+1. We then consider i f 1 = max{is : is = ℓ} and we call i k (2) 1 , . . . , i k (2) w 2 the w2 labels having common value i f 1 , always using the ordering k

(2) 1 < k (2) 2 < . . . < k (2) w 2 (i.e., k (2)
w 2 = f1). We build in this way N groups of labels and we denote q the group label such that k ( q) 1 = 1 .

We can write then

(61) 1 n p 1≤j 1 ≤n ,..., 1≤jp ≤n E n [0,T ] p p h=1 dsj h qn(xj h (sj h ) -ci h )I {c i h ∈Tε (s j h-1 ,z j h-1 )} ×I O∪O 1 ,P∪P 1 ,M ε,s jp ,z jp (c ℓ )M #{i 1 ,...,ip,ℓ}-1 1 n j p+1 ∈P 1 Tz jp ,c ℓ 0 dsqn(xj p+1 (s) -c ℓ )   ≤ E n c [M N ] [Λn] N 1 n p+1 j k (1) w 1 ∈P 1 1≤j k (1) h ≤n h=1,...,w 1 -1 1≤j k (2) h ≤n h=1,...,wq . . . 1≤j k (N ) h ≤n h=1,...,w N N q=1 Λn dci k (q) wq [0,T ] wq wq h=1   dsj k (q) h qn(xj k (q) h (sj k (q) h ) -ci k (q) wq )I {c i k (q) wq ∈ wq h=1 Tε(s j k (q) h -1 ,z j k (q) h -1
)}

  

We use then the bound

I {c i k (q) wq ∈ wq h=1 Tε(s j k (q) h -1 ,z j k (q) h -1 )} ≤ I {c i k (q) wq ∈Tε(s j k (q) 1 -1 ,z j k (q) 1 -1 )} .
so that, thanks to (42), we have:

(61) ≤ E n c [M N ] [Λn] N #P1 q ∞T a d n n N q=1 (K1T ) wq -1 n N-1 Λ N n N q=1 dci k (q) wq I {c i k ( q)
w q ∈Tε(s j 0 ,z j 0 )}

N q=1 q = q     1≤j k (q) 1 ≤n [0,T ] dsj k (q) 1 -1 qn(xj k (q) 1 -1 (sj k (q) 1 -1 ) -ci k (q) 1 -1 )I {c i k (q) wq ∈Tε(s j k (q) 1 -1 ,z j k (q) 1 -1 )}           
and, using ( 43) and E n c [M N ] ≤ K(µε[Λn]) N we get finally: 60) is valid for k = 1. For k = 2, since

(61) ≤ K #P1 q ∞ T a d n n µT C d K2 K1 N (T K1) p+1 K2 |vj 0 | so that (
I O∪O 1 ,P∪P 1 ,M ε,s jp ,z jp (c ℓ )A nM (2) O,P (O1, P1)(Tz jp ,c ℓ , c ℓ ) ≤ I O∪O 1 ,P∪P 1 ,M ε,s jp ,z jp (c ℓ )   1 n h∈P 1 Tz jp ,c ℓ 0 dsqn(x h (s) -c ℓ )+ 1 n h / ∈P 1 Tz jp ,c ℓ 0 dsqn(x h (s) -c ℓ ) m∈O I {cm∈Tε(s h ,z h )}   ,
in the same way (using Lemma 2 and Lemma 3) we get the second limit in ( 60) and

(61) ≤ KD #P1 a d n n + #O(a d n ε ζ + ε d-1 ) µT C d K2 K1 N (T K1) p+1 K2 |vj 0 | (with KD depending on q ∞ , T , K1, K2, C d ).
Let assume then that ( 60) is true for k -1, with k > 2. From the definition of A (k,O,P) n,ε,M (Tz jp ,c ℓ , c ℓ ) and the bound (41) we have, for p ≥ 0: (62)

I O∪O 1 ,P∪P 1 ,M ε,s jp ,z jp (c ℓ )A nM (k) O,P (O1, P1)(Tz jp ,c ℓ , c ℓ ) ≤ I O∪O 1 ,P∪P 1 ,M ε,s jp ,z jp (c ℓ )×    #P1 q ∞ T a d n n + 1 n j p+1 / ∈P T 0 dsqn(xj p+1 (s) -c ℓ )× M m=1 I {cm∈Tε(s,z j p+1 )} |δ (k-2,O,P) n,ε,M -δ (k-2,O∪O 1 ,P∪P 1 ) n,ε,M
|(Tz j p+1 ,cm , cm) , and from Lemma 1, for δ > 0, (63)

Eτ m δ (k-2,O∪O 1 ,P∪P 1 ) n,ε,M -δ (k-2,O,P) n,ε,M (Tz j p+1 ,cm , cm) ≤ 2δ+ 1 δ Eτ m A nM (k-2) O,P (O1 ∪ {m}, P1 ∪ {jp+1})(Tz j p+1 ,cm , cm) + 1 δ Eτ m A nM (k-2)
O∪O 1 ,P∪P 1 ({m}, {jp+1})(Tz j p+1 ,cm , cm) . When substituting ( 63) through ( 62) into the expectation values in formula (60), the contribution to the expectation value coming from the first term on the righthand side of ( 63) is bounded by 2KI δ, where KI = KI (T, C d , µ, K1, K2, p, #{i1, . . . , ip, ℓ}), while the contribution to (60) due to the last terms in (63) vanish asymptotically because of the inductive hypothesis. The remaining term coming from ( 62) is O(a d n n -1 ). The last part of the Lemma is proved replacing Tz j ,c ℓ by T .

Lemma 8. Consider the stochastic variables defined in ( 33) and ( 59). Under the same hypothesis as in Proposition 1, ∀k ≥ 1, ∀O, P ⊂ N such that 0 ≤ #O, #P < ∞, P-almost everywhere w.r.t. sequences of initial data Z∞, Proof. We can prove Lemma 8 by induction.

For k = 1 and 1 ≤ u ≤ n we have:

1 |vu| E n [I O,P,M ε,T,zu (c ℓ )|A (1,O,P) n -Â(1) n |(Tz u,cℓ , c ℓ )] ≤ (#P)C d T 2 µ q ∞ a d n n ,
and for k = 2 (using Lemma 3)

1 |vu| E n [I O,P,M ε,T,zu (c ℓ )|A (2,O,P) n -Â(2) n |(Tz u,cℓ , c ℓ )] ≤ Km(#P a d n n + #Oε d-1 + a d n ε ζ )
so that (64), because of ( 18) and ( 19), is valid (Km depends on q ∞ , T , µ, K2, C d ).

Let assume ( 64) is valid for a given k > 1 and ∀O, P.

We shall use the notation From the definitions ( 59) and ( 33) we obtain

1 |vu| E n [I O,P,M ε,T,zu (c ℓ )|A (k,O,P) n,ε,M - Â(k) n,ε,M |(Tz u,cℓ , c ℓ )] ≤ (#P)C d T 2 µ q ∞ a d n n + E n I O,P,M ε,T,zu (c ℓ ) |vu|n j / ∈P T 0 dsqn(xj(s) -c ℓ ) ξ(k-1) -γ (k-1,O,P) (s, zj) .
Since both ξ(k-1) (s, xj, vj) and γ (k-1,O,P) (s, xj , vj) are of the form (40), using (41) and then Lemma 2 and Lemma 3, P-almost everywhere w.r.t. sequences of initial data Z∞ we get (we omit the time integral):

(65) 1 n j / ∈P E n I O,P,M ε,T,zu (c ℓ ) |vu| qn(xj(s) -c ℓ ) ξ(k-1) -γ (k-1,O,P) (s, zj) ≤ T C d µ q ∞ a d n T C d (#O)( 1 n n i=1 |vi|) ε d-1 + 3Kc(ε ζ + o(a -d n )) + j / ∈P E n   I O,P,M ε,T,zu (c ℓ ) n|vu| qn(xj(s)-c ℓ ) h / ∈O∪{ℓ} I c h ∈Tε(s,z j ) η(k-2) -δ (k-1,O,P) n,ε,M (Tz j ,c h , c h )   .
We use then lemma 1 and we write, for δ > 0:

(66)

Eτ ℓ η(k-2) -δ (k-1,O,P) n,ε,M (Tz j ,c h , c h ) ≤ 2δ+ Eτ ℓ δ V (k-2) -A (k-2,O∪{ℓ},P∪{j}) n,ε,M + A nM (k-2) O,P ({ℓ}, {j}) (Tz j ,c h , c h ) , since A (k-2,O∪{ℓ},P∪{j}) n,ε,M
(Tz j ,c h , c h ) does not depend on τ ℓ . The contribution to the right-hand side of (65) coming from the last term in the last expectation value of (66) vanishes in the limit thanks to Lemma 7 and the contribution coming from the first term is bounded by 2δ

(Ka + K b a d n ε ζ + o(a -d n ))
for each δ > 0. We evaluate now the remaining term, recalling that h = ℓ. When h = ℓ, since V (k-2) -A (k-2,O∪{ℓ},P∪{j}) n,ε,M (Tz j ,c h , c h ) is independent of c ℓ , we have (again omitting the time integral):

E n     I O,P,M ε,T,zu (c ℓ ) |vu|n j / ∈P h / ∈O∪{ℓ} qn(xj(s)-c ℓ )I c h ∈Tε(s,z j ) V (k-2) -A (k-2,O∪{ℓ},P∪{j}) n,ε,M (Tz j ,c h , c h )     ≤ µε[Λn]L n|vu| j / ∈P E n [I c∈Tε(T,zu) qn(xj(s)-c)] × E n I O,P,M ε,s,z j (c1) V (k-2) -A (k-2,O∪{ℓ},P∪{j}) n,ε,M (Tz j ,c 1 , c1) .
By the triangular inequality, we may then bound the time integral in [0, T ] of this quantity, P-almost everywhere w.r.t. sequences of initial data Z∞, by the sum of:

µε[Λn]L n|vu| j / ∈P T 0 ds E n [I {c∈Tε(T,zu)} qn(xj(s) -c)] ×E n I O,P,M ε,s,z j (c1) V (k-2) - Â(k-2) n,ε,M (Tz j ,c 1 , c1) ≤ KV sup 1≤u≤n E n 1 |vu| I M ε,T,zu (c1) V (k-2) - Â(k-2) n,ε,M 2 (Tz u,c1 ,
×E n I O,P,M ε,s,z j (c1) Â(k-2) n,ε,M -A (k-2,O∪{ℓ},P∪{j}) n,ε,M (Tz,c 1 , c1) ≤ KA sup 1≤j≤n E n I O,P,M ε,T,z j (c1) |vj | A (k-2,O∪{ℓ},P∪{j}) n,ε,M - Â(k-2) n,ε,M (Tz j ,c 1 , c1) ,
vanishing because of the inductive hypothesis (the constants there depend on C d , µ, T , K1, K2). Since, because of condition a d n ε ζ → 0, all terms vanish, the Lemma is proved.

Collecting all results in this section we prove Proposition 3. Consider the stochastic variables defined in ( 31) and ( 33). Then, under the same hypothesis as in Proposition 1, P-almost everywhere sequences of initial data Z∞,

(67) lim n→∞ 1 n n j=1 E n ξ(k) (t, zj) -ξ (k) n,ε,M (t, zj) = 0 and (68) lim n→∞ E n ε d-1 M I suppφ (c ℓ )|η (k) n,ε,M (t, c ℓ ) - η(k) n,ε,M (t, c ℓ )| = 0.
Proof. We start from ( 67), bounding it first as

1 n n j=1 E n ξ(k) n,ε,M -ξ (k) n,ε,M (t, zj) ≤ 1 n n j=1 E n I M ε,t,z j (c)|η (k) n,ε,M -η(k) |(Tz j ,c, c) .
Using lemma 1 and the bound

(69) | V (k) -A (k,{h},{j}) n,ε,M | ≤ | V (k) - Â(k) n,ε,M | + | Â(k) n,ε,M -A (k,{h},{j}) n,ε,M |,
we get, ∀δ > 0:

Eτ h |η (k) n,ε,M -η(k) |(•, c h ) ≤ 2δ + 1 δ Eτ h |A nM (k) ∅,∅ ({h}, {j})|(•, c h ) + 1 δ Eτ h | Â(k) n,ε,M -A (k,{h},{j}) n,ε,M | + | Â(k) n,ε,M -V (k) | (•, c h ) . Define E (k) h,l (t) = sup z∈R d ×R d 1 |v| E n [I {h},{j},M ε,t,z (c l )A nM (k) ∅,∅ ({h}, {j})(Tz,c l , c l )] + sup 1≤i≤n 1 |vi| E n [I M ε,t,z i (c l )| V (k) - Â(k) n,ε,M | 2 (Tz i ,c l , c l )] + sup 1≤i≤n 1 |vi| E n [I {h},{j},M ε,t,z i (c l )|A (k,{h},{j}) n,ε,M - Â(k) n,ε,M |(Tz i ,c l , c l )].
We get, for all δ > 0:

1 n n j=1 E n ξ(k) n,ε,M -ξ (k) n,ε,M (t, zj) ≤ 1 n n i=1 |vi| C d T µδ + 1 δ sup t∈[0,T ] E h,h (t)
and (67) follows from Lemma 7, Proposition 2 and Lemma 8. As for (68), we use Lemma 1 to get the bounds

Eτ h |η (k) n,ε,M - η(k) n,ε,M |(t, c ℓ ) ≤ 2δ+ 1 δ Eτ h |A nM (k) ∅,∅ ({h}, {j})| + |A (k,{h},{j}) n,ε,M - Â(k) n,ε,M | (t, c ℓ ) , and, for h = ℓ, Eτ h |η (k-2) n,ε,M -δ k-2,{ℓ},{j} n,ε,M |(t, c h ) ≤ 2δ + 1 δ Eτ h A nM (k-2) ∅,∅ ({h}, {j})|(t, c h ) ,
and we bound the difference of risk functions through (41). Noticing that the quantity

A nM (k) ∅,∅ ({ℓ}, {j})(t, c h ) does not depend on c ℓ for h = ℓ when c ℓ / ∈ n i=1
Tε(t, zi), we get then, through (69), for all δ > 0:

E n ε d-1 M I suppφ (c ℓ )|η (k) n,ε,M (t, c ℓ ) - η(k) n,ε,M (t, c ℓ )| ≤ µ[suppφ]LKp(δ + a d n δn ) + K d δ 3 sup s∈[0,T ] E (k-2) ℓ,ℓ-1 (s) + a d n ε d-1 sup s∈[0,T ] E (k-2) ℓ,ℓ (s) +ε d-1 1 n 1≤i≤n sup s∈[0,T ] z∈R d ×R d 1 |v| E n [0,T ] dsiqn(xi(si) -c ℓ )I {c ℓ ∈Tε(s,z)} ×I {ℓ},{j},M ε,s i ,z i (c ℓ-1 )M A nM (k-2) ∅,∅ ({ℓ}, {j})(Tz i ,c ℓ-1 , c ℓ-1 ) ,
where Kp = Kp(T, q ∞ , K1, K2, µ, C d ) and

K d = K d (µ, Θ, |v|f0 L 1 )
. The proof of the proposition follows using Lemmas 7, 8 and Proposition 2.

5. Final proposition: asymptotic equivalence of ( 31) and ( 25)

We may now prove our final proposition, which will allow us to establish the vanishing limits of ( 34) and (37) : Proposition 4. For the life functions defined in (30) and in (31), under the same hypothesis as in Proposition 1 and for all φ ∈ C b (R d × R d ) and ψ ∈ CK (R d ), the following limits are valid, P-almost everywhere w.r.t. sequences of initial data Z∞:

(70) lim n→∞ E n | 1 n n i=1 φ(T t (zi))( ξ(k) (t, zi) -ξ (k) n,ε,M (t, zi))| = 0 and (71) lim n→∞ E n ε d-1 M i=1 I suppψ (ci)|η (k)
n,ε,M (t, ci)η(k) (t, ci)| = 0.

Proof. By the triangular inequality:

E n | 1 n n i=1 φ(T t (zi))[ ξ(k) -ξ (k) n,ε,M ](t, zi)| ≤ E n | 1 n n i=1 φ(T t (zi))[ ξ(k) - ξ(k) n,ε,M ](t, zi)| + φ ∞ n n i=1 | ξ(k) n,ε,M -ξ (k) n,ε,M |(t, zi)
and the right-hand side term vanishes because of Lemma 5 and Proposition 3.

In the same way, This proposition completes the proof of theorem 1. 6. Appendix 6.1. Existence and Uniqueness of solutions for the limit system. We give here the theorem of existence and uniqueness of solutions of system (2), which we can state as follows:

Theorem 2. Let f0 ≥ 0 and σ0 ≥ 0 be s.t

. f0 ∈ L 1 (R d v ; W 1,∞ (R d x )), vf0 ∈ L 1 (R d v ; W 1,∞ (R d x )) ∩ L ∞ (R d x × R d v ), v 2 f0 ∈ L 1 (R d v ; L ∞ (R d x )
) and σ0 ∈ W 1,∞ (R d x ). Then for each T > 0 there exists an unique solution (f, σ) to the initial value problem (2) in the interval [0, T ].

The proof of theorem ( 2) is the consequence of the following proposition: Proposition 5. Consider the space W = {F = (F1, F2, F3) : Fi ≥ 0 i = 1, 2, 3,

F1 ∈ L ∞ ([0, T ] × R d × R d ), F2, F3 ∈ L ∞ ([0, T ] × R d ) } with norm F W = F1 L ∞ ([0,T ]×R d ×R d ) + F2 L ∞ ([0,T ]×R d ) + F3 L ∞ ([0,T ]×R d )
and the map M = (M1, M2, M3) : W → W defined, for f0, σ0 ≥ 0, as: • Let M [f ] = f be the (unique) fixed point of M , then for each t ∈ [0, T ], T < T0,

f1(t, •, •) ∈ L 1 (R d v ; W 1,∞ (R d x )), vf1(t, •, •) ∈ L 1 (R d v ; W 1,∞ (R d x )) ∩ L ∞ (R d x × R d v ), v 2 f1(t, •, •) ∈ L 1 (R d v ; L ∞ (R d x )) f3(t, •) ∈ W 1,∞ (R d ).
Remark 6. The map (72) is defined so to represent the solutions to the linear problem (25) (cf. the proof of Theorem 2 on next page), and therefore it does not depend on F1 (the sources in the linear problem are indeed F2 and F3). Three components are (of course) nevertheless needed to get a map having as unique fixed point the solution to the nonlinear problem (2).

Proof of Proposition 5. Consider F, G ∈ W. Then (76)

M1[F]-M1[G] L ∞ ([0,T ]×R d ×R d ) ≤ C d T vf0 L ∞ (R d ×R d ) F3 -G3 L ∞ ([0,T ]×R d ) M2[F]-M2[G] L ∞ ([0,T ]×R d ) ≤ C d T vf0 L 1 (R d ;L ∞ (R d )) F3 -G3 L ∞ ([0,T ]×R d ) M3[F]-M3[G] L ∞ ([0,T ]×R d ) ≤ ΘT σ0 L ∞ ([0,T ]×R d ) F2 -G2 L ∞ ([0,T ]×R d ) 1 n n h=1 a d n q(anx h )φ(x h , v h ) -Θ R d f0(0, v)φ(0, v)dv ≤ 1 (2π) d
We consider then, for h = 1, . . ., the sequence of independent stochastic variables (with K depending on φ ∞ and q L 1 ); we obtain then, for κ ∈ (0, 1 2 ) and for all δ > 0, Proposition 6. In the same hypothesis as in Lemma 9, on a full measure set with respect to P qn(x)µ n 0 (x, v) ⇀ Θδ0(x)f0(x, v).

ν φ h (k) = e ik•x h φ(x h , v h ) -
Proof. We consider first functions in the separable space C0(R d ×R d ), the set of continuous functions vanishing at infinity. Let D be a countable dense set in C0(R d ×R d ) and consider the two sets Since the convergence in ( 80) is valid also for φ = 1, on the set A ∩ B, weakly in the sense of measure, qn(x)µn(x, v) ⇀ Θδ0(x)f0(x, v) (see e.g. [MA], p.90, theorem 6.8), and the proposition is proved.

Remark 8. Since f0 ∈ S (R d × R d ) and dvf0 ∈ S (R d ), the convergence proved in Proposition 6 is valid also for |v| j qnµ n 0 , j = 1, 2, . . . (i.e. |v| j qnµ n 0 ⇀ |v| j δ0f0). To show it, it suffices to rewrite the proof with ν φ h replaced with |v| j ν φ h . Under the same hypothesis, it is also possible to prove, by induction, that ⊗ M k=1 qnµn ⇀ ⊗ M k=1 δ0f0, for M = 1, 2 . . .. Remark 9. Whenever, in addition to the hypothesis in Lemma 9, we have f0

∈ L 1 (R d v ; W 1,∞ (R d x )) and v 2 f0 ∈ L 1 (R d v ; L ∞ (R d x )
), we can obtain, following the same steps, the weak convergence of (T v,s a q)nµ 0 n and |v|(T v,s a q)nµ 0 n , where we define (T v,s a g)(x) = g(x + vt + a). The convergence is uniform in a ∈ R d , as can be easily checked.

Moreover, a very simplified form of the procedure allows to prove that, given a limit density f0 ∈ S (R d × R d ) and two positive integers P, Q, on a full measure set w.r.t. P, |v| j µ n 0 ⇀ |v| j f0 and ⊗ i k=1 |v k | j µn ⇀ ⊗ i k=1 |v| j f0, for j = 0, . . . , P and i = 1, . . . , Q.

  ,M |(Tz u,cℓ , c ℓ )| = 0.

  n [I {c∈Tε(T,zu)} qn(xj(s)c)]

  ,Mη(k) |(t, ci)] ≤ E n [ε d-1 M I suppψ (c){|η ,Mη(k) |}(t, c)]and the right-hand side term vanishes because of Corollary 2 and Proposition 3.

  ](t, x, v) = f0(xvt, v)e -C d |v| t 0 dsF 3 (s,x-v(t-s)) M2[F](t, x) = R d dvf0(xvt, v)e -C d |v| t 0 dsF 3 (s,x-v(t-s)) M3[F](t, x) = σ0(x)e -Θ t 0 dsF 2 (s,x) . Assume σ0 ∈ W 1,∞ (R d ) and f0 ∈ L 1 (R d v ;W 1,∞ (R d x )), together with vf0 ∈ L 1 (R d v ;W 1,∞ (R d x ))∩ L ∞ (R d x × R d v ) and v 2 f0 ∈ L 1 (R d v ; L ∞ (R d x )). Then • M is a strict contraction on W whenT< T0, whereT0 depends on C d , Θ, vf0 L ∞ (R×R d ) , vf0 L 1 (R d v ;W 1,∞ (R d x )) , σ W 1,∞ (R d ) )

e

  ik•x h φ(x h , v h ) -R d ×R d e ik•x f0(x, v)φ(x, v)dxdv dk + R d ×R d a d n q(anx)f0(x, v)φ(x, v)dxdv -Θ R d f0(0, v)φ(0, v)dv .

R

  d ×R d e ik•x f0(x, v)φ(x, v), s.t. E P [ν φ h (k)] = 0 and E P [|ν φ h | 2 (k)] ≤ 2 φ 2∞ and for which we have, for any even j ∈ N, anx h )φ(x h , v h ) -Θ

  qn(x) = a d n q(anx) is such that qn ⇀ Θδ0,

A

  = {Z∞ : ∀φ ∈ D R d ×R d dxdvqn(x)µ n 0 (x, v)φ(x, v) → ΘBecause of Lemma 9, both sets have full measure, so as their intersection A ∩ B.For Z∞ ∈ A ∩ B the sequence {qnµ 0 n } ∞ n=1 is a sequence of finite positive measures s.t. for all φ ∈ C0(R d × R d )

Valeria Ricci acknowledges the support by the GNFM (research project 2008:"Approssimazione di equazioni alle derivate parziali tramite sistemi di particelle").

We can therefore write ( 73)

whereA is a constant depending on C d ,Θ and on the norms

) . Whenever T < T0, where T0 = 1 A , M is a contraction on the complete space W. Therefore, whenever T < 1

A , there exists a unique fixed point (f, g, σ) = M (f, g, σ) (and of course, g = dvf ).

The properties of the fixed point functions listed in the second part of the thesis of the proposition follow trivially from (72) and the assumptions on f0.

Proof of theorem 2. The map M maps the point

A . Because of the property of the fixed point (f, dvf, σ), the solution is prolongeable for any value of T > 0.

6.2. Condition for the weak convergence qnµ n 0 ⇀ δ0f0. We state here a simple condition on the growth rate of the generic term of the sequence {an} such that the product of two weakly convergent measures converges weakly to the product of the two limit measures.

Lemma 9. Let P be a probability measure defined as (4), with one particle probability density f0 s.t.

and q ∈ S (R d ) a non negative function s.t.

Take a sequence of positive real numbers {an} ∞ n=1 (an > 0) such that for some κ ∈ (0, 1 2 )

Proof. We observe first that, since q ∈ S (R d ), we can use the identity

e ik•x q(k)dk where q ∈ S (R d ) is the Fourier transform of q.

Because of q ∈ S (R d ) and ( 74), in all calculations below we can apply Fubini's theorem. We can then write: