Dr Isabelle Romieu
email: iromieu@correo.insp.mx.

Alban Fabre

Agnès Fournier

Francine Kauffmann

Raphaëlle Varraso

Sylvie Mesrine

Benedicte Leynaert

Françoise Clavel-Chapelon

Agnes Fournier

Francoise Clavel-Chapelon

Postmenopausal hormone therapy and asthma onset in the E3N cohort

Keywords: Asthma, Epidemiology, Menopausal hormone therapy (MHT, Hormone replacement therapy (HRT)

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The major increase in asthma prevalence observed in most developed countries in recent years [1] suggests that it may in part be due to environmental factors.

Differences in asthma incidence during the life cycle suggest that reproductive hormones influence the development of asthma and asthma severity. [START_REF] Barr | Prospective study of postmenopausal hormone use and newly diagnosed asthma and chronic obstructive pulmonary disease[END_REF] Asthma is more frequent in girls after menarche than before. [START_REF] Wjst | Is asthma an endocrine disease?[END_REF] Adult asthma and hospital admissions for asthma are more prevalent in women than in men and asthma severity has been shown to vary during the menstrual cycle and pregnancy. [START_REF] Postma | Gender differences in asthma development and progression[END_REF] In addition, the incidence of asthma tends to decrease after menopause. [START_REF] Troisi | Menopause, postmenopausal estrogen preparations, and the risk of adult-onset asthma: a prospective cohort study[END_REF] The increase of asthma onset in postmenopausal women with increasing weight points to an endocrine mechanism such as increased endogenous estrogen synthesis. [START_REF] Romieu | Body mass index, change in body silhouette, and risk of asthma in the E3N cohort study[END_REF] Menopausal hormone therapy (MHT) might therefore play a role in asthma onset.

Different studies have investigated the role of MHT in asthma. Cross-sectional studies have reported an association between prevalent asthma and asthma-like symptoms and MHT use in perimenopausal [START_REF] Real | Hormone replacement therapy, body mass index and asthma in perimenopausal women: a cross sectional survey[END_REF] and postmenopausal women, [START_REF] Jarvis | The association of asthma, atopy and lung function with hormone replacement therapy and surgical cessation of menstruation in a population-based sample of English women[END_REF][START_REF] Lange | Exogenous female sex steroid hormones and risk of asthma and asthma-like symptoms: a cross sectional study of the general population[END_REF] with a larger effect in lean women. However, these studies were cross-sectional and therefore could not evaluate the effect of MHT on asthma onset. None of these studies evaluated the risk related to different types of MHT. To date only one cohort study, the Nurses' Health Study, has suggested that MHT use is associated with an increased risk of developing asthma. It observed a similar significant (two fold) increase in risk with the use of estrogen alone or estrogen plus progestin. [START_REF] Barr | Prospective study of postmenopausal hormone use and newly diagnosed asthma and chronic obstructive pulmonary disease[END_REF] The mechanisms underlying the link between hormonal factors and asthma risk are still not clearly understood. [START_REF] Carey | Spontaneous airway hyperresponsiveness in estrogen receptor-alpha-deficient mice[END_REF] Knowing whether MHT affects this risk and if so, whether different preparations have a similar effect, would provide a useful insight into the mechanisms by which the hormonal milieu acts on the airways.

We used data from the E3N study, a large cohort of French women followed for over 10 years, to evaluate the association between asthma onset and MHT use with special focus on the types of MHT and the duration of use. Participants, who gave written informed consent, completed biennial selfadministered questionnaires addressing medical history, menopausal status, and a variety of lifestyle characteristics.

Methods

Study population: the E3N cohort

Definition of incident cases of asthma

Women were considered incident cases of asthma if they stated at baseline (menopause) that they had never had an asthma attack and met the ATS criteria for asthma definition at follow-up ("Have you ever had asthma attacks?" and if yes "Was this diagnosis confirmed by a doctor?" ), [START_REF] Ferris | Epidemiology standardization Project (American Thoracic Society)[END_REF] with a coherent reported age of asthma-onset (time of first attack between baseline and 2002).

Identification of MHT use

Information on lifetime use of hormonal treatments was first recorded in the 1992 questionnaire. For each episode of treatment (defined as the non-stop use of the same hormonal brand or combination), brand name, age at initiation of use and duration were recorded. The information was updated in each of the subsequent questionnaires. The complete history of MHT use was established using data from all the questionnaires. MHT use was categorized by i) type of estrogen: weak estrogens (orally or vaginally administered promestriene or estriol) or estradiol compounds (unopposed MHT consisted almost exclusively of estradiol compounds; only 1.3% of women ever used conjugated equine estrogens (CEEs) corresponding to 8% of the total person years of follow up and ii) type of oral progestagen used in association with the estrogen: none, micronized progesterone, pregnane or norpregnane derivatives, and testosterone derivatives.

Body mass index

Body mass index (BMI) was calculated from the height and weight selfreported in each questionnaire. BMI was used in the different analyses as a continuous or categorical time-dependent variable. Information on anthropometric measurements provided by our questionnaire have been shown to be highly reliable. [START_REF] Tehard | Anthropometric measurements and body silhouette of women: validity and perception[END_REF] Other information Information on tobacco smoking was reported in each questionnaire and respondents were categorized as never, past, regular or occasional smokers. The last two categories were combined into a "current smokers" category. Atopy was defined as present if the woman reported diagnosis of allergic rhinitis, eczema or any other allergic disease on the questionnaire prior to asthma 2002.

Population for analysis and follow-up

Our analysis was based on 57,664 postmenopausal women, corresponding to 495,448 person-years of follow-up. These women were free of asthma at inclusion and all subsequent self-reports of asthma onset were validated according to the ATS criteria. Subjects contributed person-time until the date of the first asthma attack or July 2002 (date of dispatch of the seventh questionnaire), whichever occurred first.

Figure 1 presents a diagram of the initial population, the reasons for exclusion and the population included in the analysis.

Statistical analysis

Hazard ratios (HRs) for asthma were estimated using Cox proportional hazards models, with age as the time scale. We adjusted our model for tobacco smoking and BMI included in the model as time-dependent[or time-varying] covariables , and for other risk factors. MHT use was included in the model as a time-dependent variable.

A woman contributes to person year of ever use from the date she starts her treatment to the end of the follow up, contributes to person year of recent use from the date she starts her treatment to 1.5 years after she stops and contributes to person year of past user thereafter. This cutoff (1.5 years) was chosen because 65% of the cases occurring after cessation of MHT use did so within a year and a half of cessation. The reference group in each model consisted of women who indicated that they had not used yet any MHT. Further analyses were conducted after stratification of the data by hysterectomy, type of menopause (surgical non surgical), smoking status as well as the atopic status of the women. All tests of statistical significance were two sided, and significance was set at the 0.05 level. All analyses were performed using SAS software, version 9.1 (SAS Institute Inc., Cary, North Carolina) (for further details of methods used, see the online supplement).

Results

Among the 57,664 women who reported being postmenopausal and had information on MHT use, there were 569 incident cases of asthma between 1992 and 2002, corresponding to an incidence of 1.15/1000 per year.

Never user of MHT accounted for 35.7% of the person-years, past users for 4.5%, recent users for 55.8%, and 4% did not provide information on duration of treatment.

The mean age of ever users and never users at the start of follow-up was 52.25 years (SD=4.00) and 54.58 years (SD=4.74) (p<0.0001) respectively. Women excluded from the analysis were less likely to have used MHT or oral contraceptives, and were slightly older and had a larger BMI (chi2 test p-value <.001) than those included.

Table 1 presents the characteristics of ever users and never users of MHT included in the analysis. After adjusting for age, ever users were more likely to have a lower BMI (p<0.001) and to have ever used oral contraceptive (p<0.0001) than never users.

Surgical menopause was reported by 10% of the women. The use of estrogen alone was reported by 9.4% of the women with natural menopause and by 28.1% of the women with surgical menopause. Hysterectomy was reported by 19% of the women.

Among women who had an hysterectomy 58% reported a natural menopause and 27% a surgical menopause (n=2986), 36% of these reported using estrogen alone (see online supplement).

The HR of asthma onset among women who had ever used MHT was 1.21 (95% CI 1.00-1.46) compared to never users after adjusting for age, smoking, BMI, oral contraceptive use, parity and total caloric intake (Table 2).

The HR Among recent MHT users the HR was 1.20 (95% CI 0.98-1.46), and was 1.16 (95% CI 0.86-1.57) among past users. Women who reported using MHT for less than 2 years had a HR of 1.25 (95% CI 1.02-1.53) and those who reported using MHT for two years or more a HR of 1.09 (95% CI 0.85-1.38), compared to never users (Table 2).

Estrogen alone was reported as the last treatment used by 11.2% of the women.

These women were at higher risk of asthma onset (HR=1.54, 95% CI 1.13-2.09) compared to never users. The risk was not significantly increased among women who reported estrogen/progesterone (18.3%), a combination of estrogen and another progestagen (54.7%) or another MHT (15.8%) as the last treatment used. In subsequent analyses, users of any type of estrogen-progestagen were therefore combined into an "estrogen/progestagen users" category. The risk of asthma onset associated with the use of estrogen alone was significantly different from that associated with the use of estrogen/progestagen (test for homogeneity p=0.04). The route of estrogen administration (oral or transdermal) did not significantly affect the risk (Table 2).

For recent users of estrogen alone the HR of asthma onset was 1.67 (95% CI 1.20-2.27), while for past users it was 1.04 (95% CI 0.51-2.12). Among women who reported using estrogen alone for less than 2 years the HR was 1.60 (95% CI 1.12-2.27) while among those who reported using estrogen alone for 2 years or more the HR was 1.39 (95% CI 0.84-2.31). These HRs were non significantly different (test for homogeneity p=0.63). The route of administration of estrogens without progestagen did not affect the risk. Among women reporting the use of estrogen/progestagen there was no indication of an increase in the risk of asthma onset whatever the duration or recency of use or route of estrogen administration (Table 3).

Among never smokers, MHT use was significantly related to the risk of asthma onset (HR=1.45, 95% CI 1.10-1.90) while among smokers the HR was 1.02 (95% CI 0.79-1.31) (test for interaction p=0.09) (see online supplement Table E1). Among women who reported using estrogen alone, the HR was 1.80 (95% CI 1.15-2.80) for never smokers and 1.31 (95% CI 0.85-2.03) for ever smokers (test for interaction p=0.37).

Never smokers using estrogen/progestagen had a HR of 1.38 (95% CI 1.02-1.86), while no effect (HR=0.92, 95% CI 0.69-1.21) was observed in ever smokers (test for interaction p= 0.16) (Table 3).

Higher BMI was associated with an increased risk of asthma onset (p for trend <0.001) but there was no indication of an interaction between MHT use and BMI on the risk of asthma (p=0.45). Among women with a BMI less than 22 the HR of estrogen alone users was 1.25 (95% 0.65-2.40); among women with a BMI from 22 to 25, the HR was 1.67 (95% CI 0.93-3.01) and among women with BMI>25 the HR was 1.54 (0.99-2.39).

After stratification by the presence of hysterectomy or the type of menopause we observed a small increased risk in non hysterectomized women using estrogen alone and among women declaring a natural menopause , although the small number of cases in some of the subgroups limited the interpretation of the results

(see online supplement tables E2 and E3)

Women who reported a diagnosis of allergic disease prior to asthma onset appeared to have a larger risk of asthma related to MHT than their counterpart.. An increased risk of asthma was present for the use of estrogen alone (HR = 1.86 (95% CI 1.18-2.93) and a marginal significant increase was observed for the use of estrogen/progestagens (HR=1.39 95% CI 1.01-1.91) in women with allergic disease as compared to women without allergic disease (Table 4).

Discussion

In this large prospective study of French women including 569 incident cases of asthma over 495,448 person-years, use of estrogen alone as last treatment was significantly associated with an increased risk of asthma onset in postmenopausal women, after adjustment for potential confounding factors. The increase in risk was mostly observed among women reporting an allergic disease prior to asthma onset and never smokers. In these subgroups, the risk of asthma onset was strongly related to the use of estrogen alone and a marginal association was observed with the use of estrogen/progestagens.

Our results agree in part with other recent reports. In the Nurses' Health Study (NHS), MHT use was significantly related to the risk of asthma onset in postmenopausal women. However, a similar increased risk was reported for both estrogen alone and estrogen/ synthetic progestin. The effect was stronger in current users, and the risk diminished with time after cessation. In addition, an interaction was observed between MHT and BMI, with a larger risk of newly diagnosed asthma in lean women. [START_REF] Barr | Prospective study of postmenopausal hormone use and newly diagnosed asthma and chronic obstructive pulmonary disease[END_REF] In the cross-sectional Copenhagen City Heart Study, the prevalence of self-reported asthma was significantly related to MHT in non-smokers. [START_REF] Lange | Exogenous female sex steroid hormones and risk of asthma and asthma-like symptoms: a cross sectional study of the general population[END_REF] Two further recent crosssectional studies reported an association between prevalent asthma and asthma-like symptoms with MHT use in women aged 46-54 [START_REF] Real | Hormone replacement therapy, body mass index and asthma in perimenopausal women: a cross sectional survey[END_REF] and postmenopausal women, [START_REF] Jarvis | The association of asthma, atopy and lung function with hormone replacement therapy and surgical cessation of menstruation in a population-based sample of English women[END_REF] but only in lean women. None of these cross-sectional studies provided information on the type of HRT used.

In our study, the increased risk of asthma onset among women using MHT was present only in users of estrogen alone. The effect was observed only in recent users including current users and women for whom time since last use was less than 1.5 years. Overall no increased risk was observed in users of estrogen/progestagen while a marginally significant increase was observed among women reporting a diagnostic of allergic disease prior to asthma onset and among non smokers.

Our study and the NHS are comparable in design, number of cases and person years of follow up. Results of both studies agree on the adverse effect of estrogen alone MHT; however we observed conflicting results with regards to estrogen/progesterone MHT in the overall analysis. The marginal increased risk observed in some subgroups of our population is difficult to compare given that no stratified data on smoking or allergic disease are presented in the report from the NHS. [START_REF] Barr | Prospective study of postmenopausal hormone use and newly diagnosed asthma and chronic obstructive pulmonary disease[END_REF] Several mechanisms have been proposed to explain the role of female hormones in asthma risk. However, there is still some uncertainty. Estrogen appears to have both anti-and pro-inflammatory effects, depending on criteria such as cell types, conditions in the milieu, estrogen concentration and the target organ.[13] While estrogen may have a pro-inflammatory effect regulating the response to allergens, it may also down-regulate the traffic of eosinophils to the lung during the effect or phase of the response to antigens. [START_REF] Riffo-Vasquez | Role of sex hormones in allergic inflammation in mice[END_REF] It has also been shown that estrogen acts to down-regulate airway hyperresponsiveness in a dose-dependent manner and relatively rapid, nongenomic manner by enhancing the activity of endothelial nitric oxide synthase. [START_REF] Lim | Sexual tension in the airways: the puzzling duality of estrogen in asthma[END_REF] Several animal studies have reported increased susceptibility to allergic airway disease in female mice compared to male mice. Estrogens appear to exert most of their effects through estrogen receptors alpha and beta, both of which are present in the lung. [START_REF] Carey | Spontaneous airway hyperresponsiveness in estrogen receptor-alpha-deficient mice[END_REF] Ovariectomized rats developed less airway inflammation than controls and estrogen replacement re-established airway inflammation to the level found in intact females. A recent study suggested that female rats have fewer Treg cells and therefore less protection against inflammatory stimuli such as allergens. [START_REF] Melgert | Female mice are more susceptible to the development of allergic airway inflammation than male mice[END_REF] Estrogen (estradiol) has also been shown to alter beta-2-adrenergic responsiveness [START_REF] Tan | Paradoxical down-regulation and desensitization of beta2-adrenoceptors by exogenous progesterone in female asthmatics[END_REF] and to activate endothelial nitrite oxide synthase, [START_REF] Chambliss | Estrogen modulation of endothelial nitric oxide synthase[END_REF] involved in the pathogenesis of asthma. Estrogen is capable of inhibiting the activity of 11β hydroxysteroid dehydrogenase type I isoenzyme; thereby reducing its ability to regenerate active cortisol from cortisone, and cortisol is an extremely powerful antiinflammatory agent that may be protective against the development of chronic inflammation. [START_REF] Cohen | Postmenopausal asthma: the estradiol 11betahydroxysteroid dehydrogenase connection[END_REF] Finally, estrogen levels have been related to lower levels of adiponectin, a protein that is secreted exclusively by adipocytes, acts as insulin sensitizer and has anti-inflammatory properties. [START_REF] Gavrila | Serum adiponectin levels are inversely associated with overall and central fat distribution but are not directly regulated by acute fasting or leptin administration in humans: cross-sectional and interventional studies[END_REF] This citokyne is inversely associated with fat mass and insulin resistance and has been proposed as a link between obesity and inflammation in humans. [START_REF] Garaulet | Adiponectin, the controversial hormone[END_REF] Most likely several of these mechanisms and pathways play a role and interact with individual susceptibility.

Unopposed estrogens are used in women who have had a hysterectomy, and hysterectomy may lead to hormonal changes and respiratory morbidity, or alternatively a common underlying pathology may lead both to gynecological morbidity resulting in hysterectomy and to asthma. [START_REF] Jarvis | The association of asthma, atopy and lung function with hormone replacement therapy and surgical cessation of menstruation in a population-based sample of English women[END_REF]In our data, among MHT never users, the HR of asthma onset associated with hysterectomy was 1.31 (95% CI 0.85-2.01) suggesting that hysterectomy could not fully explain the increased risk observed among the estrogen users. Only 38.8% of women with hysterectomy reported using estrogen alone and the reason for hysterectomy was unknown. In stratified analyses by hysterectomy or type of menopause, there was no indication of differential effects of MHT in these subgroups.

We did not observe a clear increase in the risk of asthma onset with estrogen/progestagen treatment in contrast to the results of the Nurses´ Health study. In our population, a marginal increased risk was restricted to never smokers and women who reported a diagnostic of allergic disease prior to asthma onset .

While we do not have a clear explanation for this, combined HT regimen have been shown to positively affect the lung functions of postmenopausal women while estrogen alone has no effect. [START_REF] Cevrioglu | The effects of hormone therapy on pulmonary function tests in postmenopausal women[END_REF] Progesterone has receptors in the lungs and trachea and can therefore have local effects at these sites and cause ventilatory stimulation via chemoreceptor activity. [START_REF] Hannhart | Effects of estrogen and progesterone on carotid body neural output responsiveness to hypoxia[END_REF] In a rat model of allergic lung inflammation, estradiol increased the number of neutrophils, eosinophils, and mononuclear cells in the bronchoalveolar lavage of ovariectomized allergic rats, whereas progesterone induced an additional reduction. Degranulation of bronchial mast cells from ovariectomized rats was reduced after in vitro challenge, an effect reverted by estradiol but not by progesterone. [START_REF] De Oliveira | Cellular recruitment and cytokine generation in a rat model of allergic lung inflammation are differentially modulated by progesterone and estradiol[END_REF] These authors suggest that intensity of allergic lung inflammation might be related to the estradiol-toprogesterone ratio at the time of immune sensitization and antigen challenge.

Exogenous progesterone might also down-regulate beta2-adrenoceptors, which would explain part of the difference between MHTs. [START_REF] Tan | Paradoxical down-regulation and desensitization of beta2-adrenoceptors by exogenous progesterone in female asthmatics[END_REF] A possible explanation for the difference of effect observed between the NHS and our study could be the difference in the type of progestorene used by Us and French women and /or the ratio of estrogen-to progesterone. In the US combined MHT uses mostly medroxyprogesterone acetate (MPA) while in our study this type of treatment represented only 10.2% of person years of FU. Micronized progesterone corresponding to 19.7% and other pregnane and norpregnane derivatives to 38.3%.

The association of MHT use with increased risk of asthma onset was more apparent among never smokers, as reported in other studies. [START_REF] Real | Hormone replacement therapy, body mass index and asthma in perimenopausal women: a cross sectional survey[END_REF][START_REF] Lange | Exogenous female sex steroid hormones and risk of asthma and asthma-like symptoms: a cross sectional study of the general population[END_REF] This might be due to the anti-estrogenic effect of smoking [START_REF] Key | Cigarette smoking and urinary oestrogen excretion in premenopausal and post-menopausal women[END_REF] or to the difficulty of isolating the additional effect of MHT among smokers. In contrast to other studies, we did not observe an interaction between BMI and MHT use. [START_REF] Barr | Prospective study of postmenopausal hormone use and newly diagnosed asthma and chronic obstructive pulmonary disease[END_REF][START_REF] Real | Hormone replacement therapy, body mass index and asthma in perimenopausal women: a cross sectional survey[END_REF][START_REF] Jarvis | The association of asthma, atopy and lung function with hormone replacement therapy and surgical cessation of menstruation in a population-based sample of English women[END_REF] An earlier analysis of our data found a significant association between BMI and the risk of asthma onset in both MHT ever users and never users. [START_REF] Romieu | Body mass index, change in body silhouette, and risk of asthma in the E3N cohort study[END_REF] The lack of interaction in our data might be due to the BMI distribution of our population, which was leaner than that of the Nurse's Health Study.

Women reporting a diagnostic of allergic disease prior to asthma onset had a higher risk of MHT related asthma than their counterpart. A crossectional study among English women Jarvis et al [START_REF] Jarvis | The association of asthma, atopy and lung function with hormone replacement therapy and surgical cessation of menstruation in a population-based sample of English women[END_REF] reported that current use of HRT was related to wheeze but not related to markers of allergy (IgE). No data on atopic asthma was presented. Our data suggest that women with allergic predisposition might be more susceptible to the effect of MHT in developing asthma. However, these results should be interpreted with caution given the small number of cases in some subgroups and the potential misclassification of allergic disease based only on women's reports at baseline and in the follow up questionnaires.

The strengths of our study include the large number of postmenopausal incident cases of asthma. While we imputed age at menopause for some women with missing information (n=9,585), results remained similar when these women were excluded from the analysis. Information on current and past MHT use was updated every two years, decreasing the potential risk of misclassification of MHT. In our analysis, we used the information relating to the hormone therapy at the time of asthma onset.

Fifty eight percent of women reporting the use of estrogen alone at the time of asthma onset or as last treatment before asthma onset had previously used another MHT. This supports our finding that the increased risk of asthma onset is linked to estrogen use. We also conducted additional analyses in which exposure was defined as exposure to the MHT with the longest duration of use. Results remained similar.

For 80.5% of the women, the last treatment used was also the longest. In addition, most participants in the E3N cohort were teachers, with a high level of education and health-consciousness,[26] while the prospective design and high rate of follow-up of our study (3.8% lost to follow-up) minimized the possibility of recall bias or bias due to loss of follow-up.

Our definition of asthma met the ATS criteria, and there was a consistent reported date of asthma onset (time of first attack between 1990 and 2002). In addition, we conducted an analysis including as asthma cases only women who reported having asthma in at least two questionnaires (persistent asthma). Results were similar to those observed when all cases of asthma onset were included. Some women may suffer not from asthma but from chronic obstructive pulmonary disease (COPD). We did not measure pulmonary function so the possibility of COPD in older women, particularly ex-and current smokers, cannot be excluded. However, 51% of the cases were non-smokers and were therefore unlikely to have COPD. Furthermore, asthma risk was related to MHT use in non-smokers suggesting that neither smoking nor the misclassification of asthma as COPD could explain the association observed between MHT and asthma incidence. In the Nurses' Health Study, MHT was not related to newly diagnosed COPD. [START_REF] Barr | Prospective study of postmenopausal hormone use and newly diagnosed asthma and chronic obstructive pulmonary disease[END_REF] Our results could be biased if MHT users systematically reported more asthma attacks or if asthma was diagnosed more often in MHT users because of more frequent doctor visits. During the follow up, 93% of the MHT users had a mammogram versus 78% of the non users. However, our population had access to free medical care and there is no reason to believe that MHT users make more frequent medical visits than non-users for non gynecological related health issues.

In conclusion, our study shows an association between the use of estrogen alone as MHT and asthma onset in a large cohort of postmenopausal women. This effect might be linked to an increase in airway inflammation mediated by different pathways, such as inflammatory cytokine release or stimulation of NO synthesis. The increase in asthma risk associated with MHT must be judged in the light of all other health effects of MHT use taking including its beneficial effect on the quality of life of menopausal women.

  E3N, a prospective cohort initiated in 1990, consists of 98,995 French women born between 1925 and 1950 and insured under a health insurance plan covering mostly teachers. The study was approved by the French National Commission for Data Protection and Privacy. E3N is the French component of the European Prospective Investigation into Cancer and Nutrition (EPIC).

  The E3N study is being carried out with financial support from the French League Against Cancer, the European Community, the 3M Company, the Mutuelle Générale de l'Education Nationale, the Institut de Cancérologie Gustave Roussy and the Institut National de la Santé et de la Recherche Médicale. Isabelle Romieu was supported by the National Center for Environmental Health -Centers for Disease Control and Prevention, Atlanta GA, USA. Alban Fabre was supported by the Cancéropôle Région Ile de France. Bénédicte Leynaert was supported by the GA²LEN project (EU contract FOODCT-2004-506378).

Table 1 .

 1 Characteristics of the study population (n= 57,664), E3N cohort study.

		Never user of	Ever user of
		MHT	MHT	p*
		N=16138	N=41526
	BMI (kg/m2)
			<.0001
	<20	1713 (10.72%) 5114 (12.26%)
	20 -22	3299 (20.65%) 10363 (24.85%)
	22 -22	5327 (33.35%) 15425 (36.99%)
	>25	5632 (35.28%) 10791 (25.90%)
	Tobacco use †
			0.19
	Never smoker	9601 (60.20%) 23023 (55.28%)
	Ex-smoker	4918 (30.83%) 14484 (34.78%)
	Current smoker	1429 (8.97%)	4134 (9.94%)
	Caloric intake (kcal/d) †
			0.84
	<1760	3393 (26.06%) 8196 (23.20%)
	1760 -2112	3293 (25.29%) 8918 (25.25%)
	2112 -2513	3167 (24.32%) 9128 (25.84%)
	>2513	3166 (24.33%) 9071 (25.71%)
	Pregnancy
			0.06
	Nulliparous	2293 (14.35%) 4574 (10.97%)
	<=2 children	8766 (54.88%) 25257 (60.57%)
	>2 children	4912 (30.77%) 11862 (28.46%)
	Oral contraceptive use

Table 2 .

 2 Hormone therapy and asthma onset among postmenopausal women, E3N Women whose age of start of MHT is missing are excluded. ‡ Women without any information other than age of start of MHT are excluded.

	cohort study
		Cases	Person-year	Multivariate-adjusted* hazard ratio (95% CI)
	Never use	186 177029	1
	Use †	383 318419	1.21 (1.00 -1.46)
	Recency ‡
	Recent	283 276506	1.20 (0.98 -1.46)
	Past	73 22294	1.16 (0.86 -1.57)
	Duration ‡
	< 2 yrs	236 104607	1.25 (1.02 -1.53)
	2 + yrs	120 194193	1.09 (0.85 -1.38)
	Type of treatment ‡
	Estrogen alone	55 33426	1.54 (1.13 -2.09)
	Estrogen-progesterone	67 58978	1.13 (0.85 -1.51)
	Estrogen-norPregnane derivate	52 54616	1.01 (0.74 -1.38)
	Estrogen-pregnane derivative §	103 89663	1.13 (0.88 -1.46)
	Estrogen-testosterone derivative	20 14882	1.36 (0.86 -2.15)
	Other	59 47235	1.22 (0.90 -1.66)
	Estrogen route**, ‡
	Oral	88 65476	1.26 (0.97 -1.65)
	Transdermal	209 187794	1.15 (0.93 -1.41)
	(nulliparous / 2 or fewer full term pregnancies / more than 2 full term pregnancies / no
	information), total caloric intake (kcal/day), type of menopause (natural / surgical /
	unknown), further stratified by year of birth ([1925-1929]/[1930-1934]/[1935-
	1939]/[1940-1944]/[1945-1950].
	†

* Adjusted for tobacco use (never smoker / current smoker / past smoker / missing information), BMI (kg/m2), ever use of oral contraceptive (yes / no), parity § As part of Pregnane derivatives, dydrogeterone users (HR: 1.06 95% CI: 0.75 -1.48) accounted for 41795 PY and 43 cases of asthma.

Table 3 .

 3 Hormone therapy and asthma onset among postmenopausal women by

	type of treatment, E3N cohort study
	*	Cases	Person-year	Multivariate adjusted † Hazard Ratio (95% CI)	Cases	Person-year	Multivariate-adjuste hazard ratio (95% C
			Estrogen alone		Estrogen-Progestagens ‡
	Recency
	Recent	41 29035	1.67 (1.20 -2.32)	208 210608	1.14 (0.92 -1.40)
	Past	14	4391	1.04 (0.51 -2.12)	34	7531	0.97 (0.64 -1.46)
	Duration
	< 2 yrs	38 13760	1.60 (1.12 -2.27)	159	67671	1.20 (0.96 -1.50)
	2 + yrs	17 19666	1.39 (0.84 -2.31)	83	150468	0.99 (0.76 -1.30)
	Estrogen route § ‡
	Oral	10	5367	1.78 (0.93 -3.38)	78	59791	1.21 (0.92 -1.60)
	Transdermal	45 28006	1.50 (1.07 -2.09)	163 158263	1.08 (0.86 -1.34)
	Smoking Status
	Never Smoker	28 31724	1.80 (1.15 -2.80)	128 122585	1.38 (1.02 -1.86)
	Ever smoker	27 14273	1.31 (0.85 -2.03)	114	95342	0.92 (0.69 -1.21)
	* Women without any information other than age of start of MHT are excluded.
	† Adjusted for tobacco use (never smoker / current smoker / past smoker / missing
	information), BMI (kg/m 2 ), ever use of oral contraceptive (yes / no), parity (nulliparous
	/ 2 or fewer full term pregnancies / more than 2 full term pregnancies / no
	information), total caloric intake (kcal/day), type of menopause (natural / surgical /
	unknown), further stratified by year of birth ([1925-1929]/[1930-1934]/[1935-
	1939]/[1940-1944]/[1945-1950].
	‡ Progestagens include: progesterone, pregnane derivatives, norpregnane
	derivatives and testosterone derivatives.
	§ Combined or not with a progestagen. Nasal estrogen, weak estrogen, other HT and
	unknown HT not shown

Table 4 .

 4 Hormone therapy and asthma onset among postmenopausal women by Ajusted for tobacco use (never smoker / current smoker / past smoker / missing information), BMI (kg/m 2 ), ever use of oral contraceptive (yes / no), parity (nulliparous / 2 or fewer full term pregnancies / more than 2 full term pregnancies / no information), total caloric intake (kcal/day), type of menopause (natural / surgical / unknown), further stratified by year of birth([1925-1929]/[1930-1934]/[1935-1939]/[1940-1944]/[1945-1950]. ‡ Progestagens include: progesterone, pregnane derivatives, norpregnane derivatives and testosterone derivatives.Figure 1. Diagram of the initial population, the reasons for exclusion and the population included in the analysis.

	atopic status, E3N cohort study
			Multivariate -adjusted †	Multivariate-adjusted †
	*	PY	non atopic (306 cases) cases HR 95%CI	atopic (251 cases) cases HR 95%CI
	Never use	177029	114	1	67	1
	Use	318419	192 1,04 (0,81 -1,33)	184 1,52 (1,13 -2,05
	Recency
	Recent	276506	137 1,02 (0,79 -1,33)	139 1,52 (1,11 -2,07
	Past	22294	41 1,04 (0,69 -1,56)	32 1,42 (0,90 -2,25
	Duration
	< 2 yrs	104607	123 1,11 (0,84 -1,45)	110 1,55 (1,12 -2,13
	2 + yrs	194193	55 0,88 (0,63 -1,22)	61 1,42 (0,98 -2,05
	Type of treatment Other	47235	27 0,97 (0,64 -1,47)	32 1,76 (1,16 -2,68
	Estrogen route § ‡,
	Oral	65476	44 1,19 (0,84 -1,69)	42 1,56 (1,05 -2,33
	Transdermal	187794	107 1,01 (0,77 -1,33)	97 1,43 (1,04 -1,97

‡ Estrogen alone 33426 29 1,43 (0,96 -2,11) 25 1,86 (1,18 -2,93 Estrogen-progestagens 218139 122 0,98 (0,75 -1,28) 114 1,39 (1,01 -1,91 *Women without any information other than age of start of MHT are excluded. †

Tibolone, weak estrogen (unopposed or not), estrogen/testosterone, progestin only and unknown MHT. ** Combined or not with a progestagen. P for heterogeneity 0.45. Nasal estrogen, weak estrogen, other HT and unknown HT not shown

Tibolone, weak estrogen (unopposed or not), estrogen/testosterone, progestin only and unknown MHT. § Combined or not with a progestagen. Nasal estrogen, weak estrogen, other HT and unknown HT not shown

Acknowledgements

The authors are indebted to all participants for providing data and to practitioners for providing pathology reports. The authors are grateful to R. Chaït, M. Fangon, M.

Niravong and L. Hoang for managing the data.

Funding/Support

Competing interesting: None


Kazuo Aoki

¸ois Golse
email: golse@math.polytechnique.fr

ON THE SPEED OF APPROACH TO EQUILIBRIUM FOR A COLLISIONLESS GAS

Keywords: 1991 Mathematics Subject Classification. 82C40 (35F10, 35B40) Free-molecular gas, Approach to equilibrium, Diffuse reflection, Kinetic theory of gases, Renewal equation

We investigate the speed of approach to Maxwellian equilibrium for a collisionless gas enclosed in a vessel whose wall are kept at a uniform, constant temperature, assuming diffuse reflection of gas molecules on the vessel wall. We establish lower bounds for potential decay rates assuming uniform L p bounds on the initial distribution function. We also obtain a decay estimate in the spherically symmetric case. We discuss with particular care the influence of low-speed particles on thermalization by the wall.

In memory of Carlo Cercignani (1939Cercignani ( -2010) )

Introduction

Relaxation to equilibrium is a fundamental issue in statistical mechanics. In the kinetic theory of gases, convergence to equilibrium is formulated in terms of the asymptotic behavior of solutions of the Boltzmann equation in the long time limit. In [START_REF] Cercignani | H-Theorem and trend to equilibrium in the kinetic theory of gases[END_REF], Cercignani formulated a quite remarkable conjecture, asserting the domination of Boltzmann's relative H functional by a constant multiple of Boltzmann's entropy production functional. If true, Cercignani's conjecture would imply exponential relaxation to equilibrium for space homogeneous solutions of the Boltzmann equation, and would have important consequences even for space inhomogeneous solutions. This conjecture was subsequently disproved in a series of papers by Bobylev, Cercignani and Wennberg [START_REF] Bobylev | The theory of the nonlinear, spatially uniform Boltzmann equation for Maxwell molecules[END_REF][START_REF] Bobylev | On the rate of entropy production for the Boltzmann equation[END_REF][START_REF] Wennberg | Entropy dissipation and moment production for the Boltzmann equation[END_REF]. And yet, in the words of Villani, "Cercignani's conjecture is sometimes true, and almost always true" [START_REF] Villani | Cercignani's conjecture is sometimes true and almost always true[END_REF]. Slightly later, Desvillettes and Villani [START_REF] Desvillettes | On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation[END_REF] studied the speed of convergence to equilibrium for space inhomogeneous solutions of the Boltzmann equation, under rather stringent conditions of regularity, decay at large velocities and positivity, but without assuming that the initial data is close to equilbrium. They found that the distance between the distribution function at time t and the final equilibrium state decays like O(t -m ) for each m > 0 as t → +∞. Their result makes critical use of the Boltzmann collision integral and entropy production, and of their combination with the free transport operator (the left-hand side of the Boltzmann equation.) In an earlier paper [START_REF] Desvillettes | Entropy dissipation rate and convergence in kinetic equations[END_REF], Desvillettes had proved that Boltzmann's entropy production can be regarded as a distance between the distribution function and the set of Maxwellian states. On the other hand, Guo proved in [START_REF] Guo | Decay and continuity of the Boltzmann equation in bounded domains[END_REF] that, for initial data close enough to a uniform Maxwellian, solutions of the Boltzmann equation converge to equilibrium at exponential speed.

In [START_REF] Desvillettes | On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation[END_REF], Desvillettes and Villani treated the cases where the spatial domain is the flat torus (equivalently, the Euclidian space with periodic initial data), or a smooth bounded domain with either the specular or bounce-back reflection conditions at the boundary. Therefore, convergence to equilibrium is driven by the collision process alone, and the role of the boundary condition is rather indirect. In the long time limit, the solution of the Boltzmann equation converges to a uniform Maxwellian state whose density and temperature are given in terms of the initial data by the conservation of total mass and total energy. The boundary condition is involved only through its compatibility with the conservation laws of total mass and energy. In a more recent work [START_REF] Villani | Hypocoercivity[END_REF], Villani extended their result to the case where the gas is enclosed in a vessel whose walls are maintained at a constant temperature, modeled with an accommodation or diffuse reflection condition. In that case, mass is the only quantity that is conserved under the dynamics of the Boltzmann equation, so that the final equilibrium state is the centered, uniform Maxwellian with the wall temperature and same total mass as the initial data. In that case, the boundary condition has a direct influence on the relaxation to equilibrium. Nevertheless, the combined effect of entropy production by the collision process and of free streaming, an example of what is termed "hypocoercivity", is the driving mechanism used by Villani in his proof of relaxation to equilibrium [START_REF] Villani | Hypocoercivity[END_REF].

However, it seems obvious from physical consideration that the accommodation or diffuse reflection condition on a wall maintained at some constant temperature is enough to drive the gas to an equilibrium state, and that the role of the collision process is not as important as in the periodic, or specular or bounce-back reflection cases -which are in any case less realistic.

In order to establish the role of Boltzmann's collision integral in the rate of relaxation to equilibrium, we consider the case of a collisionless gas enclosed in a vessel whose surface is maintained at a constant temperature, and investigate the speed of approach to equilibrium for such a system. The present paper provides a theoretical confirmation of the numerical results obtained by the authors in [START_REF] Tsuji | Relaxation of a Free-Molecular Gas to Equilibrium Caused by Interaction with Vessel Wall[END_REF] in collaboration with Tsuji. Basically, while the collision process is not necessary in the convergence to equilibrium, it greatly influences the speed of relaxation to equilibrium. In the same spirit, Desvillettes and Salvarani [START_REF] Desvillettes | Asymptotic behavior of degenerate linear transport equations[END_REF] have investigated the speed of relaxation to equilibrium in the case of linear collisional models where the collision frequency is not uniformly bounded away from 0, by adapting the hypocoercivity method.

We dedicate this piece of work to the memory of our teacher, friend and colleague Carlo Cercignani, in recognition of his leading role in the mathematical theory of the Boltzmann equation.

Statement of the problem

Consider a collisionless gas enclosed in a container materialized by a bounded domain Ω in R N whose boundary ∂Ω is maintained at a constant temperature

θ w > 0. Its distribution function F ≡ F (t, x, v) satisfies          ∂ t F + v • ∇ x F = 0 , (x, v) ∈ Ω × R N , F (t, x, v)|v•n x | = v ′ •nx>0 F (t, x, v ′ )v ′ •n x K(x, v, dv ′ ) , (x, v) ∈ Γ -, F t=0 = F in . (1)
The notation n x designates the unit outward normal at the point x ∈ ∂Ω, and we have used the notation

Γ + := {(x, v) ∈ ∂Ω × R N | v • n x > 0} , Γ -:= {(x, v) ∈ ∂Ω × R N | v • n x < 0} .
The boundary condition is given by the measure-valued kernel K(x, v, dv ′ ) ≥ 0 satisfying the assumptions (i) for each x ∈ ∂Ω, one has

v•nx<0 K(x, v, dv ′ )dv = dv ′ (ii) for each x ∈ ∂Ω, one has M (1,0,θ) (v)|v•n x | = v ′ •nx>0 M (1,0,θ) (v ′ )v ′ •n x K(x, v, dv ′ ) , v•n x < 0 , iff θ = θ w .
Throughout this paper, we use the following notation for the Maxwellian distribution:

M (ρ,u,θ) (v) = ρ (2πθ) N/2 e -|v-u| 2 2θ
(Henceforth, we denote for simplicity M w = M (1,0,θw) .) With these notations, the case of diffuse reflection corresponds with

K(x, v, dv ′ ) := M (1,0,θw) (v)|v • n x |dv ′ u•nx<0 M (1,0,θw) (u)|u • n x |du .
The case of an accommodation boundary condition (i.e. the so-called Maxwell-type condition) corresponds with

K(x, v, dv ′ ) := (1 -α(x))δ(v ′ -v + 2(v • n x )n x ) + α(x) M (1,0,θw) (v)|v • n x |dv ′ u•nx<0 M (1,0,θw) (u)|u • n x |du .
In the case of diffuse reflection, Arkeryd and Nouri [START_REF] Arkeryd | Boltzmann asymptotics with diffuse reflection boundary conditions[END_REF] have proved that the solution of (1) satisfies

F (t + •, •, •) → M (1,0,θw) |Ω| Ω×R N F in dxdv in L 1 ([0, T ] × Ω × R N ) as t → +∞.
In the present paper, we study in detail the following Problem: Is there a decay rate E(t) → 0 as t → +∞, for which an asymptotic estimate of the form

F (t + •, •, •) - M (1,0,θw) |Ω| Ω×R N F in dxdv L 1 ([0,T ]×Ω×R N ) = O(E(t)) (2)
holds for each solution F of (1) uniformly as F in runs through certain classes of bounded subsets of L 1 (R N × R N )?

Negative results and lower bounds on E

We first consider the problem of finding a decay rate that is uniform as F in runs through all bounded subsets of L 1 (R N × R N ). Proposition 3.1. There does not exist any decay rate function E : R + → R + such that E(t) → 0 as t → +∞ and

F (t + •, •, •) - M (1,0,θw) |Ω| Ω×R N F in dxdv L 1 ([0,T ]×Ω×R N ) ≤ E(t) F in L 1 (R N ×R N ) (3)
for each t ≥ 0.

Since the initial boundary value problem ( 1) is linear, if there is a long time decay as in (2) uniformly as the initial data F in runs through bounded subsets of L 1 , an inequality of the form (3) must hold.

Proof. Assume that the origin 0 ∈ Ω -if not, one can apply a translation in the x-variable without changing the nature of the problem -and set R = 1 2 dist(0, ∂Ω). We henceforth denote by B the closed unit ball of R N , and by 1 A the indicator function of the set A.

For 0 < ε ≪ 1, set

F in (x, v) = 1 ε 2N 1 εB (x)1 εB (v) (with εB := {εz | z ∈ B}) so that F in (x, v)dxdv = |B| 2 . Call τ (x, v) := inf{t ≥ 0 | x -tv /
∈ Ω}; obviously the solution F of (1) satisfies F ≥ Φ, where Φ is the solution of the same free transport equation, with the same initial data, but with absorbing boundary condition:

     ∂ t Φ + v • ∇ x Φ = 0 , (x, v) ∈ Ω × R N , Φ(t, x, v) = 0 , (x, v) ∈ Γ -, Φ t=0 = F in . (4)
Solving for Φ along characteristics, one finds that

Φ(t, x, v) = F in (x -tv, v)1 0≤t≤τ (x,v) . Now F (t + •, •, •) - M (1,0,θw) |Ω| Ω×R N F in dxdv L 1 ([0,T ]×Ω×R N ) = F (t + •, •, •) - |B| 2 |Ω| M (1,0,θw) L 1 ([0,T ]×Ω×R N ) ≥ T 0 Ω R N F (t + s, x, v) - |B| 2 |Ω| M (1,0,θw) (v) + dvdxds .
Henceforth, if A ⊂ R 3 and u ∈ R 3 , we denote A + u := {z + u | z ∈ A}. Since the function z → z + = max(z, 0) is nondecreasing, one has

F (t + •, •, •) - M (1,0,θw) |Ω| Ω×R N F in dxdv L 1 ([0,T ]×Ω×R N ) ≥ T 0 Ω R N Φ(t + s, x, v) - |B| 2 |Ω| M (1,0,θw) (v) + dvdxds = T 0 Ω R N 1 ε 2N 1 εB+(t+s)v (x)1 εB (v)1 0≤t+s≤τ (x,v) - |B| 2 |Ω| M (1,0,θw) (v) + dvdxds ≥ T 0 Ω R N 1 ε 2N 1 εB+(t+s)v (x)1 εB (v)1 0≤(t+s)|v|≤R-ε - |B| 2 |Ω| M (1,0,θw) (v) + dvdxds since τ (x, v)|v| ≥ 2R -ε for each x ∈ εB. Now 1 ε 2N 1 εB+(t+s)v (x)1 εB (v)1 0≤(t+s)|v|≤R-ε - |B| 2 |Ω| M (1,0,θw) (v) + = 1 ε 2N 1 εB+(t+s)v (x)1 εB (v)1 0≤(t+s)|v|≤R-ε 1 - ε 2N |B| 2 |Ω| M (1,0,θw) (v) + ≥ 1 ε 2N 1 εB+(t+s)v (x)1 εB (v)1 0≤(t+s)|v|≤R-ε 1 - ε 2N |B| 2 |Ω|(2πθ w ) N/2 + . Then T 0 Ω R N 1 ε 2N 1 εB+(t+s)v (x)1 εB (v)1 0≤(t+s)|v|≤R-ε dvdxds = |B| T 0 R N 1 ε N 1 εB (v)1 0≤(t+s)|v|≤R-ε dvds = |B| T 0 R N 1 ε N 1 0≤|v|≤min(ε,(R-ε)/t+s) dvds = |B| T 0 1 ε N min ε N , (R -ε) N (t + s) N ds ≥ |B| T ε N min ε N , (R -ε) N (t + T ) N .
Therefore, if there is a uniform decay rate E as in (3), it must satisfy

T |B| 1 - ε 2N |B| 2 |Ω|(2πθ w ) N/2 + min 1, (R -ε) ε(t + T ) N ≤ |B| 2 E(t) .
Here ε only has to satisfy 0 < ε < R. If one lets ε → 0 + , this contradicts the existence of a profile E in (3) such that E(t) → 0 as t → +∞.

The negative result in Proposition 3.1 could have been easily anticipated: as ε → 0, our choice of F in converges to |B| 2 δ x=0 δ v=0 in the weak topology of measures. Then particles distributed under δ x=0 δ v=0 do not move, hence cannot reach the boundary and therefore cannot be thermalized. Thus, in some sense, the problem of finding a decay rate as in ( 3) is trivial if stated in this way.

Before going further, let us comment on the kind of inequality considered in Proposition 3.1. In order to measure the speed of approach to equilibrium for the problem (1), it might be more natural to consider a pointwise in time inequality of the type

F (t, •, •) - M (1,0,θw) |Ω| Ω×R N F in dxdv L 1 (Ω×R N ) ≤ E(t) F in L 1 (R N ×R N ) (5)
instead of (3). Observe that the l.h.s. of ( 3) is the time-average of the l.h.s. of ( 5), so that ( 5) is a stronger statement than (3) -as a matter of fact, ( 5) entails ( 3) with E(t) changed into T E(t). Therefore, negative results obtained on timeaveraged estimates like (3) are stronger than their analogues for the corresponding pointwise estimate similar to [START_REF] Desvillettes | Entropy dissipation rate and convergence in kinetic equations[END_REF]. This is why we sought to confirm numerically pointwise in time bounds like (5) in our previous paper [START_REF] Tsuji | Relaxation of a Free-Molecular Gas to Equilibrium Caused by Interaction with Vessel Wall[END_REF], and not time-averaged estimates as in the work of Arkeryd-Nouri [START_REF] Arkeryd | Boltzmann asymptotics with diffuse reflection boundary conditions[END_REF]. Throughout the present section, we discuss negative results on the speed of approach to equilibrium and therefore consider time-averaged decay estimates.

A slightly less trivial issue is to investigate the existence of such a decay rate under the additional assumption ruling out the possibility of a complete concentration as above. This could be done by controling not only the L 1 norm of the initial data, but (for instance) its entropy, or relative entropy with respect to the final state. For instance, one could look instead at a bound of the form

F (t + •, •, •) - M (1,0,θw) |Ω| Ω×R N F in dxdv L 1 ([0,T ]×Ω×R N ) ≤ E(t) Ω×R N F in (x, v)| ln F in (x, v)|dxdv . (6)
Proposition 3.2. If there is a time decay rate E such that the solution of ( 1) satisfies [START_REF] Desvillettes | Asymptotic behavior of degenerate linear transport equations[END_REF], then it must satisfy

E(t) ≥ C ln t as t → +∞ .
Proof. For the same initial data as before

F in (x, v)| ln F in (x, v)|dxdv = 1 ε 2N 1 εB (x)1 εB (v) ln 1 ε 2N dxdv = 2N |B| 2 | ln ε| .
Proceeding as before to obtain a lower bound for the l.h.s. of [START_REF] Desvillettes | Asymptotic behavior of degenerate linear transport equations[END_REF], one arrives at the inequality

T |B| 1 - ε 2N |B| 2 |Ω|(2πθ w ) N/2 + min 1, (R -ε) ε(t + T ) N ≤ 2N |B| 2 | ln ε|E(t) .
Choosing ε 0 > 0 small enough, this inequality takes the form

C min 1, 1 εt N ≤ | ln ε|E(t)
for all t > T and all ε ∈ (0, ε 0 ). Picking then ε = 1/t with t ≫ 1 so that in particular t > max(T, 1/ε 0 ) leads to

E(t) ≥ C ln t as t → +∞ .
The constraint on the decay rate E obtained in Proposition 3.2 remains the same if one replaces the r.h.s. in the bound [START_REF] Desvillettes | Asymptotic behavior of degenerate linear transport equations[END_REF] with either

E(t) Ω×R N F in (x, v)(1 + |v| 2 + | ln F in (x, v)|)dxdv or E(t) Ω×R N F in (x, v) ln F in ρM (1,0,θw) -F in + ρM (1,0,θw) (x, v)dxdv , where ρ = 1 |Ω| Ω×R N F in dxdv .
Indeed, compared to the r.h.s. in the bound [START_REF] Desvillettes | Asymptotic behavior of degenerate linear transport equations[END_REF], both these quantities involve in addition the total energy integral

Ω×R N |v| 2 F in (x, v)dxdv .
Since the inequality ( 6) is tested with velocity distribution functions concentrated near v = 0, this additional term is negligible and its presence does not change the final result.

Of course, prescribing an entropy bound is not the only possibility for avoiding concentrations in the initial data F in . For instance, one could ask for a control of F in in some L p -norm with 1 < p ≤ ∞. In other words, one could investigate the possibility of an inequality of the form

F (t + •, •, •) - M (1,0,θw) |Ω| Ω×R N F in dxdv L 1 ([0,T ]×Ω×R N ) ≤ E(t) F in L p (Ω×R N ) . (7)
Proposition 3.3. If there is a time decay rate E such that the solution of ( 1) satisfies [START_REF] Desvillettes | On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation[END_REF], then it must satisfy

E(t) ≥ C t N min(1,2/p ′ )
as t → +∞ .

Proof. With the same initial data as before

F in L p (Ω×R N ) = |B| 2/p ε 2N/p ′ , with p ′ = p p-1 <
∞ the dual exponent of p, so that, estimating the l.h.s. by the same argument as before, one arrives at the inequality

T |B| 1 - ε 2N |B| 2 |Ω|(2πθ w ) N/2 + min 1, (R -ε) ε(t + T ) N ≤ |B| 2/p ε 2N/p ′ E(t) .
Assume that p ≥ 2. Since the growth of the l.h.s. in 1/ε is smaller than that of the r.h.s., there is no point in letting ε → 0. One therefore chooses ε > 0 small enough but fixed, so that ε < R/2 and ε 2N |B| 2 < 1 2 |Ω|(2πθ w ) N/2 , so that

T 2 |B| min 1, R 2ε(t + T ) N ≤ |B| 2/p ε 2N/p ′ E(t) .
In that case, one sees that there exists C = C(T, R, ε) > 0 such that

E(t) ≥ C t N as t ≫ 1 .
If 1 < p < 2, by picking ε 0 > 0 small enough, the inequality above takes the form

C min 1, 1 εt N ≤ |B| 2/p ε 2N/p ′ E(t) ,
for all t > 0 and all ε ∈ (0, ε 0 ). Picking ε = 1/t with t ≫ 1 so that in particular t > max(T, 1/ε 0 ), one concludes from this last estimate that

E(t) ≥ C t 2N/p ′ as t → +∞ .
At this point, it is worth comparing the result in Proposition 3.3 for the collisionless gas driven to equilbrium by the boundary, and the following important result, obtained by Ukai, Point and Ghidouche [START_REF] Ukai | Sur la solution globale du problème mixte de l'équation de Boltzmann non linéaire[END_REF], for the Boltzmann equation linearized at a uniform Maxwellian equilibrium.

Consider the Boltzmann equation linearized at the uniform Maxwellian state M = M (1,0,1) , assuming hard sphere collisions, in the case where the spatial domain is the periodic box with unit side T 3 . In other words, consider the Cauchy problem

(∂ t + v • ∇ x )g + L M g = 0 , (t, x, v) ∈ R * + × T 3 × R 3 , g t=0 = g in (8)
The linearized collision operator is given by

L M φ(v) := R 3 ×S 2 (φ(v ′ ) + φ(v ′ * ) -φ(v) -φ(v * ))|(v -v * ) • ω|M (v * )dv * dω with v ′ = v -(v -v * ) • ωω v ′ * = v * + (v -v * ) • ωω
where ω runs through the unit sphere S 2 . It is understood that the notation

L M φ(t, x, v) designates (L M φ(t, x, •))(v). Define Πg in = g in + vg in • v + ( 1 3 |v| 2 -1)g in 1 2 (|v| 2 -3) with the notation φ = R 3 φ(v)M (v)dv .
Theorem 3.4. [START_REF] Ukai | Sur la solution globale du problème mixte de l'équation de Boltzmann non linéaire[END_REF] There exist γ * > 0, and, for each γ ∈]0, γ * [, a positive constant C γ such that, for each g in ∈ L 2 (T 3 × R 3 ; M dvdx), the solution g of ( 8) satisfies the inequality

g(t, •, •) - T 3 Πg in dx L 2 (T 3 ×R 3 ;Mdvdx) ≤ C γ e -γt g in L 2 (T 3 ×R 3 ;Mdvdx)
for each t ≥ 0. This result expresses that there is a spectral gap for the linearized Boltzmann (unbounded) operator

Lg(x, v) = v • ∇ x g(x, v) + L M g(x, v) on L 2 (T 3 × R 3 ; M dvdx) with domain D(L) = {φ ∈ L 2 (T 3 ×R 3 ; M dvdx) | v •∇ x φ and (1+|v|)φ ∈ L 2 (T 3 ×R 3 ; M dvdx)} .
In other words, the spectrum of L is included in a domain of the form

spec(L) ⊂ {0} ∪ {z ∈ C | Re(z) ≥ θ}
for some θ > 0 (the spectral gap). The eigenvalue 0 corresponds to the fivedimensional nullspace of L, given by

ker L = {a + b • v + c 1 2 (|v| 2 -3) | a, c ∈ R and b ∈ R 3 } .
One reason explaining the spectral gap as above is the fact that L M is an (unbounded) Fredholm operator in the v variable, of the form Indeed, assume for simplicity that θ w = 1 in the problem [START_REF] Arkeryd | Boltzmann asymptotics with diffuse reflection boundary conditions[END_REF]. Then by the Cauchy-Schwarz inequality

L M φ(v) = ν(|v|)φ(v) -Kφ
|Ω| 1/2 F (t, •, •) M - 1 |Ω| Ω×R 3 F in dxdv L 2 (Ω×R 3 ;Mdvdx) ≥ F (t, •, •) - M |Ω| Ω×R 3 F in dxdv L 1 (Ω×R 3 ;dvdx)
so that a spectral gap inequality of the form

F (t, •, •) M - 1 |Ω| Ω×R 3 F in dxdv L 2 (Ω×R 3 ;Mdvdx) ≤ C γ e -γt F in M L 2 (Ω×R 3 ;Mdvdx) (9)
would entail

F (t + •, •, •) - M |Ω| Ω×R 3 F in dxdv L 1 ([0,T ]×Ω×R 3 ;dvdxds) ≤ T |Ω| 1/2 C γ e -γt F in M L 2 (Ω×R 3 ;Mdvdx) . Now, for F in (x, v) = 1 ε 2N 1 εB (x)1 εB (v) , one has, assuming that 0 < ε < 1, F in M L 2 (Ω×R 3 ;Mdvdx) ≤ ((2π) N e) -1/4 F in L 2 (Ω×R 3 ;dvdx) .
In other words, the existence of a spectral gap for the problem [START_REF] Arkeryd | Boltzmann asymptotics with diffuse reflection boundary conditions[END_REF] in Ω × R 3i.e. the inequality (9) would entail the inequality

F (t + •, •, •) - M (1,0,θw) |Ω| Ω×R 3 F in dxdv L 1 ([0,T ]×Ω×R 3 ;dvdxds) ≤ T |Ω| 1/2 ((2π) N e) -1/4 C γ e -γt F in L 2 (Ω×R 3 ;dvdx)
, which stands in contradiction with Proposition 3.3.

Algebraic decay

After the negative results obtained in the previous section, we now consider the problem of establishing decay estimates compatible with the results of section 3.

We shall do so on a special class of boundary value problems, satisfying the following assumptions: (a) the spatial domain Ω is a ball of the Euclidian space R 3 ; (b) the distribution function of the collisionless gas at t = 0 is a radial function of both the space and velocity variables; (c) the boundary condition is a diffuse reflection (complete accommodation), with a constant boundary temperature (no temperature gradient at the boundary of the vessel.)

None of these assumptions excludes the mechanism producing the negative results in section 3. Indeed, all these results are based on the choice of an initial distribution function that is concentrated, in the velocity variable, near v = 0, and in the space variable, near a point that is away from the boundary. This is obviously possible with the symmetries above, by choosing the initial data concentrated, in the space variable, near the center of the ball, and in the velocity variable, near v = 0 as in section 3. Particles so distributed at time t = 0 will not reach the boundary of the spatial domain until after very late, and, meanwhile, the exact nature of the boundary condition is immaterial for such particles.

Therefore, we set the spatial domain to be Ω = B(0, 1), the unit ball of R 3 . The temperature on ∂Ω is set to be θ w = 1. Consider the initial boundary value problem

         (∂ t + v • ∇ x )f = 0 , (t, x, v) ∈ R + × Ω × R 3 , f (t, x, v) = f + M + M (v) , v • n x < 0 , x ∈ ∂Ω , f t=0 = f in (|x|, |v|) , (10)
with the notation

φ + = v•nx>0 φ(v)v • n x dv ,
and we recall the notation M = M (1,0,1) for the Maxwellian distribution with density 1, zero bulk velocity and temperature 1.

The main result in this section is the following.

Theorem 4.1. Assume that 0 ≤ f in (x, v) ≤ CM (v) a.e. in (x, v) ∈ Ω × R 3 .
Then the solution of (10) satisfies

Ω×R 3 f (t, x, v) M (v) -1 |Ω| Ω×R 3 f in (|x|, |v|)dxdv p M (v)dv = O 1 t 3 + O 1 t p so that f (t, •, •) M -1 |Ω| Ω×R 3 f in (|x|, |v|)dxdv L p (Ω×R 3 ;Mdvdx) = O 1 t min(1,3/p) for all 1 ≤ p < ∞.
The reader might want to compare the upper bound in this theorem with the lower bound in Proposition 3.3: this is done by taking p = ∞ in the estimate [START_REF] Desvillettes | On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation[END_REF] and in Proposition 3.3, and p = 1 in the theorem above. The difference in the exponents observed suggests that one of these results at least is not sharp. The numerical investigations in [START_REF] Tsuji | Relaxation of a Free-Molecular Gas to Equilibrium Caused by Interaction with Vessel Wall[END_REF] suggests that it is the lower estimate in Proposition 3.3 that is sharp, while the upper bound in the theorem above is not.

Our main task in the proof of this theorem is to use the symmetries in the problem [START_REF] Tsuji | Relaxation of a Free-Molecular Gas to Equilibrium Caused by Interaction with Vessel Wall[END_REF] to reduce it to a renewal integral equation for a scalar unknown quantity, and to use classical results for such equations.

The role of symmetries in this problem is summarized in the following statement.

Lemma 4.2. If f is the solution of the initial boundary value problem [START_REF] Tsuji | Relaxation of a Free-Molecular Gas to Equilibrium Caused by Interaction with Vessel Wall[END_REF], its outgoing flux at the boundary

µ := v•nx>0 f (t, x, v)v • n x dv
is independent of the position variable x as x runs through ∂Ω, and a function of t ≥ 0 only.

Proof. If R ∈ O 3 (R) is any orthogonal matrix, the function f R (t, x, v) := f (t, Rx, Rv) obviously satisfies f R t=0 = f t=0 , and f R + (t, x) = f + (t, Rx), while

(∂ t + v • ∇ x )f R (t, x, v) = ∂ t f (t, Rx, Rv) + v • (R T ∇ x f )(t, Rx, Rv) = ∂ t f (t, Rx, Rv) + Rv • ∇ x f (t, Rx, Rv) = 0 .
By uniqueness of the solution of (10), we conclude that

f R = f for each R ∈ O 3 (R). Thus f + (t, x) = f R + (t, x) = f + (t, Rx) for each (t, x) ∈ R + × ∂Ω and R ∈ O 3 (R): hence µ := f + is independent of x as x runs through ∂Ω.
Proof of Theorem 4.1. With Lemma 4.2, we easily arrive at an integral equation satisfied by µ. Pick x ∈ ∂Ω, and set θ = (n x , v); the backward exit time

τ x,v := inf{t > 0 | x -tv / ∈ Ω}
is given by the explicit formula

τ x,v = 2 cos θ |v| , Ω v n x x O θ x,v τ v Figure 1.
The exit time for a spherical domain as can be seen on Figure 1. By the method of characteristics, for each x ∈ ∂Ω and v ∈ R 3 such that v•n x > 0, one has

f (t, x, v) = 1 t>τx,v M (v) M + µ(t -τ x,v ) + 1 t<τx,v f in (|x -tv|, |v|) .
Hence

µ(t) = v•nx>0 f (t, x, v)v • n x dv = 1 M + v•nx>0 µ(t -τ x,v )1 t>τx,v M (v)v • n x dv + v•nx>0 f in (|x -tv|, |v|)1 t<τx,v v • n x dv
Introducing spherical coordinates in v, with n x being the polar direction and θ the colatitude, we see on Figure 1 that, for each x ∈ ∂Ω, one has τ x,v |v| = 2 cos θ, so that

v•nx>0 µ(t -τ x,v )1 t>τx,v M (v)v • n x dv = 2π ∞ 0 π/2 0 µ t - 2 cos θ |v| 1 t|v|>2 cos θ M (|v|)|v| cos θ|v| 2 sin θdθd|v| ,
abusing the notation M (|v|) to designate the radial function v → M (v).

Substituting y = cos θ, this integral is expressed as

∞ 0 1 0 µ t - 2y |v| 1 t|v|>2y M (|v|)|v| 3 ydyd|v| = 1 4 ∞ 0 2/|v| 0 µ (t -τ ) 1 t>τ M (|v|)|v| 5 τ dτ d|v| = 1 4 ∞ 0 t 0 µ (t -τ ) 1 τ <2/|v| M (|v|)|v| 5 τ dτ d|v| = 1 4 t 0 µ (t -τ ) 2/τ 0 M (|v|)|v| 5 d|v| τ dτ .
We henceforth define

K(τ ) := πτ 2 M + 2/τ 0 M (|v|)|v| 5 d|v| .
The function K is continuous on R * + , satisfies K(τ ) ≥ 0 for each τ ≥ 0, and

K(τ ) → 0 as τ → 0 + , K(τ ) ∼ 8 3τ 5 as τ → +∞ . Therefore µ satisfies µ(t) = t 0 K(τ )µ(t -τ )dτ + S(t) (11)
where the source term is

S(t) := v•nx>0 f in (|x -tv|, |v|)1 t<τx,v v • n x dv .
This is a renewal equation, to which we shall apply the results of [START_REF] Feller | On the integral equation of renewal theory[END_REF].

Next we consider the source term. Define Φ(|v|) := supess x∈Ω f in (|x|, |v|) ;

assuming that 0 ≤ f in ∈ L ∞ (Ω × R 3 ) implies that Φ ∈ L ∞ (R + ). Then, S(t) ≤ v•nx>0 Φ(|v|)1 t<τx,v v • n x dv = 2π ∞ 0 Φ(|v|)|v| 3 π/2 0 1 t|v|<2 cos θ cos θ sin θdθ d|v| = 2π ∞ 0 Φ(|v|)|v| 3 1 0 1 t|v|<2y ydy d|v| = π ∞ 0 Φ(|v|)|v| 3 (1 -1 4 t 2 |v| 2 ) + d|v| ≤ π Φ L ∞ 2/t 0 |v| 3 d|v| = 4π Φ L ∞ t 4 .
Apply Theorem 4 of [START_REF] Feller | On the integral equation of renewal theory[END_REF]: with n = 3, one has

∞ 0 t m K(t)dt < +∞ for m = 0, 1, 2, 3 while
S(t) = o 1 t , ∞ t S(τ )dτ = o 1 t as t → +∞. Then µ(t) → µ ∞ = ∞ 0 S(τ )dτ ∞ 0 τ K(τ )dτ
as t → +∞, and

µ(t) -µ ∞ = o 1 t .
Now let g := f /M , where f is the solution of (10); g satisfies

       (∂ t + v • ∇ x )g = 0 , (t, x, v) ∈ R + × Ω × R 3 , g(t, x, v) = √ 2πµ(t) , v • n x < 0 , x ∈ ∂Ω , g t=0 = f in (|x|, |v|)/M (v) . ( 12
)
The function g is given by

g(t, x, v) = √ 2πµ(t -τ x,v )1 t>τx,v + f in (|x|, |v|) M (v) 1 t<τx,v , so that g(t, x, v) - √ 2πµ ∞ = √ 2π (µ(t -τ x,v ) -µ ∞ ) 1 t>τx,v + f in (|x|, |v|) M (v) - √ 2πµ ∞ 1 t<τx,v = I + II . Now τ x,v ≤ 2/|v|, so that R 3 1 t<τx,v M (v)dv ≤ |v|≤2/t M (v)dv ∼ 2 3 |B 3 | (2π) 3/2 1 t 3 . Hence R 3 |II| p M (v)dv = O 1 t 3 . ( 13
)
On the other hand

R 3 |I| p M (v)dv ≤ R 3 C1 t>τx,v (t -τ x,v + 1) p M (v)dv ≤ |v|<ε C ((t -2 |v| ) + + 1) p M (v)dv + |v|≥ε C ((t -2 |v| ) + + 1) p M (v)dv ≤ C |v|<ε M (v)dv + C ((t -2 ε ) + + 1) p |v|≥ε M (v)dv ≤ C ′ ε 3 + C ′ ( t 2 + 1) p for some other constant C ′ > 0, provided that t ≥ 4/ε. Choosing ε = 4/t < 1, we arrive at R N |I| p M (v)dv ≤ 4 3 C ′ t 3 + C ′ t p . ( 14
)
Putting together ( 13)-( 14) entails the first estimate in the theorem. The second follows from the first and interpolation with the maximum principle for the solution g of ( 12). Finally, the fact that

√ 2πµ ∞ = 1 |Ω| Ω×R 3 f in (|x|, |v|)dxdv
follows from the conservation of mass for the solution of [START_REF] Tsuji | Relaxation of a Free-Molecular Gas to Equilibrium Caused by Interaction with Vessel Wall[END_REF].

Notice that the decay requirements on the renewal kernel in Feller's Theorem 4 [START_REF] Feller | On the integral equation of renewal theory[END_REF] forbid extending our proof to space dimensions less than 3. In the one-dimensional case, Yu obtained in [START_REF] Yu | Stochastic formulation for the initial boundary value problems of the Boltzmann equation[END_REF] an algebraic rate of convergence for the outgoing flux (specifically, of order t -1/10 ), by probabilistic arguments significantly more involved than the above proof.

The monokinetic case: exponential decay

The discussion in section 3 shows the existence of too many slow particles is the reason for the slow return to equilibrium in the case of a collisionless gas in a container with constant wall temperature. In the present section, we confirm this effect by proving that monokinetic populations of particles in the same conditions converge to equilibrium exponentially fast.

Radiative transfer provides a natural example of a monokinetic transport process. Consider the following (academic) problem: at time 0, a source emits radiation in an empty container surrounded by infinitely opaque material. Photons propagate at the speed of light, and are instantaneously thermalized on the wall. Denoting I ≡ I(t, x, ω, ν) the radiative intensity at time t, position x, in the direction ω and with frequency ν, one arrives at the following initial-boundary value problem:

           1 c ∂ t I + ω • ∇ x I = 0 , x ∈ Ω , |ω| = 1 , ν > 0 , I(t, x, ω, ν) = B ν (θ I (t, x)) , x ∈ ∂Ω , ω • n x < 0 , ν > 0 , I t=0 = I in . (15)
Here c > 0 is the speed of light in vacuum, Ω is a bounded open set in R 3 with C 1 boundary, n x is the outward unit normal at x ∈ ∂Ω. The notation B ν (θ) designates Planck's function for the radiative intensity emitted by a black body at temperature θ, i.e.

B ν (θ) = 2hν 3 c 2 1 e hν/kθ -1 ,
where h and k are respectively the Planck and the Boltzmann constants. The temperature θ I is defined by

S 2 ×R+ I(t, x, ω, ν)(ω • n x ) + dωdν = S 2 ×R+ B ν (θ I (t, x))(ω • n x ) + dωdν =: σθ I (t, x) 4
where σ is the Stefan-Boltzmann constant

σ := 2πk 4 c 2 h 3 ∞ 0 x 3 dx e x -1 = 2π 5 k 4 15c 2 h 3 .
By definition of θ I , the net flux of energy at each point of the wall is

S 2 ×R+ I(t, x, ω, ν)ω • n x dωdν = 0 ,
so that the total energy in the container is conserved: for each t ≥ 0, one has

Ω×S 2 ×R+ I(t, x, ω, ν)dxdωdν = Ω×S 2 ×R+ I in (x, ω, ν)dxdωdν .
Therefore, one expects that, as t → +∞, the radiation inside the container thermalizes so that

I(t, x, ω, ν) → B ν (θ ∞ ) as t → +∞ , with 4σ|Ω|θ 4 ∞ := 4π|Ω| ∞ 0 B ν (θ ∞ )dν = Ω×S 2 ×R+ I in (x, ω, ν)dxdωdν .
In this section, we seek the rate of convergence to equilibrium for the problem above.

Theorem 5.1. Assume that I in is smooth and satisfies supp(I in ) ⊂ Ω × R + together with the bound 0 ≤ I in (x, ν) ≤ B ν (Θ) with Θ > 0, for all x ∈ Ω and ν ≥ 0.

Then, the solution I of the initial boundary value problem [START_REF] Yu | Stochastic formulation for the initial boundary value problems of the Boltzmann equation[END_REF] satisfies

I(t, x, ω, ν) -B ν (θ ∞ ) = O(e -α ′ t ) as t → +∞, uniformly on Ω × S 1 × R + , for each α ′ < α, where θ ∞ ≥ 0 is given by θ 4 ∞ := π σ|Ω| Ω×S 2 I in (|x| 2 , ν)dxdν while α is α = inf -ℜ(z) | z ∈ C * such that 1 2 2 0 e -sz sds = 1 .
Proof. The problem is simplified by considering the radiative intensity integrated over frequencies, the grey intensity u defined as

u(t, x, ω) := ∞ 0 I(t, x, ω, ν)dν .
The formula defining θ I reads

S 2 u(t, x, ω)(ω • n x ) + dω = σθ I (t, x) 4 while ∞ 0 B ν (θ I )dν = σ π θ 4
I . Therefore, choosing for simplicity units of time and space so that c = 1, the grey intensity u satisfies

                 ∂ t u + ω • ∇ x u = 0 , x ∈ Ω , |ω| = 1 , u(t, x, ω) = 1 π S 2 u(t, x, ω ′ )(ω ′ • n x ) + dω ′ x ∈ ∂Ω , ω • n x < 0 , u t=0 = u in := ∞ 0 I in dν . (16)
Henceforth, we assume the same geometric setting as in section 4: Ω = B(0, 1) ⊂ R 3 , and I in is radial in x and isotropic, meaning that I in ≡ I in (|x| 2 , ν). In that case, reasoning as in Lemma 4.2 shows that

f (t) := 1 π S 2 u(t, x, ω ′ )(ω ′ • n x ) + dω ′
is independent of the position x ∈ ∂Ω.

Proceeding as in section 4, we derive a renewal integral equation for f . Indeed, integrating the transport equation in ( 16) by the method of characteristics, we see that

u(t, x, ω) = 1 t≤τx,ω u in (|x -tω| 2 ) + 1 t>τx,ω f (t -τ x,ω ) (17)
whenever x ∈ ∂Ω with ω • n x > 0, so that

f (t) = 1 π S 2 u(t, x, ω)(ω • n x ) + dω = 1 π S 2 1 t≤τx,ω u in (|x -tω| 2 )(ω • n x ) + dω + 1 π S 2 1 t>τx,ω f (t -τ x,ω )(ω • n x ) + dω = S(t) + t 0 K(s)f (t -s)ds .
Here

t 0 K(s)f (t -s)ds = 1 π S 2 1 t>τx,ω f (t -τ x,ω )(ω • n x ) + dω = 1 π π/2 0 1 t>2 cos θ f (t -2 cos θ) cos θ • 2π sin θdθ = 2 0 1 t>s f (t -s) 1 2 sds = t 0 K(s)f (t -s)ds so that K(s) = 1 2 s1 s<2 , while S(t) := 1 π S 2 1 t≤τx,ω u in (|x -tω| 2 )(ω • n x ) + dω = 1 π S 2 1 t≤τx,ω u in (1 + t 2 -2tx • ω)(ω • n x ) + dω = 1 π π/2 0 1 t≤2 cos θ u in (1 + t 2 -2t cos θ) cos θ • 2π sin θdθ = 1 2 2 0 1 t≤s u in (1 + t 2 -ts)sds = +∞ 0 u in (1 -tτ )K(t + τ )dτ .
Since u in ∈ L ∞ (Ω), and K is bounded with support in [0, 2], so is S and one has ∞ 0 t m K(t)dt < +∞ and S(t) = O(t -m ) as t → +∞ for each m ≥ 1 .

Applying Theorem 4 of [START_REF] Feller | On the integral equation of renewal theory[END_REF] already implies that

f (t) → f ∞ := ∞ 0 S(t)dt ∞ 0 tK(t)dt as t → +∞ ,
and that f (t)f ∞ is rapidly decaying as t → +∞:

|f (t) -f ∞ | = O(t -m ) as t → +∞ for each m ≥ 1 .
We improve this estimate and obtain exponential convergence, as follows. Given φ ∈ L 2 (R + ), we denote its Laplace transform by φ(z) := ∞ 0 e -zt φ(t)dt .

We recall that the Laplace transform φ → φ is an isomorphism from L 2 (R + ) to the Hardy space H 2 (Π 0 ), where Π a := {z ∈ C | ℜ(z) > a}. We recall that an element of H 2 (Π a ) is a holomorphic function ψ ≡ ψ(z) on the half-plane Π a that satisfies

sup x>a R |ψ(x + iy)| 2 dy < +∞ .
Applying the Laplace transform to both sides of the renewal equation

f (t) = S(t) + t 0 K(s)f (t -s)ds , t ≥ 0 (18)
satisfied by f , we see that f (z)(1 -K(z)) = S(z) , ℜ(z) > 0 .

Since K ∈ L ∞ (R) with support in [0, 2], the Laplace transform of K is entire holomorphic; it is given by

K(z) = 1 -e -2z (1 + 2z) 2z 2 , z = 0 , K(0) = 1 .
Since K′ (0) = -4 3 , the only zero of 1 -K near z = 0 is 0. Besides

| K(z)| = ∞ 0 e -tℜ(z) K(t)dt < 1 whenever ℜ(z) > 0 , since K is a probability distribution on R + . Furthermore ℜ( K(iy)) = ∞ 0 cos(ty)K(t)dt < ∞ 0 K(t)dt = 1
so that z = 0 is the only zero of 1 -K on the imaginary axis. Finally, for each a ∈ R, one has

| K(z)| ≤ 1 2|z| 2 (1 + (1 + 2|z|)e -2ℜ(z) ) . (20)
In particular | K(z)| → 0 as ℑ(z) → +∞ uniformly in ℜ(z) ≥ a. Therefore, there exists α > 0 such that 1 -K(z) = 0 and ℜ(z) > -α ⇔ z = 0 .

Hence z → z 1-K(z) is holomorphic on the half-plane Π -α , and one recasts (19) as

z f (z) = z S(z) 1 -K(z) , ℜ(z) > -α .
Indeed, since S ∈ L ∞ (R) with support in [0, 2], its Laplace transform is an entire holomorphic function. Moreover, assuming that u in ∈ C 1 c ((0, 1)), we deduce from the formula

S(t) = ∞ t u in (1 + t 2 -ts)K(s)ds that S ∈ C 1 (R + ), with S ′ (t) = ∞ t (u in ) ′ (1 + t 2 -ts)(2t -s)K(s)ds
so that, for all t ≥ 0, one has

|S ′ (t)| ≤ 2t (u in ) ′ L ∞ ∞ t K(s)ds ≤ 4 (u in ) ′ L ∞ .
Since S(0) = u in (1) = 0, one has z S(z) = S ′ (z); and since S ′ is continuous with compact support on R, its Laplace transform S ′ ∈ H 2 (Π a ) for each a ∈ R.

In view of (20), for each α ′ < α, the function z →

(1 -K(z)) -1 is bounded on Π -α ′ \ B(0, α ′ /2), so that the function z → z S(z) 1 -K(z) belongs to H 2 (Π -α ′ )
for each α ′ < α. Since e at φ = φ(za), this implies the existence of a function g such that e α ′ t g ∈ L 2 (R + ) for each α ′ < α and

g(z) = z S(z) 1 -K(z) , whenever ℜ(z) > -α ′ .
Consider the function G defined by

G(t) = t 0 g(s)ds , t ≥ 0 . Then G is continuous on R with support in R + , satisfies 0 ≤ |G(t)| ≤ e α ′ t g L 2 √ 2α ′ , t ≥ 0 , so that G belongs to H 2 (Π η ) for each η > 0. Now, for each z ∈ Π η z G(z) = ∞ 0 ze -zt t 0 g(s)dsdt = ∞ 0 g(s) ∞ s ze -zt dtds = ∞ 0 g(s)e -zs ds = g(z) = z S(z) 1 -K(z) = z f (z) .
In other words, G and f are bounded continuous functions on R supported in R + , whose Laplace transforms coincide on Π 0 . Therefore

f (t) = G(t) = t 0 g(s)ds , for each t ≥ 0 ,
and since e α ′ t g ∈ L 2 (R + ) for each α ′ < α, one conclude that

|f (t) -f ∞ | = +∞ t |g(s)|ds ≤ e α ′ t g L 2 +∞ t e -α ′ s ds 1/2 = e α ′ t g L 2 √ 2α ′ e -α ′ t ≤ C α ′ e -α ′ t
for each t > 0.

Going back to the original problem [START_REF] Yu | Stochastic formulation for the initial boundary value problems of the Boltzmann equation[END_REF] or ( 16), we deduce from the method of charateristics (17) that

u(t, x, ω) = f (t) , whenever t > diam(Ω) ,
or equivalently, in the frequency dependent case,

I(t, x, ω, ν) = B ν ( π σ f (t)) 1/4 , whenever t > diam(Ω) c , since τ x,ω ≤ diam(Ω).
Before concluding, it is worth discussing the role of the monokinetic assumption in the exponential decay estimate obtained here.

Intrinsically, the integral kernel K(s)ds in the proof of Theorem 5.1 is the image under ω → τ x,ω of the probability measure proportional to (ω • n x ) + dω on S 2 . Since 0 ≤ τ x,ω ≤ 2 = diam(Ω), this kernel K is supported in [0, 2]. Therefore its Laplace transform is holomorphic entire, and since solving the renewal integral equation is equivalent to the possibility of inverting 1-K, the only constraint is the potential presence of zeros of 1 -K; besides, since K has integral 1, the half-plane ℜ(z) ≥ 0 is zero-free, except for z = 0, and zeros of 1 -K cannot accumulate on the imaginary axis. Hence the solution of the renewal equation is obtained by inverting the Laplace transform, deforming the contour of integration to be ℜ(z) = -α ′ for some α ′ > 0.

In the case of the collisionless gas in Theorem 4.1, the integral kernel K(s)ds is the image under v → τ x,v of the probability measure proportional to (v•n x ) + M (v)dv on R 3 . Because of the low-speed gas molecules, its behavior near s = 0 is the same as that of the image measure of, say, (v 1 ) + dv on the unit ball B(0, 1) under v → τ x,v -the Maxwellian weight M (v) obviously plays no role near v = 0. Thus, the decay of K for s → +∞ only depends on the Jacobian weight |v| N -1 that appears when integrating radial functions with the Lebesgue measure in R N . This results in only algebraic decay for K in the limit s → +∞, so that the Laplace transform of K does not have a holomorphic extension in any half-plane of the form ℜ(z) > -α with α > 0. Therefore, the same deformation of contour as in the proof of Theorem 5.1 is impossible in that case. In order to be able to push the integration contour in the Laplace inversion formula to the left of the imaginary axis, K should decay exponentially fast as s → +∞. This would be the case if the Jacobian weight |v| N -1 appearing in the integration of radial functions with the Lebesgue measure in space dimension N could be replaced with e -a/|v| for some a > 0. Thus, the difference between the speed of approach to equilibrium observed in a collisionless gas with diffuse reflection of gas molecules on vessel walls kept at constant temperature, and the spectral gap property for the linear Boltzmann equation on the other hand, can be measured in terms of the vanishing order of |v| N -1 and e -a/|v| as |v| → 0.

Conclusion

We have discussed in this paper the speed of approach to equilibrium for a collisionless gas enclosed in a vessel whose wall is kept at a uniform and constant temperature, assuming diffuse reflection on the vessel wall. We propose lower bounds for this convergence rate, depending on the possibility of initial distribution functions that are concentrated at low particle speeds. Not only do these lower bounds exclude exponential decay to equilibrium, they also constrain the power in any potential algebraic decay rate. On the other hand, assuming spherical symmetry of the initial velocity distribution function and of the vessel, we establish an algebraic decay rate by reducing the problem to a renewal equation for the flux of outgoing gas particles. All these results are consistent with the asymptotic behavior obtained in [START_REF] Tsuji | Relaxation of a Free-Molecular Gas to Equilibrium Caused by Interaction with Vessel Wall[END_REF] with numerical simulations. To clarify the role of low-speed particles, we have studied the case of radiative transfer in the vacuum, in a container with infinitely opaque boundary, so that the total incoming radiation flux on the boundary is reemitted as a Planck distribution. We establish exponential convergence of the radiative intensity to a Planck distribution in the long time limit, assuming the same spherical symmetries as in the case of a collisionless gas.

There remain several open problems in this direction, such as a) establishing decay rates without any symmetry assumptions, and b) obtaining the optimal decay rates, even in the spherically symmetric case. Both problems might involve mathematical techniques quite different from the ones used here. We hope to return to these questions in subsequent publications.

  where ν(|v|) is a multiplication operator by the collision frequency satisfying the inequality ν * (1 + |v|) ≤ ν(|v|) ≤ ν * (1 + |v|) for some positive constants ν * and ν * . Proposition 3.3 for p = 2 rules out the possibility of a similar spectral gap property for the problem (1).

Acknowledgements. We express our gratitude to Claude Bardos and Laurent Desvillettes for valuable discussions about the problem considered here, and to the Isaac Newton Institute for Mathematical Sciences for its support and hospitality during the preparation of this paper. This work was partially supported by grantin-aid for scientific research No. 20360046 from JSPS.