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AUX MEMBRES DU JURY DE THESE

Monsieur le Professeur Lionel CAMILLERI

Cher Professeur, c'est un véritable honneur que vous me faîtes en acceptant de faire partie de ce jury de thèse. Ce fût un grand plaisir de vous avoir comme enseignant pendant ces années d'internat. Je vous adresse mes remerciements les plus sincères.

Monsieur le Professeur Johan GAGNIERE

Cher Johan, je te remercie d'avoir accepté de participer à mon Jury. Je te remercie pour tout ce que tu as pu essayer de m'apprendre dans ton domaine. Je te remercie également grandement pour nous avoir choisi pour te succéder au 22 rue des chaussetiers.

Monsieur le Docteur Grégoire MASSOULIE

Cher Grégoire, je te remercie d'avoir accepté de faire partie de mon Jury. Je te remercie pour l'enthousiasme dont tu as fait preuve lorsque je t'ai évoqué mon sujet, ainsi que pour ta disponibilité et ton temps.

A NOTRE DIRECTEUR DE THESE

Madame le Docteur Laura FILAIRE

Chère Laura, je te remercie grandement pour ton soutien et ta confiance pour préparer cette thèse. J'espère avoir été à la hauteur, et faire honneur à ta première thèse en tant que directrice.

Dédicaces validées le 5 septembre 2023

Le Directeur de thèse, Laura FILAIRE UFR de Médecine et des Professions Paramédicales Service de la formation -Pôle 4 TSA 50400 28, Place Henri-Dunant 63001 Clermont-Ferrand Cedex 1

TABS AND FIGURES

INTRODUCTION

Catheter ablation (CA) of atrial fibrillation (AF) is a widespread procedure since it has proven its benefit in term of reduction of arrhythmia burden and improvement of quality of life [START_REF] Packer | Effect of Catheter Ablation vs Antiarrhythmic Drug Therapy on Mortality, Stroke, Bleeding, and Cardiac Arrest Among Patients With Atrial Fibrillation: The CABANA Randomized Clinical Trial[END_REF].

Although many techniques are used to decrease its morbidity, esophageal lesions are reported up to 40% and classified as erythema, ulceration, or perforation [START_REF] Halbfass | Progression From Esophageal Thermal Asymptomatic Lesion to Perforation Complicating Atrial Fibrillation Ablation: A Single-Center Registry[END_REF]. The physiopathology is not entirely understood but esophageal lesion seems to be a prerequisite for the development of a fistula from the esophagus towards the atrium (2). Atrio-esophageal fistula (AEF) is the most feared and life-threatening complication, occurring with an incidence of 0.02 to 0.04% [START_REF] Gandjbakhch | Incidence, epidemiology, diagnosis and prognosis of atrio-oesophageal fistula following percutaneous catheter ablation: a French nationwide survey. Eur Eur Pacing Arrhythm Card Electrophysiol J Work Groups Card Pacing Arrhythm Card[END_REF][START_REF] Yousuf | Management of Atrio-Esophageal Fistula Following Left Atrial Ablation[END_REF]. This incidence is probably underestimated due to the non-specificity of symptoms and delayed diagnosis. Overall mortality has been reported from 55 to 65% [START_REF] Tilz | A worldwide survey on incidence, management, and prognosis of oesophageal fistula formation following atrial fibrillation catheter ablation: the POTTER-AF study[END_REF][START_REF] Han | Atrioesophageal Fistula: Clinical Presentation, Procedural Characteristics, Diagnostic Investigations, and Treatment Outcomes[END_REF]. Many therapeutic options have been described: medical treatment, endoscopic treatment alone or associated with surgery and upfront surgical treatment. Some authors have shown a clear benefit in survival of a surgical management for AEF in comparison to endoscopic or medical treatment alone [START_REF] Gandjbakhch | Incidence, epidemiology, diagnosis and prognosis of atrio-oesophageal fistula following percutaneous catheter ablation: a French nationwide survey. Eur Eur Pacing Arrhythm Card Electrophysiol J Work Groups Card Pacing Arrhythm Card[END_REF][START_REF] Tilz | A worldwide survey on incidence, management, and prognosis of oesophageal fistula formation following atrial fibrillation catheter ablation: the POTTER-AF study[END_REF][START_REF] Rocca | Clinical presentation, diagnosis, and treatment of atrioesophageal fistula resulting from atrial fibrillation ablation[END_REF][START_REF] Jehaludi | Retrospective review of 65 atrioesophageal fistulas post atrial fibrillation ablation[END_REF][START_REF] Amirkhosravi | Surgical Treatment of Atrioesophageal Fistula: A Systematic Review[END_REF]. However, multiple surgical strategies have been reported concerning the management of the esophageal and left atrium perforations, the use of cardiopulmonary bypass (CPB), or a staged surgery. To date, no guidelines are available on the safest and most effective surgical strategies. Thus, we aimed to evaluate the efficacity of different surgical strategies in the management of AEF after CA of AF in French national survey.

MATERIEL AND METHODS

a. Ethics

The study is in accordance with the declaration of Helsinki and was approved by the Institutional Review Board of the French Society of Thoracic and Cardiovascular Surgery on the 10 th of November 2022 (approval number: CERC-SFCTCV-2023-04-25_28215). A written informed consent was sent to each patient.

b. Population

A retrospective observational cohort study was conducted from January 2010 to April 2023.

Inclusion criteria were patients over 18-year-old with a diagnosis of AEF or pericardoesophageal fistula (PEF) after CA of AF who were treated with a surgical approach. Because patient's data were extracted from the French surgical data base EPITHOR, patients treated medically or with endoscopic strategies alone were not included.

c. Data collection

Patient's screening was realized by sending an email to 68 French cardiothoracic surgical units, all participating in the EPITHOR database [START_REF] Dahan | Not Available[END_REF]. Twenty-one centers replied (30.8 %) among them, only 6 centers had eligible patients (8.8%).

Baseline characteristics and comorbidities, AF treatment-related variables, AEF subtypes, clinical and paraclinical exams, surgical strategies (surgical approach, used of CBP, management of the esophageal perforation, flap interposition), and follow-up data were collected by each participating center.

The primary outcome was the survival rate after AEF treated surgically. The secondary outcomes were the assessment of the different surgical strategies to highlight risk factors of mortality.

d. Statistical analysis

All statistical analysis was performed using STATA software version 15. Given the exploratory nature of the analysis, no adjustment for Type I error was made. A bilateral Type I error risk of 5% was considered in all analyses.

Categorical variables are described by frequencies and associated percentages, while quantitative variables will be presented as mean and standard deviation or median and interquartile range, depending on their statistical distribution. Normality is assessed through graphical analysis. A survival analysis is conducted on the time before death. A Cox-type model is used to assess the main dependencies of survival time. The center effect is also studied.

RESULTS

a. Population

Eighteen patients met eligible criteria. The mean age was 58 ± 13-year-old. Sixteen patients were male (88.9%). Fifteen patients had a diagnosis of AEF (83.3%), 3 others had a PEF (16.7%). Baseline characteristics are shown in Table I. No difference was observed between centers. The overall survival rate at 180 days was 55%.

Table I : Clinical description of atrio-esophageal fistula cases (n=18)

N (%) or mean

Figure 4 : Survival curve according to the esophageal perforation management

Survival at 180 days after esophagectomy is 86% versus 33% in case of other surgical treatments for esophageal perforation. The red line represents the survival curve of patients who underwent an esophagectomy and the blue line represents the survival curve of patients treated with another surgical options. There was no statistical difference in terms of survival at 180 days (p=0.092). In multivariate analysis, significant independent risk factors for death were sepsis (HR 6.25, 95% CI 1.17-33.3; p=0.032) and coma (HR 4.81, 95% CI 0.85-27.12; p=0.075). According to symptoms at diagnosis (sepsis and/or coma) and surgical management (esophagectomy or not), patients who underwent esophagectomy still tended to have a better survival than patients with sepsis and/or coma who did not underwent esophagectomy (Figure 5 and6).

DISCUSSION

This study aimed to evaluate the different surgical strategies available in the management of AEF after CA for AF in France. To the best of our knowledge, this is the largest cohort on this topic in the literature. Our main findings were: (i) surgical management varied from one center to another; (ii) aggressive treatment of the esophageal perforation (esophagectomy) tended to offer the best survival benefit; (iii) the use of cardiopulmonary by-pass did not worsen the mortality rate; (iv) sepsis and/or neurological failure at the initial clinical presentation were poor prognostic factors affecting the risk of death.

AEF incidence is 0.025% over a 13-year period that includes 93% of French center [START_REF] Gandjbakhch | Incidence, epidemiology, diagnosis and prognosis of atrio-oesophageal fistula following percutaneous catheter ablation: a French nationwide survey. Eur Eur Pacing Arrhythm Card Electrophysiol J Work Groups Card Pacing Arrhythm Card[END_REF].

From 2010 to 2023, we identified a theoretical number of 33.4 cases of AEF, with 50% eligible to a surgical treatment. Our cohort of 18 patients appears to be true-to-life representation of reality.

Diagnosis of AEF is challenging as clinical and biological presentation are non-specific.

A high degree of suspicion is needed by the medical team to avoid or limit the delay in the treatment which is known to increase mortality rate. In our cohort, clinical presentation was in accordance with other studies, chest pain, fever, and neurological symptoms were the most frequent [START_REF] Gandjbakhch | Incidence, epidemiology, diagnosis and prognosis of atrio-oesophageal fistula following percutaneous catheter ablation: a French nationwide survey. Eur Eur Pacing Arrhythm Card Electrophysiol J Work Groups Card Pacing Arrhythm Card[END_REF][START_REF] Tilz | A worldwide survey on incidence, management, and prognosis of oesophageal fistula formation following atrial fibrillation catheter ablation: the POTTER-AF study[END_REF][START_REF] Amirkhosravi | Surgical Treatment of Atrioesophageal Fistula: A Systematic Review[END_REF]. Neurological symptoms was associated with a latter AEF onset compared to other patients (11) and a higher risk of mortality [START_REF] Han | Atrioesophageal Fistula: Clinical Presentation, Procedural Characteristics, Diagnostic Investigations, and Treatment Outcomes[END_REF]. It highlights that AEF with initial neurological symptoms are the most severe and deadly esophageal perforation secondary to potential systemic embolism and major inflammatory reaction.

Mean time onset from CA to first symptoms was 18.8 days and even longer to surgery (25 days) which in is in line with literature [START_REF] Han | Atrioesophageal Fistula: Clinical Presentation, Procedural Characteristics, Diagnostic Investigations, and Treatment Outcomes[END_REF][START_REF] Rocca | Clinical presentation, diagnosis, and treatment of atrioesophageal fistula resulting from atrial fibrillation ablation[END_REF][START_REF] Amirkhosravi | Surgical Treatment of Atrioesophageal Fistula: A Systematic Review[END_REF][START_REF] Moiroux-Sahraoui | Atriooesophageal fistula following atrial fibrillation ablation: how to manage this dreaded complication?[END_REF]. AEF are iatrogenic esophageal perforation managed lately, as reflected by these delays. This should be kept in mind to adapt the surgical treatment.

All patients underwent chest CT allowing the diagnosis of AEF in 61%, which was inferior to other studies where chest CT had a sensibility around or over 80% [START_REF] Gandjbakhch | Incidence, epidemiology, diagnosis and prognosis of atrio-oesophageal fistula following percutaneous catheter ablation: a French nationwide survey. Eur Eur Pacing Arrhythm Card Electrophysiol J Work Groups Card Pacing Arrhythm Card[END_REF][START_REF] Tilz | A worldwide survey on incidence, management, and prognosis of oesophageal fistula formation following atrial fibrillation catheter ablation: the POTTER-AF study[END_REF][START_REF] Amirkhosravi | Surgical Treatment of Atrioesophageal Fistula: A Systematic Review[END_REF][START_REF] Liu | Clinical Manifestations, Outcomes, and Mortality Risk Factors of Atrial-Esophageal Fistula: A Systematic Review[END_REF] AEF are esophageal perforation not eligible for non-operative management according to Pittsburgh classification (13) despite a few case reports or short series [START_REF] Back Sternick | Esophago-pericardial fistula after catheter ablation of atrial fibrillation: A review[END_REF]. Many reviews have shown a clear survival benefit of surgical management [START_REF] Gandjbakhch | Incidence, epidemiology, diagnosis and prognosis of atrio-oesophageal fistula following percutaneous catheter ablation: a French nationwide survey. Eur Eur Pacing Arrhythm Card Electrophysiol J Work Groups Card Pacing Arrhythm Card[END_REF][START_REF] Tilz | A worldwide survey on incidence, management, and prognosis of oesophageal fistula formation following atrial fibrillation catheter ablation: the POTTER-AF study[END_REF][START_REF] Rocca | Clinical presentation, diagnosis, and treatment of atrioesophageal fistula resulting from atrial fibrillation ablation[END_REF][START_REF] Jehaludi | Retrospective review of 65 atrioesophageal fistulas post atrial fibrillation ablation[END_REF][START_REF] Amirkhosravi | Surgical Treatment of Atrioesophageal Fistula: A Systematic Review[END_REF]. Although many reviews on AEF are available, studies focusing on their surgical management are rare. Surgical approaches differed from patients to another and center from another but had no consequences on mortality [START_REF] Amirkhosravi | Surgical Treatment of Atrioesophageal Fistula: A Systematic Review[END_REF]. From a surgical standpoint, these results were logical as surgical approach is only a way to have the best exposure, allowing surgeon to realize their surgical plan and fix surgical issue.

In our study, CPB was used in two-third of patients and did not significantly impact survival contrary to Amirkhosravi et al. who have shown that the CPB were associated with better survival (84.8% vs 15.2%; p<0.001) [START_REF] Amirkhosravi | Surgical Treatment of Atrioesophageal Fistula: A Systematic Review[END_REF]. In AEF, CBP allows a good exploration and reparation of the left atrium perforation either through a left atrium opening technique or through a posterior approach of the left atrium after cardiac mobilization. Benefits of CPB in AEF rely on the prevention of air or septic emboli during surgical repair and manipulation, cross-clamping ( 9) and resection of necrotic left atrium tissue to avoid secondary septic emboli or endocarditis [START_REF] Kim | 'exploiterai pas le pouvoir hérité des circonstances pour forcer les consciences. Je donnerai mes soins à l'indigent et à quiconque me les demandera[END_REF]. In our study, cannulation strategies were not significantly different between centers and were not found as risk factor for mortality. Cannulation strategies depend on surgical approaches, patient's hemodynamic instability, and surgeon's preferences [START_REF] Amirkhosravi | Surgical Treatment of Atrioesophageal Fistula: A Systematic Review[END_REF][START_REF] Kim | 'exploiterai pas le pouvoir hérité des circonstances pour forcer les consciences. Je donnerai mes soins à l'indigent et à quiconque me les demandera[END_REF].

Groin cannulation at the beginning of the procedure allows correction of hemodynamic instability, an easy fistula dissection by minimizing the risk of hemorrhage and can be converted into central cannulation.

Over 18 patients, eight (44,4%) had a primary tension-free esophageal repair reinforced by flap interposition. Only two of them (25%) underwent redo surgery but six patients died (75%). Although general principal of esophageal perforation recommends a first tension-free direct [START_REF] Chirica | Esophageal emergencies: WSES guidelines[END_REF] repair, this technique might not be appropriate for AEF. Indeed, AEF are secondary to thermic or ischemic lesion in patients with cardiovascular comorbidities and delayed as shown by the time laps between the first symptoms' occurrence and treatment (around 7 days;

Figure 2). Micro vascularization might be of poor quality and necrotic tissue around the fistula impaired a direct repair with good tissues quality. Thus, in our opinion, primary repair should be attempted only on patient with non-delayed surgical management, in patient without hemodynamic instability and a minor sepsis.

All patients, except those who underwent an esophagectomy, had a flap interposition.

Flap interposition allows to buttress the esophageal repair, to bring viable tissue, and to make a separation between the esophageal repair and the left atrial repair avoiding secondary fistula.

Flap interposition in the surgical management of AEF relies on general surgical principal to protect or reinforced suture. Thus, primary esophageal repair should be associated with flap interposition.

Esophageal stenting was used in 5 patients (27,8%) either as bridge-to-esophagectomy or in association with esophageal repair. Two patients died (40%). Three patients (60%) are still alive but needed a redo-surgery for complications. Esophageal stenting was not associated with an increased risk of mortality. Recently, in the POTTER-AF study, patients receiving an interventional treatment (stenting and/or surgery) had a lower mortality rate and less stroke during clinical course [START_REF] Tilz | A worldwide survey on incidence, management, and prognosis of oesophageal fistula formation following atrial fibrillation catheter ablation: the POTTER-AF study[END_REF]. However, worse survival were reported after esophageal stenting alone [START_REF] Gandjbakhch | Incidence, epidemiology, diagnosis and prognosis of atrio-oesophageal fistula following percutaneous catheter ablation: a French nationwide survey. Eur Eur Pacing Arrhythm Card Electrophysiol J Work Groups Card Pacing Arrhythm Card[END_REF][START_REF] Han | Atrioesophageal Fistula: Clinical Presentation, Procedural Characteristics, Diagnostic Investigations, and Treatment Outcomes[END_REF][START_REF] Rocca | Clinical presentation, diagnosis, and treatment of atrioesophageal fistula resulting from atrial fibrillation ablation[END_REF][START_REF] Amirkhosravi | Surgical Treatment of Atrioesophageal Fistula: A Systematic Review[END_REF][START_REF] Liu | Clinical Manifestations, Outcomes, and Mortality Risk Factors of Atrial-Esophageal Fistula: A Systematic Review[END_REF]. No difference in survival rate was found between endoscopic treatment in association with surgery in comparison to surgery alone. Thus, if esophageal stenting is performed, it should be associated with a surgical management.

Esophagectomy was realized in 7 patients (38,9%), among them 5 (71,4%) underwent a restorative surgery and are all still alive, the two other patients (28,4%) died. Three patients (42,8%) needed a redo-surgery in addition to the restorative surgery. Although, more reinterventions were found in the group of patients with esophagectomy, esophagectomy tended to improve survival. In fact, as AEF are potentially late esophageal perforation with a high inflammatory reaction in patients with hemodynamic instability, esophagectomy might be a good option as recommended by the World Society of Emergency Surgery (WSES) guidelines [START_REF] Chirica | Esophageal emergencies: WSES guidelines[END_REF]. Restorative surgery is required 6 -12 months after complete diversion or resection of the thoracic esophagus by colonic interposition. This radical procedure has a certain impact on the quality of life but most of patients had a restorative surgery (71,4%) within 6 months. In our cohort, four patients underwent esophagectomy under CPB with left atrial repair, there was no postoperative mortality, and 3 patients were still alive during follow-up, with a mean follow-up of 5.8 years. Amirkhosravi et al. did not find any difference in survival according to esophageal perforation management but esophagectomy and esophageal direct repair had the lower mortality rate (15.4%) [START_REF] Amirkhosravi | Surgical Treatment of Atrioesophageal Fistula: A Systematic Review[END_REF]. Thus, esophagectomy appears to be the best damage control surgery in the management of esophageal perforation in AEF with the best chance of survival although it is a radical option with the need for further surgery.

Overall survival rate was 50% which is in line with the literature (5,9). However, a higher mortality rate was reported in the first 90 postoperative day as shown by the survival rate at 15 days (60%) and 84 days (55%). Risk factors of mortality were sepsis and neurological symptoms. In our series, only esophagectomy tended to influence positively the survival. Others risk factors of death have been found: stenting alone and medical management alone [START_REF] Tilz | A worldwide survey on incidence, management, and prognosis of oesophageal fistula formation following atrial fibrillation catheter ablation: the POTTER-AF study[END_REF][START_REF] Liu | Clinical Manifestations, Outcomes, and Mortality Risk Factors of Atrial-Esophageal Fistula: A Systematic Review[END_REF][START_REF] Liu | Clinical Manifestations, Outcomes, and Mortality Risk Factors of Atrial-Esophageal Fistula: A Systematic Review[END_REF], the absence of CBP ( 9), gastro-intestinal bleeding, coma, and age over 60 year-old (7).

STRENGTH AND LIMITATIONS

This study has some limitations: only 18 patients were included which can be explained by the rare incidence of AEF. Many surgical strategies were assessed with only few patients in each group, decreasing statistical power and limiting statistical analysis. Comparison of surgical results on quality of life was not possible as no data were available for patients who were still alive. However, apart from review of literature, our study is the largest nationwide cohort on atrio-esophageal fistula.

CONCLUSION

Atrio-esophageal fistula should be highly suspected in patients readmitted after catheter ablation of atrial fibrillation as diagnosis and treatment is a race against time.

FIGURE 1 :

 1 FIGURE 1 : CLINICAL PRESENTATION .....................................................................................

FIGURE 2 :

 2 FIGURE 2 : TEMPORAL TRENDS FROM CATHETER ABLATION TO SURGERY ...........................

FIGURE 3 :

 3 FIGURE 3 : OVERALL SURVIVAL CURVE AT 180 DAYS ............................................................

FIGURE 4 :

 4 FIGURE 4 : SURVIVAL CURVE ACCORDING TO THE USE OF ESOPHAGECTOMY ......................

FIGURE 5 :

 5 FIGURE 5 : SURVIVAL CURVE ACCORDING TO THE PRESENCE OF SEPSIS AND THE USE OF

FIGURE 6 :

 6 FIGURE 6 : SURVIVAL CURVE ACCORDING TO THE PRESENCE OF COMA AND THE USE OF

N e u r o l o g i c a l s y m p t o m s F e v e r C o n f u s i o n R e c c u r e n t A F S e p s i s P e r i c a r d i a l e f f u s i o n G I b l e e d i n g T a m p o n n a d e

Figure 3 :

 3 Figure 3 : Overall survival curve at 180 days

Figure 5 :

 5 Figure 5 : Survival curve according to the presence of sepsis and the use of esophagectomy

Figure 6

 6 Figure 6 Survival curve according to the presence of coma and the use of esophagectomy

NOTRE PRESIDENT DE THESE Monsieur le Professeur Marc FILAIRE Cher

  Professeur, c'est un honneur que vous me faîtes en présidant ce jury de thèse. Un grand merci pour votre disponibilité et votre aide durant toutes ces années. Soyez assuré de ma reconnaissance et de mon respect.

				N°
	UNIVERSITÉ CLERMONT AUVERGNE UNIVERSITE CLERMONT AUVERGNE Mme ESCHALIER Bénédicte Médecine Générale Mme RICHARD Amélie Médecine Générale A
	___________________ Médecine Générale UFR DE MÉDECINE ET DES PROFESSIONS PARAMÉDICALES TESSIERES Frédéric Mme ROUGE Laure M. Médecine Générale
	Mme BERTRAND-JARROUSSE Véronique	Médecine Générale
	PRESIDENTS HONORAIRES Mme VICARD-OLAGNE Mathilde		: JOYON Louis Médecine Générale
	UNIVERSITE D'AUVERGNE M. MENINI Thibault	: TURPIN Dominique : DOLY Michel Médecine Générale THÈSE D'EXERCICE
		pour le		: VEYRE Annie : DULBECCO Philippe
	DIPLÔME D'ÉTAT DE DOCTEUR EN MÉDECINE : ESCHALIER Alain
	PRESIDENTS HONORAIRES UNIVERSITE BLAISE PASCAL	par		: CABANES Pierre : FONTAINE Jacques
				: BOUTIN Christian
		DUPAUTET Ludovic	: MONTEIL Jean-Marc : ODOUARD Albert
				: LAVIGNOTTE Nadine
	PRESIDENTDE L'UNIVERSITE			: BERNARD Mathias
	PREMIERE VICE-PRESIDENTE
	Présentée et soutenue publiquement le 6 Octobre 2023 CHARGEE DU PILOTAGE ET DES MOYENS : FOGLI Anne
	VICE PRESIDENTE CHARGEE DE LA FORMATION	: PEYRARD Françoise
	VICE-PRESIDENTE CHARGEE DE LA RECHERCHE	: PREVOT Vanessa
	DIRECTEUR GENERAL DES SERVICES		: PAQUIS François
	TRAITEMENT CHIRURGICAUX DES FISTULES ATRIO-OESOPHAGIENNES
	APR S ABL TION PER U AN E D FIBR L ATION ATR AL : TU E NATIONALE FRANCAISE ²²²²²
	UFR DE MEDECINE
	ET DES PROFESSIONS PARAMEDICALES
	Directeur de thèse : Madame Laura FILAIRE, Docteur, Centre Jean PERRIN, Service
	de chirurgie thoracique et endocrinienne
	DOYENS HONORAIRES			: DETEIX Patrice
				: CHAZAL Jean
	Président du jury : Monsieur Marc FILAIRE, Professeur, UFR de Médecine et des
	Professions paramédicales de Clermont-Ferrand DOYEN-DIRECTEUR	: CLAVELOU Pierre
	DIRECTRICE ADMINISTRATIVE			: ROBERT Gaëlle
	Membres du jury :
	Monsieur Lionel CAMILLERI, Professeur, UFR de Médecine et des Professions
	paramédicales de Clermont-Ferrand
	Monsieur Johan GAGNIERE, Professeur, UFR de Médecine et des Professions
	paramédicales de Clermont-Ferrand
	Monsieur Grégoire MASSOULIE, Docteur, Service de Cardiologie et médicale et
	médecine vasculairE

Secrétariat Doyen-Directeur / Scolarité -Mise à jour janvier 2023

MAITRES DE CONFERENCES ASSOCIES DES UNIVERSITES

TABLE I : CLINICAL DESCRIPTION OF ATRO-ESOPHAGEAL FISTULA CASES (N=18)

 I ..............

TABLE II :

 II PARACLINICAL EXAMS ..........................................................................................

TABLE III :

 III SURGICAL STRATEGIES AND OUTCOMES .............................................................

Table

  II details the paraclinical exams. All patients underwent a chest computed tomography (CT) with intravenous contrast and/or contrast esophagogram. Diagnosis of AEF or PEF was made for 10 patients (55.5%) based on the chest CT. Reported abnormalities were free air between the left atrium and the esophagus, or free air within the pericardium associated with pericardial effusion. Transthoracic echocardiogram (TTE) findings were similar to that of chest CT. Cerebral imaging (brain CT or Magnetic Resonance Imaging) found ischemic lesions secondary to systemic embolism. Esophageal endoscopy exams like Transesophageal echocardiogram (TEE) or Esophagogastroduodenoscopy (EGD) were performed in more than Figure 2 illustrates the different time laps from CA to surgery. Diagnosis was first evocated in 44.4% of the patients (n = 8). Patients were first admitted in reference center in 66.7% of cases (n = 12).

	c. Surgical outcomes
	Surgical strategies for each patient are detailed in Table III. Five patients (27.7%) underwent
	esophageal stenting. For 2 of them (11.1%) (patients 4 and 13), the diagnosis of AEF was made
	by EGD. Esophageal stenting was used as bridge to surgery and led to an esophagectomy in 2
	Figure 2 : Temporal trends from catheter ablation to surgery patients (11.1%) (patients 13 and 15). Both patients needed a redo surgery for postoperative
	complications. One patient died, and one is still alive. In patients 4, 5 and 18, esophageal
	30 stenting was associated with a left atrium repair and flap interposition. Patients 4 and 18 were
		25			24,1	25
		20	18,8
	Days	15
		10	19,5	6,83 Table II : Paraclinical exams
					n (%)	Se (%)
		5
			Chest CT with contrast	17 (94) 3,5	65 0,94	0,5
		0	Chest CT with contrast CA to symptoms onsent First symptoms to 3 (16.7) esophagogram Cerebral imaging 12 (67) diagnosis Time trends	66,6 Diagnosis to surgery 17
	TTE Mean The mean length of stay (LOS) in the intensive care unit and hospitalization were respectively 16 (89) 6 Median Total time (mean)
	TEE Yellow line represents the mean overall time from catheter ablation to symptoms onset, 1 (5.6) 0 27.9 ± 24.6 days and 53.9 ± 46.1 days.
	EGD A center effect was observed concerning the different surgical approaches and esophageal 7 (39) 57 diagnosis, and surgery.
	Se: sensibility; EGD: Esophagogastroduodenoscopy; TEE: Transesophageal echocardiogram; perforation management: thoracotomy (p=0.002), sternotomy (p=0.008), clamshell (p=0.0029)
	TTE: transthoracic echocardiography and esophagectomy (p=0.002). No center effect was observed concerning the use of CBP

one-third of patients and allowed diagnosis in 4 patients (22.2%). No clinical degradation was reported after any endoscopic procedure. Retrospectively, reinterpretation of the first paraclinical exam could have set the diagnosis of AEF in 7 patients (38.9%). Among them, 2 patients (11.1%) had the diagnosis obtained by esophageal endoscopy. both reoperated on but are still alive whereas patient 5 died 15 days later. Seven patients (38.9%) underwent esophagectomy, among which a restorative surgery was performed in five patients (71.4%). Six patients (85.7%) needed to be reoperated on either for complications (n=2) and/or a restorative surgery. Two patients died (28.6%).

Eight patients (44.4%) underwent a surgical treatment with esophageal direct repair and flap interposition. Six of them (75%) had also a left atrium repair either by direct suture, or patch repair. Only two patients (25%) underwent a redo-surgery for complications. Six patients (75%) died postoperatively. Flap interposition was systematically realized in absence of esophagectomy. CPB was used in twelve patients (66.6%) whatever surgical option was considered for esophageal repair.

(p=0.471), and AEF subtypes (p=0.082). Time to diagnosis was not influenced by a center effect or a suspected diagnosis at admission.

TABLE III : SURGICAL STRATEGIES AND OUTCOMES

 III The mean follow-up time was 768.2 days with a median of 89.5 days. Overall survival rate was

						Clamshell, CBP, LA patch repair,
	11	F	AEF	EGD	-	Esophagus direct suture, Flap	NR	-	Alive
						interposition
	N Age Sex 12 M 1 M 13 M 50% (n = 9). The survival rate at 15 days was 60%, and 55% at 84 days. As almost half of the AEF sub-types Imaging for diagnosis Esophageal stenting Surgical strategies Right thoracotomy, CBP, LA patch AEF Chest CT -repair, AEF Chest CT Esophagectomy -Sternotomy, CBP, LA patch repair, Esophagus direct suture, Flap interposition AEF EGD Yes Right thoracotomy, LA patch repair, Esophagectomy	Complications NR Re Do surgery Re Do surgery	Restorative Yes surgery --	Vital Status Alive Deceased Deceased
	2 14 event happened during the first 90 days, we decided to analyze the survival rate during the first M PEF EGD -Sternotomy, CBP, LA direct suture Esophagus direct suture M AEF Chest CT -Right thoracotomy, LA patch repair, Esophagectomy,	Re Do surgery NR	-Yes	Deceased Alive
	Flap interposition Right thoracotomy, CBP, LA patch repair, 180 days for more relevancy. Figure 3 illustrates the overall survival rate at 180 days and figure 3 M AEF Brain CT -15 M AEF Chest CT Yes Sternotomy, Esophageal direct suture Flap interposition 4 M PEF Chest CT Yes Esophagectomy 4 the survival rate according to the esophageal perforation management at 180 days. No Right thoracotomy, CBP, LA patch Sternotomy, CBP, Pericardium direct suture, Esophagus direct suture, 16 M AEF Chest CT -repair, Esophagectomy differences in survival rate were seen among the different surgical approaches. The use of an Flap interposition 5 M AEF Chest CT contrast esophagogram Yes Sternotomy, CBP, LA direct suture, Right thoracotomy, Esophagus direct suture, Flap interposition 6 M AEF TTE -Right thoracotomy, Esophagus direct suture, Flap interposition 18 M AEF contrast esophagogram Yes suture, Flap interposition Right thoracotomy, Esophagus direct Chest CT with 17 M AEF Chest CT -Right thoracotomy, CBP, LA patch repair, Esophagectomy esophageal stent tends to improve overall survival (HR 0.258; 95% CI 0.03-2.1; p=0.2)	Re Do surgery NR Re Do surgery NR NR NR Re Do surgery Neurological failure	Yes --Yes ----	Alive Deceased Alive Alive Deceased Alive Deceased Deceased
	7	F	PEF	Chest CT	-	Thoracotomy, Esophagectomy, Flap interposition	NR	Yes	Alive
						Thoracotomy, CBP, LA direct suture,
	8	M	AEF	Brain CT	-	Esophagus direct suture, Flap	NR	-	Deceased
						interposition
						Clamshell, CBP, LA direct suture,
	9	M	AEF	EGD	-	Esophagus direct suture, Flap	NR	-	Alive
	interposition Clamshell, CBP, LA direct suture, relevant: sepsis (HR 6.398, 95% CI 1.265-32.359; p=0.025) and neurological failure (HR 4.905,
	10 95% CI 0.954-25.218; p=0.0057). GI bleeding (HR 0.453, 95% CI 0.056-3.70; p=0.46), M AEF Chest CT -Esophagus direct suture, Flap interposition	NR	-	Deceased
	pericardial tamponade (HR 1.593, 95% CI 0.321-7.918; p=0.569), mediastinitis (HR 0.552,
	95% CI 0.111-2.747; p=0.468) were not identified as risk factors.				25

AEF: Atrio-esophageal fistula; PEF: pericardo-esophageal fistula; EGD: esogastroduodenoscopy; TTE: Trans Thoracic echocardiogram; CBP: Cardiopulmonary By-pass; LA: Left Atrium; NR: Not reported d. Survival whatever the surgical management. Esophagectomy tended to be associated with an improved survival (HR 0.163, 95% CI 0.019-1.340; p=0.092) with 86% rate of survival at 180 days (Figure

4

). The use of CBP did not impact survival (HR 1.953, 95% CI 0.392-9.719; p=0.413).

Overall survival tended to be better in case of PEF than AEF (HR 1.9; 95% CI 0.23-15.81; p= 0.54). In univariate analysis for risk factors for death, only 2 symptoms were statistically

  enhanced if repeated. After retrospective chest CT reinterpretation, diagnosis was missed in 7 patients.Retrospective chest CT sensibility could have been 77.7%. The low sensibility of chest CT for AEF diagnosis might be explained by the high incidence of EGD (more than one-third of patients) with a sensibility of 57%. Although, EGD can allow esophageal perforation diagnosis and potential endoscopic treatment, it is not a recommended diagnosis tool in case of suspected AEF as it can increase systemic air emboli through the fistula, and increase clinical symptoms in 28%[START_REF] Han | Atrioesophageal Fistula: Clinical Presentation, Procedural Characteristics, Diagnostic Investigations, and Treatment Outcomes[END_REF]. Repeated chest CT and a better knowledge of AEF by radiologist could reduce the time to diagnosis and ensure an urgent management.

  En présence des Maîtres de cette FACULTE et de mes chers CONDISCIPLES, je promets et je jure d'être fidèle aux lois de l'Honneur et de la Probité dans l'exercice de la Médecine.Je donnerai mes soins gratuits à l'indigent et je n'exigerai jamais un salaire au-dessus de mon travail. Admis dans l'intérieur des maisons, mes yeux ne verront pas ce qui s'y passe, ma langue taira les secrets qui me seront confiés et mon état ne servira pas à corrompre les moeurs ni à favoriser le crime.Respectueux et reconnaissant envers mes MAÎTRES, je rendrai à leurs enfants l'instruction que j'ai reçue de leurs pères.Que les HOMMES m'accordent leur estime si je suis fidèle à mes promesses. Que je sois couvert d'OPPROBRE et méprisé de mes confrères si j'y manque. Les fistules atrio-oesophagiennes (FAO) sont des complications rares mais mortelles après ablation percutanée de fibrillation atriale (FA). Bien que le traitement chirurgical apporte un bénéfice en termes de survie, aucune recommandation existe concernant ces modalités.Objectif : L'objectif de notre étude est d'évaluer, à partir d'une cohorte nationale française, l'efficacité des différentes stratégies chirurgicales utilisées dans la prise en charge des FAO après ablation percutanée de FA.Méthode : Entre 2010 et 2023, les patients de plus 18 ans avec un diagnostic de FAO ou de fistule péricardo-oesophagienne après ablation percutanée de FA, traités chirurgicalement ont été rétrospectivement sélectionnés par contact mail des centres participants à la base de données EPITHOR.Résultats : 18 patients ont été inclus, 15 avec un diagnostic de FAO et 3 avec un diagnostic de fistule péricardo-oesophagienne. Cinq patients ont bénéficié d'un stenting oesophagien en association à une chirurgie (réparation directe ou oesophagectomie), dont deux sont décédés (40%). Sept patients (38,9%) ont bénéficié d'une oesophagectomie dont cinq avec chirurgie de rétablissement à 6 mois. Deux patients sont décédés (28,6%). Une réparation directe de l'oesophage en association avec un lambeau d'interposition a été réalisé chez huit patients dont six sont décédés (75%). Un effet centre est observé sur la voie d'abord chirurgicale et la réalisation d'une oesophagectomie (p=0,002). Le suivi médian était de 89,5 jours avec une survie à 90 jours de 55%. L'oesophagectomie semble être associée à une meilleure survie (HR 0,163; 95% IC 0,019-1,340; p=0,092). En analyse univariée, les facteurs de risques de décès étaient le sepsis (HR 6,398 ; 95% IC 1,265-32,359; p=0,025) et les symptômes neurologiques (HR 4,905 ; 95% IC 0,954-25,218; p=0,006). La prise en charge chirurgicale des FAO est un vrai challenge où la réalisation d'une oesophagectomie semble apporter les meilleures chances de survie, notamment chez les patients en sepsis et présentant des signes neurologiques. La réparation directe de l'oesophage avec ou sans stenting préalable doit être réserver aux patients les moins sévères.

	TRAITEMENT CHIRURGICAUX DES FISTULES ATRIO-OESOPHAGIENNES
	APRES ABLATION PERCUTANEE DE FIBRILLATION ATRIALE : ETUDE SERMENT D'HIPPOCRATE NATIONALE FRANCAISE
	Information of the patient and medical staff on AEF by interventional cardiologist might reduce the delay and improve survival. Surgical treatment offers the best chance of survival. Among the different surgical strategies, esophagectomy should be considered as first strategy especially in patients with hemodynamic Pierre CLAVELOU Marc FILAIRE Nom, Prénom Signature Rationnel : Conclusion :

instability, sepsis, or neurological symptoms. The use of cardiopulmonary bypass can be reasonably recommended. As an alternative, primary direct esophageal repair might be attempted in less severe patients with or without esophageal stenting, in association with flap interposition. Le Doyen de l'UFR de Médecine, Le Président du Jury, Mots clefs : fistule atrio-oesophagienne, perforation oesophagienne, fibrillation atriale, OEsophagectomie, stenting oesophagien


Didier Henrion

Mauricio Junca

Mauricio Velasco

Moment-SOS hierarchy and exit location of stochastic processes *

The moment sum of squares (moment-SOS) hierarchy produces sequences of upper and lower bounds on functionals of the exit time location of a polynomial stochastic differential equation with polynomial constraints, at the price of solving semidefinite optimization problems of increasing size. In this note we use standard results from elliptic partial differential equation analysis to prove convergence of the bounds produced by the hierarchy. We also use elementary convex analysis to describe a superand sub-solution interpretation dual to a linear formulation on occupation measures. Finally, we introduce a novel Christoffel-Darboux approach for the recovery of the exit location and occupation measures. The practical relevance of these results is illustrated with numerical examples.

Introduction

This paper deals with the numerical evaluation of functionals of solutions of nonlinear stochastic differential equations (SDE). Our approach consists of constructing a family of convex optimization problems (semidefinite programming problems, SDP) of increasing size whose solutions yield bounds on the given functional. This is an application of the so-called Lasserre or moment sum of squares (SOS) hierarchy [START_REF] Lasserre | Moments, positive polynomials and their applications[END_REF][START_REF] Henrion | The moment-SOS hierarchy -Lectures in probability, statistics, computational geometry, control and nonlinear PDEs[END_REF]. We are especially concerned about proving convergence of the bounds to the value of the functional.

The moment-SOS hierarchy was already developed and used in [START_REF] Lasserre | LP relaxations for the moment approach in some performance evaluation problems[END_REF][START_REF] Lasserre | Pricing a class of exotic options via moments and SDP relaxations[END_REF] for obtaining bounds on SDEs coming from finance. However, only lower and upper bounds were obtained, and the question of convergence was left open.

A key step to construct the moment-SOS hierarchy is the reformulation of the original, typically nonlinear problem, as a linear problem on occupation measures. This linear reformulation is classical in Markov decision processes (MDP) [START_REF] Bhatt | Occupation measures for controlled Markov processes: characterization and optimality[END_REF][START_REF] Kurtz | Existence of Markov control and characterization of optimal Markov controls[END_REF]. In order to prove convergence of the bounds obtained with the moment-SOS hierarchy, one has to prove that there is no relaxation gap between the original nonlinear problem and the linear problem on measures. This was already achieved in [START_REF] Bhatt | Occupation measures for controlled Markov processes: characterization and optimality[END_REF][START_REF] Kurtz | Existence of Markov control and characterization of optimal Markov controls[END_REF] in the context of controlled MDP, but the proofs are lengthy and technical. Zero relaxation gap for optimal control of SDEs was proven in [START_REF] Buckdahn | Stochastic optimal control and linear programming approach[END_REF] with the help of viscosity solutions to Hamilton-Jacobi-Bellman partial differential equations (PDE).

In [START_REF] Kashima | On weak approximation of stochastic differential equations through hard bounds by mathematical programming[END_REF][START_REF] Kawai | Explicit hard bounding functions for boundary value problems for elliptic partial differential equations[END_REF], bounds on functionals of solutions of SDEs were obtained by a dual approach, seeking test functions satisfying inequalities. When the functions and the SDE coefficients are polynomial, the inequalities are replaced by SOS constraints and solved numerically with SDP. Our occupation measure formulation can be interpreted as a primal approach, from which the dual on test functions follows from elementary convex analysis arguments. More recently, a primal-dual moment-SOS hierarchy approach to optimal control of SDEs was followed in [START_REF] Jiao | Computable primal and dual bounds for stochastic optimal control[END_REF], as a stochastic counterpart of [START_REF] Henrion | Linear conic optimization for nonlinear optimal control[END_REF], and no relaxation gap was ensured by approximating the value function solving the dual HJB PDE.

In this paper, we focus on a specific class of SDE functional evaluation problems, namely the exit location of an uncontrolled SDE. The exit location is a random variable that can be characterized by its moments. As shown in [START_REF] Helmes | Computing moments of the exit time distribution for Markov processes by linear programming[END_REF], the exit time moments can be approximated numerically with occupation measures and linear programming (LP), with convergence guarantees based on the zero relaxation gap proof of [START_REF] Kurtz | Existence of Markov control and characterization of optimal Markov controls[END_REF].

Our contribution is as follows:

• we provide a new proof of the equivalence, or zero relaxation gap, between the infinitedimensional linear formulation on occupation measures and the original nonlinear SDE; the proof, much shorter and simpler in our opinion than the MDP proofs of [START_REF] Bhatt | Occupation measures for controlled Markov processes: characterization and optimality[END_REF][START_REF] Kurtz | Existence of Markov control and characterization of optimal Markov controls[END_REF] or the HJB proof of [START_REF] Buckdahn | Stochastic optimal control and linear programming approach[END_REF][START_REF] Jiao | Computable primal and dual bounds for stochastic optimal control[END_REF], relies on standard results from elliptic PDE analysis;

• we describe a neat primal-dual linear formulation with no duality gap, allowing readily the application of the moment-SOS hierarchy.

• we propose a novel use of Christoffel-Darboux polynomial kernels [START_REF] Lasserre | The Christoffel-Darboux kernel for data analysis[END_REF] to approximate the densities of occupation measures and characterizing the exit location measure.

The paper is organized as follows. The exit location problem is defined in Section 2. Its linear reformulation with occupation measures is described in Section 3. Our main result on zero relaxation gap is described and proved in Section 4. The dual linear formulation is described in Section 5 and generalized in Section 6. Application of the moment-SOS hierarchy and numerical examples are described in Section 7. Section 8 describes how Christoffel-Darboux polynomial kernels can be used to approximate the densities of the occupation measures.

Concluding remarks are gathered in Section 9 ..,n : R n → R n×n is positive definite. Assume b and B are continuous on R n and growing at most linearly outside of X , so that by standard arguments [START_REF] Evans | An introduction to stochastic differential equations[END_REF]Chapter 5] there is a unique solution to the SDE, the stochastic process X(t).

Let g : ∂X → R be a given continuous function. We want to evaluate the function

v * (x) = E[g(X(τ x ))] (1)
where τ x is the first time X(.) hits ∂X , see e.g. [7, Example 2, Section 6.2.1].

Linear reformulation

The generator of the stochastic process is the linear partial differential operator

-Lf := n i,j=1 a ij ∂ i ∂ j f + n i=1 b i ∂ i f
where ∂ i denotes the derivative with respect to the i-th variable. With this sign convention, and since the matrix A is positive definite, linear operator L is uniformly elliptic [6, Section 6.1.1].

Given a function f ∈ C 2 (X ), Itô's chain rule [START_REF] Evans | An introduction to stochastic differential equations[END_REF]Chapter 4] implies that

f (X(τ x )) = f (X(0)) - τx 0 Lf ds + τx 0 Df • B dW
where

Df • B dW = m k=1 n i=1 ∂ i f b ik dW k .
Taking the expected value yields Dynkin's formula [START_REF] Evans | An introduction to stochastic differential equations[END_REF]Section 6.1.3]:

E[f (X(τ x ))] = E[f (X(0))] -E τx 0
Lf ds that we rearrange as follows

E[f (X(τ x ))] + E τx 0 Lf ds = f (x). (2)
Given x ∈ X , define the expected occupation measure µ of the process X up to time τ x , such that µ(A ) := E for every set B in the Borel sigma algebra of ∂X . An equivalent analytic definition is

f, ν := E [f (X(τ x ))]
for every test function f . Then Dynkin's formula (2) becomes a linear partial differential equation on measures

f, ν + Lf, µ = f (x) (3)
which can be written in the sense of distributions as

ν + L µ = δ x ( 4
)
where L is the linear operator adjoint to L and δ x is the Dirac measure concentrated at x. This equation is called the Kolmogorov or Fokker-Planck equation.

Following [START_REF] Lasserre | LP relaxations for the moment approach in some performance evaluation problems[END_REF][START_REF] Lasserre | Pricing a class of exotic options via moments and SDP relaxations[END_REF], now define

v min (x) := min µ,ν g, ν s.t. ν + L µ = δ x (5)
and

v max (x) := max µ,ν g, ν s.t. ν + L µ = δ x (6)
which satisfy by construction

v min (x) ≤ v * (x) ≤ v max (x)
for each x ∈ X . Note that (5) and ( 6) are linear optimization problems over measures µ and ν supported on X and ∂X respectively.

No relaxation gap

Let us make the following regularity assumptions on the problem data, see [START_REF] Urbas | Lectures on second order linear partial differential equations[END_REF]Sections 3 and 4] for definitions.

Assumption 1

The boundary ∂X has class C 2,α and the coefficients a ij , b i have class C 0,α (X ) for some α ∈ (0, 1).

Theorem 1 Under Assumption 1, there is no relaxation gap between the nonlinear function evaluation problem (1) and the linear optimization problems (5) and (6), i.e.

v min (x) = v * (x) = v max (x)
for each x ∈ X .

The proof of Theorem 1 is based on the following result.

Lemma 1 Under Assumption 1, for each x ∈ X there exists a unique exit location measure ν = ν x on ∂X and a unique expected occupation measure µ = µ x on X solving the Kolmogorov equation (4).

Proof of Lemma 1: Let α ∈ (0, 1) be the regularity exponent of Assumption 1. Given

any function p ∈ C 2,α (X ), let f p ∈ C 2,α (X ) be a solution to the Dirichlet boundary value problem Lf = 0 in X f = p on ∂X . (7)
Existence, uniqueness and regularity of the solution follows from [START_REF] Urbas | Lectures on second order linear partial differential equations[END_REF]Theorem 4.5]. Plugging

f p into (3) yields p, ν = f p (x). ( 8
)
Since X is compact, the space C(X ) is separable and by choosing countably many functions p ∈ C 2,α (X ) -e.g. polynomials -we can generate countably many linear relations (8) that uniquely specify the measure ν that we denote ν x .

Uniqueness of the measure µ also holds. Indeed, if h ranges over a countable dense set of functions in C 2,α (X ) and the f h are the unique solutions in C 2,α (X ) of the Poisson problem

Lf h = h in X f h = 0 on ∂X then the Kolmogorov equation (4) yields the equality h, µ = f h (x) -f h , ν = f h (x)
and the moments of µ with respect to the functions in h are uniquely specified. By density, this implies the uniqueness of µ that we denote µ x .

Proof of Theorem 1: Let x ∈ X and let ν x denote the exit location measure on ∂X solving the Kolmogorov equation ( 4). Notice that the objective function in problems ( 5) and ( 6) depends only on ν x . It follows that g, ν x = v min (x) = v max (x).

Remark 1 Existence and uniqueness of a regular solution to the Dirichlet problem (7) also follows from the uniformly ellipticity of L and weaker conditions on the data, as stated in [

Duality

In this section we use elementary notions from convex duality to derive the dual problem to the minimization resp. maximization problem on measures. We show that admissible solutions to the dual problem are subsolutions resp. supersolutions to the boundary value PDE solved by the value function. In particular we show that the concept of supersolution (resp. subsolution) arises naturally from elementary duality theory.

Lemma 2

The linear problem dual to (5) reads as follows

max v v(x) s.t. Lv ≤ 0 in X , v ≤ g on ∂X (9)
where the maximization is with respect to functions v ∈ C 2 (X ). There is no duality gap, i.e. the value of (9) is equal to the value of (5).

Proof: Let us denote by v ∈ C(X ) the Lagrange multiplier corresponding to the equality constraint in primal problem [START_REF] Dahl | Semidefinite optimization using MOSEK[END_REF], and build the Lagrangian (µ, ν, v)

:= g, ν + v, δ x - ν -L µ = g -v, ν + v, -L µ + v, δ x = g -v, ν + -Lv, µ + v(x).
The Lagrange dual function is then min µ,ν (µ, ν, v) = v(x) provided v ≤ g on ∂X , the support of ν, and Lv ≤ 0 on X , the support of µ. The dual problem (9) then consists of maximizing the dual function subject to these inequality constraints. To prove that there is no duality gap, we use [1, Theorem IV.7.2] and the fact that the image through the linear map ( g, ν , ν + L µ) of the cone of measures µ resp. ν supported on X resp. ∂X is nonempty and bounded in the metric inducing the weak-star topology on measures.

As recalled in [7, Example 2, Section 6.B], the value function v * is the solution of the boundary value problem

Lv = 0 in X v = g on ∂X . ( 10
)
Lemma 3 Any admissible function v for linear problem (9) is a subsolution of boundary value problem [START_REF] Henrion | GloptiPoly 3: moments, optimization and semidefinite programming[END_REF], in the sense that v * ≥ v on X .

Proof: Let v be admissible for [START_REF] Henrion | The moment-SOS hierarchy -Lectures in probability, statistics, computational geometry, control and nonlinear PDEs[END_REF]. Function u := v -v * is such that Lu ≤ 0 in X and u ≤ 0 on ∂X . By the weak maximum principle [6, Theorem 1 page 327], if Lu ≤ 0 in X then max X u = max ∂X u. Since u ≤ 0 on ∂X , this implies that u ≤ 0 and hence v * ≥ v on X .

Linear problem (9) selects the subsolution that touches the value function from below at x.

Similarly, the linear problem dual to (6) reads as follows

min v v(x) s.t. Lv ≥ 0 in X , v ≥ g on ∂X (11)
where the maximization is with respect to functions v ∈ C 2 (X ). There is no duality gap, i.e. the value of ( 11) is equal to the value of ( 6). Any admissible function v for linear problem ( 11) is a super-solution of boundary value problem [START_REF] Henrion | GloptiPoly 3: moments, optimization and semidefinite programming[END_REF], in the sense that v * ≤ v on X and ∂X . Linear problem [START_REF] Henrion | Linear conic optimization for nonlinear optimal control[END_REF] selects the supersolution that touches the value function from above at x.

Random initial condition

All the above developments generalize readily to the case that the initial condition X(0) in the SDE is a random variable whose law is a given probability measure ξ on X . The previous results can then be retrieved with the particular choice ξ = δ x for a given x ∈ X .

The quantity to be evaluated becomes

v * ξ := X E[g(X(τ x ))]dξ(x).
The Kolmogorov equation (4) becomes

ν + L µ = ξ
and exactly the same arguments of the proof of Lemma 1 can be used to prove that it has a unique solution ν ξ depending on ξ. As in Theorem 1, it follows that the linear problems min

µ,ν g, ν s.t. ν + L µ = ξ (12)
and max

µ,ν g, ν s.t. ν + L µ = ξ (13)
have the same value v * ξ , i.e. there is no relaxation gap. The respective dual problems

max v v, ξ s.t. Lv ≤ 0 in X , v ≤ g on ∂X (14)
and min

v v, ξ s.t. Lv ≥ 0 in X , v ≥ g on ∂X (15)
have the same value v * ξ , i.e. there is no duality gap.

Moment-SOS hierarchy and examples

If the SDE coefficients b and B and the functional g are semialgebraic 1 in a semialgebraic set X , we can apply the moment-SOS hierarchy on ( 5) and ( 6) with convergence guarantees. In the primal, we obtain approximate moments (also called pseudo-moments or pseudo-expectations) of the occupation measures. They are not necessarily moments of the occupation measures as we are solving relaxations. In the SOS dual, we obtain polynomial sub-resp. super-solutions of increasing degrees of the boundary value problem [START_REF] Henrion | GloptiPoly 3: moments, optimization and semidefinite programming[END_REF]. Each primal-dual problem is a semidefinite optimization problem.

Semidefinite relaxations

Let us briefly describe the construction of the moment-SOS hierarchy.

A bounded closed semialgebraic set Z of R n can be written as the union of finitely many basic semialgebraic sets Z i := {z ∈ R n : p i,j (z) ≥ 0, j = 1, . . . , m i }, i = 1, . . . , m, described by finitely many polynomials p i,j . Note that polynomial equations can be modeled by two inequalities of reverse signs. Since Z is bounded, without loss of generality, for each i = 1, . . . , m, one of the polynomials defining each Z i can be chosen equal to R 2 -n k=1 z 2 k for R sufficiently large, and for notational convenience we let p i,0 (z) := 1

There are several algebraic characterizations of the set of positive polynomials on Z i . To describe one such characterization, let r be a positive integer and for each i = 1, . . . , m define the truncated quadratic module of degree r of Z i , denoted Q(Z i ) r , to be the set of polynomials which can be written as m i j=0 p i,j s i,j where the s i,j are sums of squares (SOS) of polynomials such that 2degs i,j + degp i,j ≤ r. Every polynomial in the Minkowski sum m i=1 Q(Z i ) r is obviously nonnegative on Z . Putinar's Positivstellensatz [START_REF] Putinar | Positive polynomials on compact semi-algebraic sets[END_REF] is the much deeper statement that every polynomial strictly positive on Z lies in m i=1 Q(Z i ) r for some r.

We will now describe a hierarchy of semidefinite optimization problems which depend on the degree r and provide us with upper and lower bounds on the value v * ξ . We will show that as r → ∞ these bounds converge to the value v * ξ . Let us assume that X is a bounded basic semi-algebraic set. Its boundary ∂X is then a union of finitely many bounded basic semi-algebraic sets X ∂ i , i = 1, . . . , m. The primal (moment) problems are given by

p min r := min m i=1 ν i (g) s.t. µ (v) ≥ 0, ∀v ∈ Q(X ) r ν i (v) ≥ 0, ∀v ∈ Q(X ∂ i ) r µ (Lv) + m i=1 ν i (v) = v, ζ , ∀v ∈ P r (16)
1 A function is semialgebraic if its graph is a semialgebraic set. A semialgebraic set is defined by a finite sequence of polynomial equations and inequalities. and

p max r := max m i=1 ν i (g) s.t. µ (v) ≥ 0, ∀v ∈ Q(X ) r ν i (v) ≥ 0, ∀v ∈ Q(X ∂ i ) r µ (Lv) + m i=1 ν i (v) = v, ξ , ∀v ∈ P r (17)
where the unknowns are linear operators µ , ν 1 , . . . , νm from P r to R, for P r denoting the vector space of n-variate real polynomials of degree up to r. The dual (SOS) problems are given by

d min r := max v, ξ s.t. -Lv ∈ Q(X ) r g -v ∈ m i=1 Q(X ∂ i ) r (18)
and

d max r := min v, ξ s.t. Lv ∈ Q(X ) r v -g ∈ m i=1 Q(X ∂ i ) r (19)
where the unknowns are polynomials v ∈ P r .

Theorem 2 Problems (16), ( 17), ( 18) and ( 19) are semidefinite programming problems. For each r > 0, it holds

d min r ≤ p min r ≤ v * ξ ≤ p max r ≤ d max r . Moreover lim r→∞ (d max r -d min r ) = 0.
Proof: To show that problems ( 18) and ( 19) are semidefinite programming problems, just observe that a polynomial p(z) of degree at most 2r is a sum of squares of polynomials if and only if there is a positive semidefinite symmetric matrix S such that p(z) = b(z) Sb(z) where b(z) is a vector of polynomials spanning P r . It follows that the truncated quadratic modules in [START_REF] Lasserre | The Christoffel-Darboux kernel for data analysis[END_REF] and ( 19) are projections of the semidefinite cone, and optimizing linear functions over them is an instance of semidefinite programming.

Problems ( 16) and ( 18) are in duality. This follows easily from computing the Lagrangian as in Lemma 2. The constraint in problem ( 16) that a linear operator is non-negative for all polynomials in a truncated module can be expressed as a linear matrix inequality, i.e. it forms a spectrahedron, a linear slice of the semidefinite cone. From weak duality we conclude that p min r ≥ d min r . Similarly, problems [START_REF] Lasserre | Moments, positive polynomials and their applications[END_REF] and [START_REF] Lasserre | LP relaxations for the moment approach in some performance evaluation problems[END_REF] are in duality and d max r ≥ p max r .

To prove that p min r ≤ v * ξ observe that if (ν, µ) are measures satisfying ν + L µ = ξ then defining µ (v) := v, µ and ν i (v) := v, ν i where ν = m i=1 ν i for each ν i supported on X ∂ i , we obtain admissible linear operators for the primal problem [START_REF] Kurtz | Existence of Markov control and characterization of optimal Markov controls[END_REF]. Coefficients of the linear operators are moments of the respective measures. It may however happen that linear operators admissible for problem [START_REF] Kurtz | Existence of Markov control and characterization of optimal Markov controls[END_REF] do not correspond to measures. Since we minimize a linear function on a possibly larger set, we obtain a lower bound. Similarly, we can prove that v * ξ ≤ p max r .

The most substantial claim is the convergence result. Given > 0 we will show that there exists an integer r such that 0 ≤ v * ξ -d min r ≤ for r ≥ r . Similar arguments imply an analogous convergence result for d max r .

Recall from Section 5 that the unique function f ∈ C 2 (X ) which satisfies Lf = 0 in X and f = g on ∂X is the value function f = v * of the problem and it satisfies v * ξ = v * , ξ . The proof proceeds by showing that this value function can be approximated by a sequence of elements v n which are feasible for problem [START_REF] Lasserre | The Christoffel-Darboux kernel for data analysis[END_REF]. Since operator L is uniformly elliptic, there is a polynomial w such that -Lw > 0 in X , take e.g. a quadratic polynomial with sufficient large leading coefficients. By substracting a sufficiently large constant from w, we can also ensure that w < g on ∂X . Now let > 0 be given and let (w n ) n∈N be a sequence of polynomials which approximate v * and its derivatives of order up to 2 uniformly on X . Let η > 0 be a real number with η < 2 w-v * and let v n := (1 -η)w n + ηw. Due to the uniformity of the convergence of w n and the definition of v n the following statements hold for all n sufficiently large: w n -v * < η w -v * , -Lv n > 0 in X and v n < g on ∂X . By Putinar's Positivstellensatz [START_REF] Putinar | Positive polynomials on compact semi-algebraic sets[END_REF], for any such n there exists a degree r n such that -Lv n ∈ Q(X ) rn and g -v n ∈ Q(∂X ) rn and therefore v n is feasible for problem [START_REF] Lasserre | The Christoffel-Darboux kernel for data analysis[END_REF] for r = r n . Furthermore, for any such n we have

| v n -v * , ξ | ≤ (1 -η) w n -v * + η w -v * ≤ 2η w -v * < so we conclude that v * ξ -d min rn ≤
proving the claim since > 0 was arbitrary.

Solving the relaxations

Moment relaxations of these linear problems can be modeled as generalized problems of moments with the GloptiPoly interface [START_REF] Henrion | GloptiPoly 3: moments, optimization and semidefinite programming[END_REF] for Matlab. The relaxations are solved with the SDP solver in MOSEK [START_REF] Dahl | Semidefinite optimization using MOSEK[END_REF].

Scalar example

Let us illustrate the application of the moment-SOS hierarchy with an elementary exit time problem (1) considered in [START_REF] Helmes | Computing moments of the exit time distribution for Markov processes by linear programming[END_REF]Example 5.1]. The SDE is dX(t) = (1 + 2X(t))dt + √ 2X(t)dW(t) on the domain X := (0, 1) with initial condition x = 1/2. This process always exits at the point {1}, so ν = δ 1 solves (3) and the value of linear problems ( 5) and ( 6) is equal to g [START_REF] Barvinok | A course in convexity[END_REF]. For the choice g(z) = z 2 we report in Table 1 the values of the lower and upper bounds obtained by solving the moment relaxations to problems ( 5) and ( 6), for increasing relaxation degrees. The degree 10 relaxation was solved in 0.15 seconds on our laptop. The corresponding GloptiPoly script in Matlab can be downloaded at homepages.laas.fr/henrion/software/exittime.m

Multivariate example

Consider problem (1) with g(z) := n k=1 z 2 k for the n-dimensional Brownian motion X(t) = W(t), i.e. b = 0 and B = I n , in the convex semi-algebraic domain X := {z ∈ R n : n k=1 z 4 k ≤ 1} with initial condition x = 0. Table 2: Computational burden for relaxation degree 8 and increasing dimensions.

On Figure 1 we plot for n = 2 and n = 3 the lower resp. upper bounds obtained by minimizing resp. maximizing the functional, for increasing relaxation degrees. We observe a fast convergence of the bounds.

In Table 2 we report the number of moments as well as the computational time required to solve the moment relaxation of degree 8, for increasing values of the dimension n. For this relaxation degree the gap between the lower and upper bounds on the functional is less than 2%.

Approximating occupation and exit location densities

As described in the previous section, the moment-SOS hierarchy allows us to obtain estimates of the moments of our occupation and exit location measures, together with error bars on those estimates. For many applications however, it could be more useful to know the densities of the occupation and exit location measures with respect to some natural measures on X and ∂X respectively. In this section we outline a method, based on Christoffel-Darboux kernels [START_REF] Lasserre | The Christoffel-Darboux kernel for data analysis[END_REF], which allows us to construct sequences of functions converging to such densities.

For simplicity we will state our results when X is the unit ball, but the same method can be applied more generally as discussed below.

ρ we have lim d→∞ N (d)Λ µ,V d (u) = ρ(u) ω(u)
where the equality holds uniformly on compact subsets of the interior of B n .

Computing occupation and exit location densities

To estimate the occupation density on the unit ball and the exit location density on its boundary we propose the following procedure. As before, let V d denote the subspace of polynomials of degree up to d in R n . Let W d be the vector space of functions obtained by restricting the elements of V d to the unit sphere S n-1 ⊆ R n .

Procedure 1 Given a diffusion with generator L and initial location X(0) = x ∈ B n ⊆ R n , a total degree d > 0, an approximation degree r > 0 and an integer offset s > 0, carry out the following steps:

1. Fix a basis g 1 , . . . , g ng for W d . For each pair of elements g i , g j , let g := g i g j and compute our estimate for g, ν = gν as the average Âij := (p min r + p max r )/2 of the values of the optimization problems ( 16) and (17). (g 1 (z), . . . , g ng (z)) T Â-1 (g 1 (z), . . . , g N (z))

Fix a basis

and our estimate for the occupation density on the unit ball by

β(z) = dim V d-s ω(z) (h 1 (z), . . . , h n h (z)) T B-1 (h 1 (z), . . . , h n h (z))
with ω defined in Example 2. Proof: If µ, ν denote the occupation measure and the exit location measure respectively and X(0) = x then the Kolmogorov equation implies that for any twice-differentiable function f we have

f, ν + Lf, µ = f (x) If LV d ⊇ V d-

Some computational examples in the unit disc

In this section we illustrate the usefulness of Procedure 1 for estimating occupation densities in the unit ball and exit location densities on the unit sphere for some diffusions. The Julia code used for computing the examples in this Section is available via github at github.com/mauricio-velasco/Exit location density estimation.git.

Example 3 (Brownian motion in the unit disc). Let W (t) be the two-dimensional brownian motion, let X(t) be a solution of the SDE

X(0) = (0, 1/2) dX(t) = dW (t) for t ≥ 0.
The generator of the process is Lf = -1 2 ∆f . We study the behavior of X(t) on or before its exit time from the unit disc in Figure 2. Example 4 (Brownian motion with drift in the unit disc). Let W (t) be the two-dimensional brownian motion, let X(t) be a solution of the SDE

     (X 1 (0), X 2 (0)) = (0, 1/2) dX 1 (t) dX 2 (t) = 0 2 dt + 1 0 0 1 dW (t).
The generator of the process is Lf = -2 ∂f ∂x 2 -1 2 ∆f . We study the behavior of X(t) on or before its exit time from the unit disc in Figure 3. The generator of the process is Lf = -∂f ∂x 2 -1 2

∂ 2 f ∂x 2 1 + 4x 2 ∂ 2 f ∂x 2 2
. We study the behavior of X(t) on or before its exit time from the unit disc in Figure 3.

Conclusion

Using elementary analytic arguments, we proved that there is no relaxation gap between the original problem and the linear problem on occupation measures in the special case of evaluating functionals of the exit time of stochastic processes on bounded domains. If the domain is basic semialgebraic and the SDE coefficients and the functional are polynomial or semi-algebraic, we can then readily apply the moment-SOS hierarchy with convergence guarantees. Tight bounds on the functionals can be obtained with off-the-shelf SDP solvers at a moderate cost.

Of particular practical interest are approximations to the moments of the exit time distribution, as studied in [START_REF] Helmes | Computing moments of the exit time distribution for Markov processes by linear programming[END_REF]. As we have shown, any sufficiently accurate method of moment We would also like to extend these techniques to optimal stopping times [START_REF] Cho | Linear programming formulation for optimal stopping problems[END_REF] and stochastic optimal control problems with expectation constraints [START_REF] Buckdahn | Stochastic optimal control and linear programming approach[END_REF], and to develop theoretical guarantees for success of the proposed Christoffel-Darboux kernel approach on general diffusions.

2 2 m

 2 Exit location problem Let X be a given open connected bounded set of R n with smooth boundary ∂X and compact closure X := X ∪ ∂X . Let W(t) = (W k ) k=1,...,m denote the m-dimensional Brownian motion and let X(t) denote the solution of the stochastic differential equation (SDE) dX = b(X)dt + B(X)dW, X(0) = x starting at x ∈ X where drift functions b = (b i ) i=1,...,n : R n → R n and diffusion functions B = (b ij ) i=1,...,n, j=1,...,m : R n → R n×m are given. We assume that B has full rank, so that the matrix A = (a ij := 1 k=1 b ik b jk ) i,j=1,.

τx 0 I

 0 A (X(s))ds for every set A in the Borel sigma algebra of X , where I A denotes the indicator function equal to one in A and zero outside. An equivalent analytic definition is f, µ :=E τx 0 f (X(s))ds for any test function f , where f, µ := f µ denotes the duality pairing of a continuous function f and a measure µ. Define the exit location measure ν as the law of X(τ x ) i.e. ν(∂B) := E[I ∂B (X(τ x ))]

Figure 1 :

 1 Figure 1: Lower and upper bounds on the functional for increasing relaxation degrees and dimension n = 2 (left) and n = 3 (right).

  h 1 , . . . , h n h for V d-s . For each pair of elements h i , h j , find a polynomial g which solves the equation Lg = h i h j and compute our estimate for g, µ = gµ as the shifted average Bij = g(x) + (p min r + p max r )/2 of the values of the optimization problems[START_REF] Kurtz | Existence of Markov control and characterization of optimal Markov controls[END_REF] and[START_REF] Lasserre | Moments, positive polynomials and their applications[END_REF].

3 .

 3 Define our estimate for the exit location density on the unit sphere by σ(z) := dim W d

Theorem 4 1 . 2 . 4 .

 4124 If the following conditions hold 1. The exit location measure has a positive and continuous density σ with respect to the Lebesgue measure on the unit sphere S n-The occupation measure has a positive and continuous density β with respect to the Lebesgue measure on the unit ball B n . 3. There exists a sequence s(d) of positive integers with d -s(d) → ∞ such that, for all sufficiently large d, the inclusion LV d ⊇ V d-s(d) holds. The chosen approximation degrees r(d) grow sufficiently quickly so as to guarantee lim d→∞ sup g∈V d |p max r(d) (g) -p min r(d) (g)| → 0 then the estimates σ resp. β defined in Procedure 1 converge uniformly to the true exit location density σ resp. occupation density β as d → ∞.

Figure 2 ,

 2 parts (a) and (b) show five and one hundred approximate sample paths of the process.

Figure 2 ,

 2 part (c) shows the unnormalized

Figure 2 :

 2 Figure 2: Example 3: brownian motion

Figure 2 ,

 2 part (d) shows the Christoffel-Darboux kernel estimates in degree r = 2 for the occupation density of X(t) before reaching the boundary in the unit disc.

Figure 3 ,

 3 parts (a) and (b) show five and one hundred approximate sample paths of the process.

Figure 3 ,

 3 part (c) shows the unnormalized Christoffel-Darboux kernel estimates in degrees r = 4, 6 for the exit location density in the unit circle represented as the axis -π/2 ≤ θ ≤ 3π/2.

Figure 3 ,

 3 part (d) shows the Christoffel-Darboux kernel estimates in degree r = 2 for the occupation density of X(t) before reaching the boundary of the unit disc.Example 5 (Square root Bessel process in the unit disc). Let W (t) be the two-dimensional brownian motion, let X(t) be a solution of the SDE     (X 1 (0), X 2 (0)) = (0, 1/2) dX 1 (t) dX 2 (t) 2 (t)dW (t).

Figure 3 :

 3 Figure 3: Example 4: brownian motion with upward drift

Figure 4 ,

 4 parts (a) and (b) show five and one hundred approximate sample paths of the process. The example is particularly interesting because, by the presence of the square root the process is forced to remain in the upper half of the disc producing a discontinuity in the exit location density. Although outside the domain of our theoretical guarantees, Figure4, part (c) shows the unnormalized Christoffel-Darboux kernel estimates in degrees r = 4, 6 for the exit location density which seem to have the correct qualitative behavior. Extending our results to cover such densities will be topic of further work.

Figure 4 ,

 4 part (d) shows the Christoffel-Darboux kernel estimates in degree r = 2 for the occupation density of X(t) before reaching the boundary of the unit disc.

Figure 4 :

 4 Figure 4: Example 5: square root Bessel process

  21, Theorem 9.2.14]. Uniqueness of the solution follows from [21, Theorem 9.2.13] under Hunt's condition and also under the assumption that all points of ∂X are regular, see [21, Definition 9.2.8]. Hunt's condition holds for all Itô diffusions whose diffusion coefficient matrix has a bounded inverse and whose drift coefficient satisfies the so-called Novikov condition. For our purpose however, we prefer to stick with stronger assumptions and the self-contained statement and proof of[START_REF] Urbas | Lectures on second order linear partial differential equations[END_REF] Theorem 4.5].

Table 1 :

 1 Scalar example -bounds for increasing relaxation degrees.

	relaxation degree	2	4	6	8	10
	lower bound	0.65000 0.92157 0.98118 0.99503 0.99827
	upper bound	1.00000 1.00000 1.00000 1.00000 1.00000

  s(d) then this equation allows us to compute the polynomial moments of the occupation measure with respect to the functions inV d-s(d) . More concretely, if h ∈ V d-s(d)then there exists p ∈ V d , which can be easily computed with linear algebra, with Lp = h. As a result h, µ = p(x) -p, ν so estimates of the exit location moments p, ν can be used to obtain certified estimates for the occupation moments as in Procedure 1. Such computations, which can become as good as we want by increasing the approximation degree r(d), allow us to estimate the corresponding Christoffel functions as accurately as we wish. The result now follows from Theorem 3 and [15, Theorem 1.3]. Procedure 1 can be carried out to estimate occupation densities for sets whose Equilibrium measure is well understood, playing the role of the density ω from Example 2. See [18, Section 4.2] for several such instances. Similarly, the assumption of Theorem 3 holds for several geometries beside spheres, see [18, p. 71] for several interesting instances.

	Remark 2

Acknowledgement

We are grateful to Milan Korda, Jean Bernard Lasserre and Matteo Tacchi for their feedback on this work.

* M. Velasco was partially supported by Proyecto INV-2018-50-1392 from Facultad de Ciencias, Universidad de and M. Velasco were partially supported by ColCiencias Colombia-France cooperation Ecos Nord grant "Problemas de momentos en control y optimización" (C19M02).

Background on Christoffel-Darboux kernels

Let U be a compact topological space and let µ be a probability measure supported on U . The set C(U ) of continuous, real-vaued functions on U becomes a Hilbert space via the inner product f, g U := E µ [f g] = U f (u)g(u)dµ(u). If V ⊆ C(U ) is a finite-dimensional vector space of functions then this space inherits a Hilbert space structure and in particular, for each u ∈ U there exists a unique element φ u ∈ V which reproduces the evaluation at u in the sense that the equality f, φ u U = f (u) holds for every f ∈ V . Such operators determine the reproducing kernel function K : V × V → R, defined via the formula K(u, v) := φ u , φ v U . This function satisfies, and is determined by, the following key reproducing property:

), f U valid for all f ∈ V and all u ∈ U. The kernel is completely determined by the measure µ and the vector space V and we write K µ,V instead of K when trying to make this dependence explicit. The following two formulas are very useful for computing and understanding the basic properties of reproducing kernels: 1) If g 1 , . . . , g m is any µ-orthonormal basis for the subspace V then K µ,V (u, v) = m j=1 g j (u)g j (v) and furthermore 2) If h 1 , . . . , h m is any basis for the subspace V and M is the corresponding moment matrix given by

which follows readily from the Cauchy-Schwartz inequality and the reproducing property.

In the special case when U ⊆ R n and V d is the subspace of polynomials of degree up to d in R n restricted to U , the asymptotic behavior of the Christoffel function can often be used for density estimation [START_REF] Lasserre | The Christoffel-Darboux kernel for data analysis[END_REF]. A concrete set-up where these estimations can be carried out is described in [START_REF] Lasserre | The Christoffel-Darboux kernel for data analysis[END_REF]Theorem 5.3.4] as an extension of [START_REF] Kroó | Christoffel functions and universality in the bulk for multivariate orthogonal polynomials[END_REF].

Example 1 A basic example for which the hypotheses of Theorem 3 hold is the sphere S n-1 ⊆ R n with its normalized area measure λ. In this case the invariance of the measure with respect to rotations implies that the function K(u, u) is constant. The expression for K(u, u) in terms of λ-orthonormal polynomials shows that the constant is the dimension of the space of polynomials restricted to the unit sphere. Since the sphere is defined by a single quadric, this dimension is given by the formula N (d

so N (d)Λ λ,V d (u) = 1 for all u ∈ U and we can use w(u) = 1 in the formula of Theorem 3 to conclude that for any measure µ which is absolutely continuous with respect to λ with a positive and continuous density ρ, this density can be estimated uniformly via the limit lim d→∞ N (d)Λ µ,V d (u) = ρ(u).