Paul Maurice

Alexandra Letrouneau

Alexandra Benachi

Jean-Marie Jouannic
email: jean-marie.jouannic@aphp.fr

Feticide in second-and third-trimester termination of pregnancy for fetal anomalies: results of a national survey Running Head: Feticide in second and third trimester TOPFA

Keywords: termination of pregnancy for fetal anomalies, feticide, second-and third trimester, fetal analgesia

What's already known about this topic?

Although the WHO recommends feticide for elective termination of pregnancy (TOP) after 20 weeks, in France, as in most other European countries, there is no legal obligation to perform this procedure in the framework of TOP for fetal anomalies.

What does this study add?

First study on a national scale of feticide in the setting of termination of pregnancy for fetal anomalies showing that most TOP performed after 22 weeks in France are associated with feticide, usually after fetal anesthesia.

Introduction

The growth of prenatal diagnosis over the last 3 decades has greatly increased the rate of prenatal detection of fetal anomalies. This rate varies among European countries, but ranges between 30 and 80% according to data from the European registers [START_REF] Garne | Prenatal diagnosis of severe structural congenital malformations in Europe[END_REF][START_REF] Ferrier | Trends in resource use and effectiveness of ultrasound detection of fetal structural anomalies in France: a multiple registry-based study[END_REF][START_REF] Braz | The impact of prenatal diagnosis on congenital anomaly outcomes: Data from 1997 to 2016[END_REF] . Certain fetal anomalies can lead to the organization of specific perinatal management, sometimes with the possibility of specific fetal therapy [START_REF] Adzick | A randomized trial of prenatal versus postnatal repair of myelomeningocele[END_REF][START_REF] Zwiers | The near disappearance of fetal hydrops in relation to current state-of-the-art management of red cell alloimmunization[END_REF] . In most European countries, termination of pregnancy for fetal anomalies (TOPFA) may be requested by the parents when the anomalies are severe, considered incurable, and associated with a high risk of severe disability. However, the legislative provisions, notably concerning a time limit beyond which TOPFA is no longer authorized, vary from one country to another.

French law authorizes TOPFA when there is a strong probability that the fetus has a particularly serious anomaly recognized as incurable at the time of diagnosis [START_REF]Code de la santé publique -Article L2213-1[END_REF] .

In France, there is no gestational age cut-off. When a pregnant woman opts for TOPFA, two fetal medicine specialists who are members of a prenatal diagnosis (PD) center must attest that the fetal anomaly is covered by the legal provisions in force.

Although the World Health Organization recommends feticide for elective termination of pregnancy after 20 weeks, in France, as in most other European countries, there is no legal obligation to perform this procedure in the framework of TOPFA. Several studies have examined the technical aspects of feticide, notably concerning the lethal substances used and their route of administration, ie, fetal or intra-amniotic injection [START_REF] Senat | Funipuncture for fetocide in late termination of pregnancy[END_REF][START_REF] Senat | The use of lidocaine for fetocide in late termination of pregnancy[END_REF][START_REF] Evans | Efficacy of second-trimester selective termination for fetal abnormalities: international collaborative experience among the world's largest centers[END_REF][START_REF] Sharvit | Intra-amniotic digoxin for feticide between 21 and 30 weeks of gestation: a prospective study[END_REF][START_REF] Pasquini | Intracardiac injection of potassium chloride as method for feticide: experience from a single UK tertiary centre[END_REF] . Some studies have examined the psychological experiences of both the parents and the healthcare professionals involved [START_REF] Garel | French midwives' practice of termination of pregnancy for fetal abnormality. At what psychological and ethical cost?[END_REF][START_REF] Dommergues | Feticide during second-and thirdtrimester termination of pregnancy: opinions of health care professionals[END_REF][START_REF] Alsulaiman | Attitudes to prenatal diagnosis and termination of pregnancy for 30 conditions among women in Saudi Arabia and the UK[END_REF] .

Expert opinions have also been formulated concerning the ethical issues associated with whether or not feticide should be performed in the setting of TOP, whatever the indications. In contrast, no study to date has reported the incidence of feticide in the setting of TOPFA. Such an overview of the practice of feticide could serve as a basis for reflection by healthcare professionals and governmental decision-makers on standardization of practices and on the organization of TOPFA.

The main objective of our study was to describe the practice of feticide in TOPFA during the second and third trimesters of pregnancy in (PD) centers in France. The secondary objective was to study methods of feticide and why it may not be performed.

Material and methods

The Fédération Française des Centres Pluridisciplinaires de Diagnostic Prénatal conducted a national questionnaire survey in 2018 concerning TOPFA performed in 2017. The questionnaire was drawn up by a committee appointed by the Federation that comprised 8 center coordinators representative of physicians involved in TOPFA, taking into account the gender and age of the physicians and the number of TOPFA performed in their center, while ensuring a balanced geographical distribution throughout France. This questionnaire, which was approved by means of a consensus of all 8 coordinators, was limited to feticide in a setting of TOPFA, and excluded termination of pregnancy for maternal reasons (life-threatening to pregnant woman). The questions related notably to whether or not feticide was universally performed, the gestational age at which feticide was practiced, the timing of feticide in TOPFA, and the technical modalities of feticide: site of injection, use of prior fetal anesthesia, type of lethal substance used. The questionnaire also included questions on the existence of a protocol at the center, the medical resources available for ultrasound-guided termination of pregnancy, and the number of TOPFA done at the centers during the previous year and reported to the Agence de la biomédecine, the public body charged with oversight of the CPNs. Lastly, the questionnaire recorded the reasons why some centers do not practice feticide universally. After the questionnaire was sent out, two follow-ups were sent over a period of 6 months. A member of the administrative staff of the Fédération Française des Centres Pluridisciplinaires de Diagnostic Prénatal collected and anonymized the questionnaire answers.

Results

Of 49 PD centers contacted, 39 completed the questionnaire (80%). A total of 5350 TOPFA were done by these 39 centers in the year corresponding to the survey (2017). According to data collected by the Agence de la biomédecine for this same year, a total of 6938 TOPFA were reported in France as a whole. Our survey therefore covered 77% of the TOPFA performed. The median [range] number of TOPFA performed in these centers was 121 [36-339]. The gestational age after which feticide was performed was 20 weeks in 2 centers (5%), 22 weeks in 28 centers (72%), 23 weeks in 4 centers (10%), and 24 weeks in 5 centers (13%). In each center, the median (extreme) number of experienced operators with ability to perform feticide was 4 (1-8). Of the 39 centers, 24 (62%) reported systematic use of feticide beyond the gestational age chosen by the center (Table 1). Fifteen of the 39 centers reported non-systematic use of feticide, the reason being a fetal anomaly with a strong probability of rapid neonatal death (13 centers), refusal by the pregnant woman (11 centers), and a technical impossibility of performing feticide (1 center). In all, 23 TOPFA were performed without feticide during the survey period, but this information was not provided by 17/39 centers. The lethal substance used for feticide was 1% or 2% xylocaine in 38/39 centers and KCl in one center. All but one centers used fetal anesthesia before feticide. The center that did not use fetal anesthesia universally performed feticide by intracardiac injection of the lethal substance. Fetal anesthesia was achieved by injection of fentanyl or pentothal or of a mixture of the two, at a dose adapted to the ultrasound estimate of fetal weight. In 30 centers (82%), fetal anesthesia was routinely achieved by ultrasound-guided injection into the umbilical vein (Table 2). At the other centers, the injection site was intracardiac or subcutaneous. Feticide was done on the day of induction of labor in 35/39 centers (90%), after maternal epidural analgesia in 33 centers, and after maternal subcutaneous anesthesia in 2 centers. Feticide was done the day before the induction of labor, after local maternal anesthesia, in 2 centers (5%).

All CPNs reported that they worked in a network with maternity units outside the tertiary institution of which they are a part, with the possibility of organizing TOPFA in these maternity units. However, 19/39 centers (49%) reported that TOPFA were only performed in the maternity unit of the tertiary institution to which the CPN was attached when the feticide was done, because the other maternity units did not have staff with the technical skills required for this ultrasound-guided procedure (Table 1). All but one of the CPNs that completed the questionnaire had the medical resources required for emergencies 24/7 in the case of unexpected labor when TOPFA is scheduled with feticide.

Discussion

Ours is the first French study describing of the use of feticide in TOPFA. Its results suggest that in France most TOPFA during the second and third trimesters of pregnancy are practiced after fetal anesthesia and feticide. In most centers that completed our questionnaire, feticide is almost always done after 22 weeks of gestation, and very few TOPFA are organized without feticide after 22-24 weeks.

While the World Health Organization recommends feticide for late TOP, in

France, unlike other countries, there is no legal obligation to perform feticide. Yet, feticide appears to be ethically justified when it prevents neonatal demise associated with a risk of suffering or pain [START_REF] Spielman | Certainty and agnosticism about lethal injection in late abortion[END_REF][START_REF] Callahan | Ensuring a stillborn: the ethics of fetal lethal injection in late abortion[END_REF][START_REF] Isada | Fetal intracardiaque potassium chloride injection to avoid the hopeless resuscitation of an abormal abortus: I. Clinical issues[END_REF][START_REF] Fletcher | Fetal intracardiaque potassium chloride injection to avoid the hopeless resuscitation of an abormal abortus: II. Ethical issues[END_REF] . Studies in the literature on late TOP concern in large part second-trimester TOP, and notably TOP performed according to the wishes of the parents, in the context of a normal pregnancy. These TOP are authorized in several countries up until 24 weeks, as in the UK. Given that almost all these TOP are performed using a surgical technique, the question of fetal survival does not arise. The context of TOP for fetal anomalies is different because TOP is legal after 22 weeks, notably in several European countries. In TOPFA, in most cases, it may be helpful to perform a postmortem examination of the fetus to define the fetal disease and to facilitate genetic counseling for subsequent pregnancies. It is against this backdrop of induction of labor that feticide is posed. However, there is the question of its acceptability to pregnant women and to healthcare professionals. Most studies of feticide have focused on its technical aspects: route of fetal injection, type of lethal substance used, and so forth [START_REF] Senat | Funipuncture for fetocide in late termination of pregnancy[END_REF][START_REF] Senat | The use of lidocaine for fetocide in late termination of pregnancy[END_REF][START_REF] Evans | Efficacy of second-trimester selective termination for fetal abnormalities: international collaborative experience among the world's largest centers[END_REF][START_REF] Sharvit | Intra-amniotic digoxin for feticide between 21 and 30 weeks of gestation: a prospective study[END_REF] . These studies mainly concentrated on maternal risks. It was thus shown that injection in the umbilical vein involved the least risk to the pregnant woman [START_REF] Senat | Funipuncture for fetocide in late termination of pregnancy[END_REF] . It seems that xylocaine is associated with the lowest maternal risk and achieves fetal asystole in most cases [START_REF] Senat | The use of lidocaine for fetocide in late termination of pregnancy[END_REF] . Some authors consider that intracardiac injection is surer, and in some cases failure of injection in the umbilical cord has led to the birth of live infants [START_REF] Pasquini | Intracardiac injection of potassium chloride as method for feticide: experience from a single UK tertiary centre[END_REF][START_REF] Gill | Induction of fetal demise in advanced pregnancy terminations: report on a funic potassium chloride protocol[END_REF][START_REF] Bhide | Comparison of feticide carried out by cordocentesis versus cardiac puncture[END_REF] .

It is possible, moreover, that such failures are underreported in the literature. Even if exceptional, such failures expose the parents, and also the healthcare professionals delivering these neonates, to the risk of psychological trauma. To avoid such situations, it seems prudent to recommend testing for fetal asystole using ultrasound, at least 10-15 minutes after feticide by injection in the umbilical cord.

Although little discussed in the literature, feticide requires experience of invasive, ultrasound-guided procedures. This may be a limiting factor in the organization of some teams. In our study, 19/39 centers reported that TOPFA requiring feticide was performed in the maternity unit of the PD center because other maternity units in the region had no physician able to perform the procedure.

Interestingly, the intra-amniotic injection of digoxin in TOP has been reported to achieve fetal demise within about 24 hours [START_REF] Sharvit | Intra-amniotic digoxin for feticide between 21 and 30 weeks of gestation: a prospective study[END_REF] . This injection is technically simpler than injection in the umbilical cord or intracardiac injection. In a recent study, this technique achieved fetal demise in up to 93% of reported cases [START_REF] Sharvit | Intra-amniotic digoxin for feticide between 21 and 30 weeks of gestation: a prospective study[END_REF] . The risk of failure seemed to be greater in the case of maternal obesity. The results of this study are interesting as they could argue in favor of easier access to this procedure, notably for couples living far from tertiary centers. However, the precise time lapse between intra-amniotic injection and fetal death is unknown. Our study shows that most French CPNs perform fetal anesthesia before feticide in accordance with the guidelines of the French College of Obstetricians and Gynecologists [START_REF]Interruption médical de grossesse[END_REF] . Interestingly, fetal pain is little considered in studies on late TOP.

The strength of our study is that fetal medicine in France is organized in approved specialized centers, which were set up following the bioethics laws of 1994.

At the time of our study, there were 49 such centers throughout mainland France and French overseas territories. The expert members of these centers are accredited by the Agence de la biomédecine, the public body that has oversight of the running and organization of these 49 centers. This facilitates centralization of the organization of TOPFA. French legislation imposes no limit on the term of gestation for TOPFA when there is a strong probability that the unborn child has a serious disease acknowledged to be incurable, at the time of diagnosis. Even though legal provisions differ from one country to another, late TOPFA is authorized in several countries, notably in Europe.

The main limitation of our study is that not all the PD centers contacted completed our questionnaire. Those that did, however, accounted for 77% of TOPFA.

In addition, our figures are in line with the proportion of feticides reported in the EPIPAGE cohort, which studied TOP prevalence and indications in 7804 premature births (between 22 weeks and 31 weeks plus 6 days) [START_REF] Monier | Indications leading to termination of pregnancy between 22+0 and 31+6 weeks of gestational age in France: A population-based cohort study[END_REF] . In this study, for TOPFA, the rate of feticide was 80% at 24 weeks of gestation and over 90% from 26 weeks. Moreover, among the 11/49 centers that failed to respond in our study, there was no regional or workload disparity that could have introduced bias. Another limitation of our survey is that it was self-report in nature, which could be a source of bias.

However, most of the information corresponds to data that must be reported every year to the Agence Nationale de la Biomédécine, which regularly conducts quality controls of transmitted data.

Ours is the first study on a national scale of feticide in the setting of TOPFA. It shows that most TOPFA performed most TOPFA performed in second-and third trimester are associated with feticide, usually after fetal anesthesia administered by injection in the umbilical cord.

Table 1 :

 1 Organization of feticide during second-and third-trimester termination of pregnancies among the 39 prenatal diagnosis centers

	Yes

Table 2 :

 2 Site of injection of the fetal anesthesia and lethal substance among the 39 prenatal diagnosis centersUmbilical cord

	Intracardiac Subcutaneous Depending on
	the conditions

The data that support the findings

Conflict of interest: none

Funding: none


M Hongisto

A Veber

N G Boetti

S Danto

V Jubera

L Petit

Transparent Yb 3+ doped phosphate glass-ceramics

Keywords: Phosphate glass, Glass-ceramic, Yb 3+, Luminescence, XRD

Yb 3+ doped oxyfluorophosphate glasses with the composition (98.75) [90NaPO3-(10-x) Na2O-xNaF] -1.25Yb2O3 (in mol%) with x = 0, 2.5, 5, 7.5 and 10 were prepared using a standard melting process. The progressive replacement of Na2O by NaF leads to an increase in the number of Q 2 units at the expense of the Q 1 units. This increase in the polymerization of the glass network leads to a shift of the optical band gap to lower wavelength, to a slight increase in the intensity of the emission at 1000 nm and more importantly to a change in the glass crystallization process. Indeed, both surface and bulk crystallization were observed in the glass with x = 0 while surface crystallization only occurs when NaF is added in the phosphate network. The heat treatment leads to the precipitation of at least three crystalline phases: as x increases, the NaPO3 phase grows at the expense of Na5P3O10. All glasses precipitate the Yb containing crystal, NaYbP2O7 which leads to an increase in the intensity of the emission at 1000 nm compared to the emission at 975 nm. We show for the first time to the best of our knowledge that transparent Yb 3+ doped phosphate glass-ceramics can be obtained within this glass system when free of NaF.

Introduction

Materials doped with Yb 3+ have been of great interest as Yb 3+ has a very simple energy level structure consisting of the ground ( 2 F7/2) and excited ( 2 F5/2) states. As these two states are well separated, multiphonon relaxation and excited state absorption (ESA) are limited. Yb 3+ doped materials can find applications in solid-state lasers [START_REF] Hönninger | Ultrafast ytterbium-doped bulk lasers and laser amplifiers[END_REF], light emitting diodes [START_REF] Zhu | Eu-, Tb-, and Dy-doped oxyfluoride silicate glasses for LED applications[END_REF], and more recently in solidstate laser induced cooling [START_REF] Epstein | Observation of laser-induced fluorescent cooling of a solid[END_REF], just to cite a few examples. The Yb 3+ doped materials can be pumped with wavelength in the range of ~0.9-1.0 μm using diode lasers, which are commercially available with high power. The emission then takes place in the ~1 μm wavelength region. Additionally, the efficiency of Yb doped fiber is not degraded due to concentration quenching and excited state absorption for example, as high Yb 3+ doping can be used without a reduction of the upper-state lifetime.

Although silicate glasses remain the most commonly used glass hosts for Yb 3+ , efforts have been focused for the past few decades on phosphate glasses due to their unique characteristics which can include, depending on their composition, low melting point, high thermal stability, high transparency, high gain density that is due to high solubility for lanthanide ions, low refractive index and low dispersion as compared to other glasses such as silicate glasses [START_REF] Jiang | Ytterbium-doped phosphate laser glasses[END_REF][START_REF] Pugliese | Concentration quenching in an Er-doped phosphate glass for compact optical lasers and amplifiers[END_REF]. Yb 3+ ions in phosphate glasses can also possess high emission cross-section, broad absorption and emission bands and long fluorescence lifetime (1-2 ms) [START_REF] Hughes | BarLaser diode pumped solid state lasers[END_REF]. Fluoride glasses are also good Yb 3+ hosts due to their low phonon energy when compared with oxide glasses. However, their use is limited due to their potential poor mechanical and chemical resistance. Therefore, oxyfluoride glasses have been under investigation as they possess higher chemical stability and better mechanical properties than fluoride glasses and lower phonon energy than oxide glasses [START_REF] Goncalves | Rare-earth-doped transparent glass ceramics[END_REF]. In addition, some oxyfluoride glasses can produce transparent glass-ceramics (GCs) containing fluorite nanocrystals after heat treatment [START_REF] Hayashi | 1.4 μ m band emission properties of Tm 3+ ions in transparent glass ceramics containing PbF2 nanocrystals for S-band amplifier[END_REF].

A GC contains at least one type of functional crystalline phase and a residual glass. The first glass-ceramic (GC) was reported in the 1960s [START_REF] Stookey | Catalyzed crystallization of glass in theory and practice[END_REF]. The GCs are usually prepared using a two-step process: the first step being the preparation of the glass. In the second step, the glass is heat treated under specific heating conditions in order to create the nuclei and to grow them into crystals. The heat treatment can lead to surface and/or volume precipitation of the crystals. To be considered as promising optical materials, the GCs should be composed of crystals homogeneously distributed in the volume glass matrix. The crystals should also contain the rareearth ions in order to enhance the spectroscopic properties of the GCs compared to the parent glass [START_REF] Wang | New transparent vitroceramics codoped with Er 3+ and Yb 3+ for efficient frequency up-conversion[END_REF]. Additionally, the GCs should also be transparent. Therefore, it is crucial to tailor both the crystal size and the refractive index difference between the crystals and the glass in order to minimize light scattering. Indeed, to ensure the optical transparency of the glass-ceramic, the size of the crystals needs to be smaller than the incident light wavelength [START_REF] Beall | Nanophase glass-ceramics[END_REF]. The first transparent oxyfluoride glass-ceramic with enhanced luminescence properties was reported in 1993 [START_REF] Wang | New transparent vitroceramics codoped with Er 3+ and Yb 3+ for efficient frequency up-conversion[END_REF].

The crystallization behavior of glasses within the system P2O5-SrO-Na2O was reported in Ref. [START_REF] Massera | Crystallization behavior of phosphate glasses and its impact on the glasses' bioactivity[END_REF] and within the system P2O5-SrO-Na2O-Al2O3-Y2O3-Yb2O3 in Ref. [START_REF] Sen | Optical, structural and luminescence properties of oxyfluoride phosphate glasses and glass-ceramics doped with Yb 3+[END_REF]. Although the heat treatment of the glasses in these systems leads to surface crystallization of multiple crystals, the heat treatment of the Yb 3+ doped glass within the system P2O5-SrO-Na2O-Al2O3-Y2O3 leads to the precipitation of the NaYbP2O7 crystals increasing the excited state 2 F5/2 lifetime of Yb 3+ and also the bandwidth of the Yb 3+ emission band centered at 1 μm [START_REF] Sen | Optical, structural and luminescence properties of oxyfluoride phosphate glasses and glass-ceramics doped with Yb 3+[END_REF]. These changes in the spectroscopic properties induced by the precipitation of NaYbP2O7 crystals are in agreement with [START_REF] Ternane | Spectroscopic properties of Yb 3+ in NaYbP2O7 diphosphate single crystals[END_REF]. Indeed, the Yb 3+ ions in NaYbP2O7 have typical fluorescence lifetime of 1-2 ms and a broad emission band at room temperature, which is different from the typical luminescence of Yb 3+ doped glasses [START_REF] Ternane | Spectroscopic properties of Yb 3+ in NaYbP2O7 diphosphate single crystals[END_REF].

Therefore, the goal of our study is to process new Yb 3+ doped oxyfluoride glasses, in which crystals similar to the NaYbP2O7 ones would precipitate in the volume upon heat treatment. Here, new Yb 3+ -doped oxyfluorophosphate glasses in the NaPO3-Na2O-NaF system were prepared and characterized. The study on the impact of replacing Na2O with NaF on the physical, thermal, optical and structural properties of Yb 3+ free glasses can be found in Ref. [START_REF] Ojha | Upconversion from fluorophosphate glasses prepared with NaYF4:Er 3+ ,Yb 3+ nanocrystals[END_REF]. We first present the spectroscopic properties of the Yb 3+ doped glasses in this system. Then, we discuss the changes in the optical properties induced by the heat treatment of the glasses.

Experimental

Glasses with the composition (98.75) [90NaPO3-(10-x)Na2O-xNaF] -1.25 Yb2O3 (in mol%) with x = 0, 2.5, 5, 7.5 and 10 were prepared using the standard melt-quench method. The batches were prepared using NaPO3 (Alfa Aesar, tech.), Na2CO3 (Sigma-Aldrich, 99.8%), NaF (Sigma-Aldrich, 99.99%) and Yb2O3 (Sigma-Aldrich, 99.9%) as the raw materials. The glasses were melted in a Pt crucible for 5min between 950 and 975 °C, depending on the glass composition. After quenching, the glasses were annealed at 200 °C for 6 h to release the stress from the quench.

The Panalytical EMPYREAN multipurpose X-Ray Diffractometer with an iron filtered cobalt Kα radiation (λ = 1.78897 Å) was used to check that the as-prepared glasses were amorphous and to identify the phases present in the glass-ceramics. The data were collected from 2Ɵ = 15°-60° in 0.026° intervals. The glasses and heat treated glasses were crushed into powder and spread over a "zero-background holder" Si-plate for the measurement of the XRD spectra.

The density of the glasses were measured using Archimedes' method. Ethanol was used as the immersion liquid. The accuracy of the measurements are ±0.02 g/cm 3 .

The thermal properties of the glasses, such as the glass transition temperature (Tg) and crystallization temperature (Tp) of the samples were measured using differential thermal analysis (DTA) using the Netszch JUPITER F1 instrument. The measurements were carried out in a Pt crucible using a heating rate of 10 °C/min. The Tg was determined as the inflection point of the endotherm obtained by taking the first derivative of the DTA curve, while Tp was taken at the maximum of the exothermic peak. The onset of the crystallization peak, Tx, was also measured. All measurements were performed with an accuracy of ±3 °C.

The absorption spectra of the polished glasses were measured using UV-Vis-NIR spectrophotometer (UV-3600 Plus, Shimadzu) from 250 to 1100 nm. The absorption cross-section σabs (λ) was calculated using Equation ( 1) :

where, log (I0/I) is the absorbance, L the thickness of the sample (in cm) and N the concentration of rare-earth ions (ions/cm 3 ). The concentration of Yb 3+ ions was calculated from the density of the glasses and expected composition. The accuracy of absorption cross-section determination was ±10%.

The emission spectra of the investigated glasses were measured from powder at room temperature using Spectro 320-131 Vis-NIR emission spectrometer (Instrument Systems, Germany). The excitation was achieved using a monochromatic single-mode fiber pigtailed laser diode (CM962UF76P-10R, Oclaro) temperature tuned to 960 nm in order to compare emission intensity between samples. Additionally, the shape of the emission was measured from the surface of the bulk glass using FLS1000, Edinburgh Instrument. The excitation was 900 nm. The shape of the emission was merged with emission intensity data in order to compare the emission shapes and intensities. The emission cross-section (σemi) of the investigated glasses was calculated using the equations [START_REF] Jiang | Tellurogermanate laser glass with high emission cross section[END_REF]:

where AR is the spontaneous emission probability, f emidλ is the integrated emission cross section , Δλeff the fluorescence effective linewidth, n is the refractive index of the medium expected to be ~1.54 [START_REF] Lopez-Iscoa | Effect of the addition of Al2O3, TiO2 and ZnO on the thermal, structural and luminescence properties of Er 3+ -doped phosphate glasses[END_REF], c is the speed of light, J', J are the total momentums for the upper and lower levels, respectively and λp is the absorption peak wavelength. The effective bandwidth of the emission band (Δλeff) was calculated using the following equation [START_REF] Jiang | Tellurogermanate laser glass with high emission cross section[END_REF]:

where I (λ) is the emission intensity and the intensity at the peak wavelength λmax.

The fluorescence lifetime of the Yb 3+ : 2 F5/2 level was measured at room temperature. The measurements were performed at the very edge of the bulk glasses in order to minimize reabsorption using a light pulses of the 915 nm laser diode. The decay traces were fitted using single exponential. The detector (PDA10CS, Thorlabs) used for this measurement was equipped with a passband filter (FB980-10, Thorlabs). Estimated error of the lifetime measurement was ±0.20 ms. The radiative lifetime (τrad) of the glasses were calculated using the following equation [START_REF] Weber | Dependence of the stimulated emission cross section of Yb 3+ on host glass composition[END_REF]:

The integration is performed over the entire transition centered at λmax.

The IR spectra were measured using a PerkinElmer Spectrum One FTIR (Fourier-transform infrared spectroscopy) spectrometer in Attenuated Total Reflectance (ATR) mode in mid infrared region 650-1400 cm -1 . The resolution of the measurement was 1 cm -1 and the spectra were obtained from the accumulation of 8 scans. Additional IR measurements were performed with a Bruker Equinox 55 FTIR spectrometer in transmission mode from 1000 cm -1 to 5000 cm -1 . The resolution of these measurements was 2 cm -1 and the spectra were obtained from 32 scans performed in an inert atmosphere.

The measurements of the Raman spectra were performed using Renishaw InVia Qontor Raman microscope, using a 532 nm laser for the excitation. The data were obtained with the accumulation of 3 scans.

Results and discussion

Structural and photoluminescence investigation

Glasses with the composition (98.75) [90NaPO3-(10-x) Na2O-xNaF] -1.25 Yb2O3 (in mol%) with x = 0, 2.5, 5, 7.5 and 10 were successfully prepared using standard melting process. The impact of the replacement of Na2O by NaF on the thermal, optical and structural properties of the Yb free glasses in the same glass system can be found in Ref. [START_REF] Ojha | Upconversion from fluorophosphate glasses prepared with NaYF4:Er 3+ ,Yb 3+ nanocrystals[END_REF]. The changes in the glass composition has no impact on the density of the Yb 3+ doped glasses (Table 1), which was consistent with the results reported in Ref. [START_REF] Ojha | Upconversion from fluorophosphate glasses prepared with NaYF4:Er 3+ ,Yb 3+ nanocrystals[END_REF], where similar results were observed without Yb 3+ . Comparing the density of our doped glasses to that of the undoped ones reported in Ref. [START_REF] Ojha | Upconversion from fluorophosphate glasses prepared with NaYF4:Er 3+ ,Yb 3+ nanocrystals[END_REF], the addition of Yb2O3 increases the density of the glass. As in Ref. [START_REF] Ojha | Upconversion from fluorophosphate glasses prepared with NaYF4:Er 3+ ,Yb 3+ nanocrystals[END_REF], an increase in x decreases the Tg due to the presence of weaker ionic cross-linking between the modifier fluoride cations and non-bridging oxygen as suggested in Ref. [START_REF] Brauer | Fluoride-containing bioactive glass-ceramics[END_REF]. One should point out that the glasses exhibit a ΔT, an empirical parameter often used for fiber drawing, greater than 90 °C, indicating that the investigated glasses have a good resistance towards crystallization [START_REF] Çelikbilek | Thermal and microstructural investigation of the TeO2-WO3 system[END_REF].

The IR and Raman spectra of the glasses are presented in Fig. 1a andb, respectively. They are similar to those of Yb 3+ free glasses reported in Ref. [START_REF] Ojha | Upconversion from fluorophosphate glasses prepared with NaYF4:Er 3+ ,Yb 3+ nanocrystals[END_REF]. The investigated glasses have a metaphosphate structure, free of Q 3 groups as suspected due to the absence of IR and Raman bands at 1300 cm -1 , typical of the ν(P O) of Q 3 groups. A complete attribution of the IR and Raman bands can be found in Ref. [START_REF] Ojha | Upconversion from fluorophosphate glasses prepared with NaYF4:Er 3+ ,Yb 3+ nanocrystals[END_REF]. The changes in the intensity of the IR and Raman bands with an increase in x are similar to those reported in Ref. [START_REF] Ojha | Upconversion from fluorophosphate glasses prepared with NaYF4:Er 3+ ,Yb 3+ nanocrystals[END_REF] indicating that an increase in x in Yb containing glasses leads to an increase in the number of Q 2 units at the expense of Q 1 units. Also, the shift of the band gap to a lower wavelength is observed as x increases (Fig. 2a). As seen in Fig. 2b, the replacement of Na2O by NaF has no impact on the shape of the absorption band of the Yb 3+ 4f -4f transition between 2 F7/2 and 2 F5/2 levels. The shape is similar to that of other Yb 3+ doped fluorophosphate glasses [START_REF] Choi | Spectroscopic properties of Yb 3+ in heavy metal contained fluorophosphate glasses[END_REF]. As seen in Table 1, all glasses possess similar absorption and emission cross-sections at 975 nm which are in a good agreement with the commercial QX-Kigre glass [22].

The normalized emission spectra are presented in Fig. 3a. The emission band is typical of the emission of Yb 3+ in an amorphous network. An increase of x decreases the intensity of the shoulder at ~1000 nm as compared to the main band at 975 nm, reducing the effective bandwidth of the emission band (Table 2). The changes in the shape of the Yb 3+ emission may be related to radiation trapping and/or re-absorption as well as energy migration among the Yb 3+ ions due an increase in the network connectivity [START_REF] Venkata Krishnaiah | Spectroscopy and radiation trapping of Yb 3+ ions in lead phosphate glasses[END_REF]. As depicted in Fig. 3b, an increase in x increases slightly the area of the emission band. The increase in the integral intensity of the emission with an increase in x can be related to the change in the network connectivity and also to the reduction in the number of OH -groups, which are known quenchers of Yb 3+ ions [START_REF] Ohkawa | Influence of water on non-radiative decay of Yb 3+ -2 F5/2 level in phosphate glass[END_REF]. The OH content in the different glasses can be estimated from IR absorption spectrum (Fig. 3c). The IR absorption spectra exhibit a broad band which can be related to the "free" OH groups (weakly associated) at 3500 cm -1 and to OH groups which are strongly associated at around 2800 ("strongly associated") and 2350 cm -1 ("very strongly associated") [START_REF] Ohkawa | Influence of water on non-radiative decay of Yb 3+ -2 F5/2 level in phosphate glass[END_REF]. As expected from the composition of the glasses, the absorption coefficient between 2250 and 3750 cm -1 is high indicating that the glasses are hygroscopic. Indeed, the richer the composition in P2O5 and in Na2O, the more unstable the glass in atmospheric conditions [START_REF]PalavitPhosphate glasses and water[END_REF]. Fluorine is often used as a drying agent to displace bound OH --groups in the glass network [START_REF] Nazabal | CardinalOxyfluoride tellurite glasses doped by erbium: thermal analysis, structural organization and spectral properties[END_REF], and thus an increase in x leads to a decrease in the intensity of the absorption bands related to the OH groups. We suspect that the addition of NaF at the expense of Na2O increases the intensity of the Yb 3+ emission as fewer OH -groups are present to quench the Yb 3+ ions (Fig. 3b).

The lifetime values of the Yb 3+ : 2 F5/2 level are listed in Table 3. Within the accuracy of the measurement, all the glasses exhibit similar lifetime values which are comparable to those reported previously for phosphate glasses [START_REF] Sen | Optical, structural and luminescence properties of oxyfluoride phosphate glasses and glass-ceramics doped with Yb 3+[END_REF][START_REF] Nandi | Ytterbium-doped P2O5-TeO2 glass for laser applications[END_REF] but longer than the lifetime reported in silicate [START_REF] Cavillon | Ytterbium-doped multicomponent fluorosilicate optical fibers with intrinsically low optical nonlinearities[END_REF] and tellurite [START_REF] Feng | Tungsten-tellurite glass: a new candidate medium for Yb 3+ -doping[END_REF][START_REF] Merzliakov | Spectroscopy of Yb-doped tungsten-tellurite glass and assessment of its lasing properties glasses[END_REF]. As seen in Table 2, the measured lifetimes are less than the radiative lifetimes calculated using Equation ( 5) either due to energy transfer among the Yb 3+ ions (diffusion limited) [START_REF] Venkatramu | Optical properties of Yb 3+ -doped phosphate laser glasses[END_REF] and/or direct coupling with OH -groups [START_REF] Venkata Krishnaiah | Spectroscopy and radiation trapping of Yb 3+ ions in lead phosphate glasses[END_REF]. It is worth mentioning that no cooperative luminescence was detected from the investigated glasses indicating that there is no or very few Yb 3+ clustering in the glass. Therefore, we suspect the short lifetime value to be due to the OH -groups in the glasses. We expect to be able to increase the lifetime values of the Yb 3+ : 2 F5/2 level of the investigated glasses by reducing the OH -content in the glasses, possible either by heat treating the raw materials prior to melting, adding dehydration agent in the glass batch prior to the melting and/or melting the glasses under a reactive atmosphere as in Ref. [START_REF] Joshi | Reduction of OHions in tellurite glasses using chlorine and oxygen gases[END_REF].

In summary, the progressive replacement of Na2O by NaF leads to a more polymerized phosphate network. This reduces the amount of OH -groups while having no significant impact on the site of Yb 3+ .

Crystallization investigation

The crystallization properties of the investigated glasses were studied using different thermal treatments on a Pt foil in air, detailed in Table 4. First, the glasses were treated at (Tg+20 °C) for 17 h to form the nuclei and then at (Tp-40 °C) for 30 min to grow the nuclei into crystals. This is subsequently denoted as heat treatment A (HT-A). After heat treatment, the glasses are opaque which is a clear sign of crystallization. The surface of the HT-A glasses has a ceramic appearance after heat treatment while the inner part of the glasses exhibits a glassy appearance indicating that surface crystallization dominates. Surface crystallization was confirmed using SEM. The images the heat treated glasses with x = 0, 5 and 7.5, taken as examples, are shown in (Fig. 4). The SEM images were taken from the glasses polished from the side and with the surface at the top of the image to clearly image the surface crystallization. It can be seen that crystals with a needle like shape are growing perpendicular to the surface of the glass. The thickness of the crystallized layer decreases as x increases indicating that the progressive replacement of Na2O by NaF reduces the crystallization tendency of the glasses. This is probably due to the increase in the glass connectivity with an increase in x as discussed above. Similar correlation between the crystallization tendency and network connectivity was reported in Refs. [START_REF] Sen | Optical, structural and luminescence properties of oxyfluoride phosphate glasses and glass-ceramics doped with Yb 3+[END_REF][START_REF] Lopez-Iscoa | Effect of the addition of Al2O3, TiO2 and ZnO on the thermal, structural and luminescence properties of Er 3+ -doped phosphate glasses[END_REF]. According to the Energy Dispersive Spectroscopy (EDS) mapping, Na rich crystals are expected in the glasses during the heat treatment.

In order to identify the crystals, the XRD pattern of the HT-A glasses were measured and can be found in Fig. 5. The XRD pattern of heat treated glasses at (Tg+20 °C) for 17 h and then at Tp for 2 h are also shown. This heat treatment was performed to fully crystallize the glasses and so to ease the identification of the crystals and is denoted as heat treatment B (HT-B). The XRD patterns show the presence of at least 3 crystalline phases: NaPO3 [ICDD 00-011-0648], Na5P3O10 [ICDD 00-011-0652 and 00-010-0179] and NaYbP2O7 [ICDD 04-014-6349]. As x increases, the NaPO3 phase grows at the expense of Na5P3O10 and a small amount of Na4P4O12(H2O) is suspected to also partially transform into NaH2(PO3)4.

The heat treatments A and B have an impact on the spectroscopic properties of Yb 3+ . Changes not only in the shape of the emission but also in the area of the Yb 3+ emission were noticed after heat treatment as depicted in Fig. 3, Fig. 6b, respectively. As seen in Fig. 6, the heat treatment A increases the intensity of the shoulders at 1000, 1040 and 1055 nm as compared to the band at 975 nm leading to an increase of the bandwidth of the emission band (Table 2). One should point out that the changes in the shape of the emission after HT-A are less visible with an increase in x as less crystals precipitate when replacing Na2O by NaF. A decrease in the area of the Yb 3+ emission occurs after heat treatment, the decrease being more pronounced as x increases (Fig. 3b). One should mention that the heat treatment A does not lead to changes in the Yb 3+ excited state 2 F5/2 lifetime within the accuracy of the measurement although a reduction in the lifetime is suspected to occur as the emission intensity decreased after the heat treatment, probably due to the reduction in the Yb-Yb distance (Table 3).

After the heat treatment B, the glasses exhibit lower overall emission areas compared to the as-prepared glasses (Fig. 3b). Additionally, as illustrated in Fig. 6, the position of the band at 975 nm shifts to 972 nm and the relative intensity of the shoulder at 1000 nm increases significantly leading to a decrease in the bandwidth of the emission band. Similar changes in the shape of the Yb 3+ emission after heat treatment were reported in Ref. [START_REF] Sen | Optical, structural and luminescence properties of oxyfluoride phosphate glasses and glass-ceramics doped with Yb 3+[END_REF] and were related to the precipitation of Yb containing crystals. The change of the local Yb 3+ environment is associated with a modification of the crystal field strength which explained both the shift of the zero-phonon line at 972 nm and the light modification of the 2 F7/2 fundamental multiplet splitting. One should point out that the emission spectra of the fully crystallized materials match well with the Yb 3+ emission observed in NaYbP2O7 crystals [START_REF] Ternane | Spectroscopic properties of Yb 3+ in NaYbP2O7 diphosphate single crystals[END_REF]. This allows one to suggest that Yb 3+ ions are mainly present in NaYbP2O7 and in the residual glassy phase.

Crystals with a needle-like shape as in Fig. 7 were easily found in the volume of the heat treated glass with x = 0 while no crystals could be seen in the volume of the other heat treated glasses. Small crystals, visible as bright spots in Fig. 7, can also seen in the SEM image. According to EDS analysis, the large needle-like crystals contain Na and P while the small crystals contain Yb. Therefore, these small crystals are thought to be NaYb(P2O7). In order to check if the glass with x = 0 is the only glass exhibiting bulk crystallization, the XRD pattern of the glasses after heat treatment A were measured after polishing their surface to remove the crystals at the surface. Peaks with very small intensity were found only in the XRD pattern of the heat treated glass with x = 0 as depicted in Fig. 8a confirming that the heat treatment leads to bulk and surface crystallization in the NaF-free glass and to only surface crystallization in the other glasses.

In order to check the possibility to control the crystallization of the glass with x = 0, this glass was heat treated at (Tg+20 °C) for 17 h and at (Tp-40 °C) for 1 h, denoted as heat treatment C (HT-C), and at (Tg+20 °C) for 66 h and at (Tp-40 °C) for 30 min, denoted as heat treatment D (HT-D). As depicted in Fig. 9, an increase in the heat treatment duration from 30 min to 1 h at (Tp-40 °C) and from 17 h to 66 h at (Tg+20 °C) increases the size of the crystals from ~175 μm to ~250 μm and their number, respectively and so the intensity of the peaks in the XRD pattern, especially for the HT-D glass (Fig. 8a). According to the XRD pattern, we suspect the Na5P3O10 I phase to precipitate in the volume of the glass. Based on the SEM image of the crystals found in the volume of the glass with x = 0, we also suspect the presence of NaYbP2O7 crystals in the glass. The amount of this phase is probably too small and/or too few to be detected with XRD. As seen in Fig. 8b, the heat treated glasses after removing the surface layer are transparent confirming that it is possible to prepare Yb 3+ doped transparent glass-ceramics with the 90NaPO3-10 Na2O glass composition. The heat treated glasses have slightly lower transmittance compared to the as-prepared glass probably due to scattering caused by density fluctuations due to the presence of micro-crystals inside the glass matrix (Fig. 8c). The transition between Yb 3+ 5 F7/2 and 2 F5/2 energy levels is clearly visible. One should mention that the 3 transparent glass-ceramics exhibit similar intensity and shape of emission (and so bandwidth) than the HT-A glass with x = 0 depicted in Fig. 3b, Fig. 6a, respectively. Therefore, it is shown here that a heat treatment of the glass with x = 0 can be used to increase slightly the bandwidth of the emission band by increasing the intensity of the shoulder at 1 μm while maintaining the spectral distribution of the emission.

Conclusion

In this study, novel Yb 3+ doped glasses within the NaPO3-Na2O-NaF system were prepared and characterized. An increase in the NaF content at the expense of Na2O leads to a polymerization of the glass network and so to changes in the crystallization process of the glass. All the glasses exhibit surface crystallization after heat treatment, the NaF-free glass being the only glass having also bulk crystallization. We showed, for the first time to the best of our knowledge, that a heat-treatment of the glass with the 90NaPO3-10 Na2O composition can produce transparent glass-ceramic with crystals also distributed in the volume of the glass. This transparent glass-ceramic was found to have a slightly larger bandwidth of the Yb 3+ emission band than the as-prepared glass while having slightly lower intensity of the emission indicating that this newly developed glass-ceramic can be applicable in solid state lasers and other optical devices.

Declaration of competing interest :

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgement : Academy of Finland (Flagship Programme, Photonics Research and Innovation PREIN-320165 and Academy Project -326418) and the Région Nouvelle-Aquitaine (Project n° 2019-1R1MO1) are greatly acknowledged for the financial support.


Julien Berestycki
email: julien.berestycki@stats.ox.ac.uk

Éric Brunet
email: eric.brunet@lps.ens.fr

Sarah Penington
email: ‡penington@maths.ox.ac.uk

Global existence for a free boundary problem of Fisher-KPP type

Motivated by the study of branching particle systems with selection, we establish global existence for the solution (u, µ) of the free boundary problem

We construct the solution as the limit of a sequence (u n ) n≥1 , where each u n is the solution of a Fisher-KPP equation with same initial condition, but with a different non-linear term.

 show that this global solution can be identified with the hydrodynamic limit of the so-called N -BBM, i.e. a branching Brownian motion in which the population size is kept constant equal to N by killing the leftmost particle at each branching event.

Main results and introduction

We establish global existence for a free boundary problem of Fisher-KPP type: Theorem 1.1. Let v : R → [0, 1] be a non-increasing function such that v(x) → 0 as x → ∞ and v(x) → 1 as x → -∞. Let µ 0 = inf{x ∈ R : v(x) < 1} ∈ {-∞} ∪ R. Then there exists a unique classical solution (u, µ) with u ∈ [0, 1] to the following free boundary problem:

             ∂ t u = ∂ 2
x u + u for t > 0 and x > µ t , u(x, t) = 1 for t > 0 and x ≤ µ t , ∂ x u(µ t , t) = 0 for t > 0, u(x, 0) = v(x) for x ∈ R.

(FBP)

Furthermore, this unique solution satisfies the following properties:

• For every t > 0, u(•, t) ∈ C 1 (R), and ∂ x u ∈ C(R × (0, ∞)).

• If v (1) ≤ v (2) are two valid initial conditions and (u (i) , µ (i) ) is the solution with initial condition v (i) , then u (1) ≤ u (2) and µ (1) ≤ µ (2) .

We say that (u, µ) is a classical solution to (FBP) above if µ t ∈ R ∀t > 0, t → µ t is continuous, u : R × (0, ∞) → [0, 1], u ∈ C 2,1 ({(x, t) : t > 0, x > µ t }) ∩ C(R × (0, ∞)), (u, µ) satisfies the equation (FBP), and u(•, t) → v(•) in L 1 loc as t ց 0. Remark 1. We will show that u(t, x) → v(x) at all points of continuity of v as t ց 0 (since v is non-increasing, it is differentiable almost everywhere).

Remark 2. If instead v(x) → l > 0 as x → ∞, then a classical solution (u, µ) of (FBP) exists for t < t c = -log l, with µ t → ∞ as t ր t c . Remark 3. As discussed below, the condition that v is non-increasing can be relaxed to some extent.

Our motivation for studying the problem (FBP) stems from its connection with the so-called N -BBM, a variant of branching Brownian motion in R in which the number of active particles is kept constant (and equal to N ) by removing the leftmost particle each time a particle branches. More details are given in Section 2 below, but in a nutshell, De Masi et al. [START_REF] De Masi | Hydrodynamics of the N -BBM process[END_REF] show that as N → ∞, under appropriate conditions on the initial configuration of particles, the N -BBM has a hydrodynamic limit whose cumulative distribution can be identified with the solution of (FBP), provided such a solution exists.

The overall idea behind the proof is to construct u as the limit of a sequence of functions u n , where, for each n, u n satisfies an n-dependent non-linear equation, but where all the u n have the same initial condition. More precisely, let v : R → [0, 1] be a measurable function and, for n ≥ 2, let (u n (x, t), x ∈ R, t ≥ 0) be the solution to

∂ t u n = ∂ 2
x u n + u n -u n n for x ∈ R and t > 0, u n (x, 0) = v(x) for x ∈ R.

(1.1)

For each n ≥ 2, this is a version of the celebrated Fisher-KPP equation about which much is known (see e.g. [START_REF] Kolmogorov | Etude de l'équation de la diffusion avec croissance de la quantité de matiere et son application à un problème biologique[END_REF][START_REF] Aronson | Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation[END_REF][START_REF] Mckean | Application of Brownian motion to the equation of Kolmogorov-Petrovskii-Piskunov[END_REF][START_REF] Uchiyama | The behavior of solutions of some non-linear diffusion equations for large time[END_REF][START_REF] Hamel | A short proof of the logarithmic Bramson correction in Fisher-KPP equations[END_REF][START_REF] Nolen | Refined long time asymptotics for the Fisher-KPP fronts[END_REF]). In particular,

• u n exists and is unique,

• u n (x, t) ∈ (0, 1) for x ∈ R and t > 0 (unless v ≡ 0 or v ≡ 1).

Since the comparison principle applies, we see furthermore that for every x ∈ R, t > 0 fixed, the sequence n → u n (x, t) is increasing. Therefore, the following pointwise limit is well defined:

u(x, t) := lim n→∞ u n (x, t), ( 1.2)
with u(x, t) ∈ (0, 1] for t > 0 (unless v ≡ 0). Indeed, in most of the cases we are interested in, there are regions where u(x, t) = 1.

We have the following results on u:

Theorem 1.2. Let v : R → [0, 1] be a measurable function. The function u(x, t) as defined by (1.1) and (1.2) satisfies the following properties:

• u is continuous on R × (0, ∞) and, for t > 0, u(•, t) is Lipschitz continuous.

• u(•, t) → v(•) in L 1 loc as t ց 0, and if v is continuous at x then u(t, x) → v(x) as t ց 0. • At any (x, t) with t > 0 such that u(x, t) < 1, the function u is continuously differentiable in t and twice continuously differentiable in x, and satisfies

∂ t u = ∂ 2 x u + u.
• u satisfies the following semigroup property: for any t > 0 and any t 0 ≥ 0, u(•, t + t 0 ) can be obtained as the solution at time t to (1.1) and (1.2) with an initial condition u(•, t 0 ).

• If v (1) ≤ v (2) are two measurable functions and u (i) is the solution to (1.1) and (1.2) with initial condition v (i) , then u (1) ≤ u (2) .

The existence result in Theorem 1.1 is then a consequence of the following result:

Proposition 1.3. Suppose that v (and µ 0 ) is as in Theorem 1.1, and define u(x, t) as in (1.1) and (1.2). Then there exists a map t → µ t with µ t ∈ R ∀t > 0 and µ t → µ 0 as t ց 0 such that

u(x, t) = 1 ⇔ x ≤ µ t for t > 0. (1.3) Furthermore, t → µ t is continuous and u(•, t) ∈ C 1 (R) for t > 0 with ∂ x u ∈ C(R × (0, ∞)).
By combining Theorem 1.2 and Proposition 1.3, we have that if v is as in Theorem 1.1 then (u, µ) is a classical solution of (FBP).

Remark 4.

For an arbitrary measurable initial condition v, for t > 0, u(•, t) is obviously C 1 in the interior of the region where u = 1, and by Theorem 1.2 it is C 1 in the region where u < 1. The difficulty in proving Proposition 1.3 is to show that u(•, t) is also C 1 at the boundary between these two domains. Remark 5. It turns out that the proof that u(•, t) is C 1 holds whenever the topological boundary between the (two-dimensional) domains {u = 1} and {u < 1} has measure zero. (In the case where v is non-increasing, this is implied by the existence of a continuous map t → µ t satisfying (1.3).) This means that it should be possible to show that u(•, t) is C 1 for any "reasonable" initial condition.

Remark 6.

The condition that v is non-increasing in Theorem 1.1 is only used in the proof of Proposition 1.3 to show the existence of a continuous boundary t → µ t as in (1.3). The idea of using the limit of (u n ) n≥1 as the solution to (FBP) first appeared in [START_REF] Berestycki | Exact solution and precise asymptotics of a Fisher-KPP type front[END_REF] and the present article puts this intuition on a rigorous footing.

The rest of the article is organised as follows: the next section is devoted to putting our result in the context of several recent works on related problems. Next, in Section 3, we present the precise versions of the Feynman-Kac representation that we shall use in the rest of the proof. The proof of one of these Feynman-Kac results will be postponed until Section 7. We establish Theorem 1.2 in Section 4, and in Section 5 we prove Proposition 1.3. In Section 6, we complete the proof of Theorem 1.1 by proving the uniqueness of the classical solution of (FBP).

Context

Let ω be a probability measure on R. Then define v : R → [0, 1] by setting

v(x) = ω[x, ∞).
Note that v is non-increasing, and that v(x) → 0 as x → ∞ and v(x) → 1 as x → -∞. Therefore, by Theorem 1.1, there exists a unique classical solution (u, µ) to the free boundary problem (FBP), and

∂ x u is continuous on R × (0, ∞). Let ρ = -∂ x u.
The following result is an easy consequence of Theorem 1.1 and its proof.

Corollary 2.1. Let ω be a probability measure on R and let

µ 0 = inf{x ∈ R : ω[x, ∞) < 1} ∈ R ∪ {-∞}.
Then (ρ, µ) constructed as above from the solution of (FBP) with initial condition v(x) = ω[x, ∞) is the unique classical solution with ρ ≥ 0 to the following free boundary problem:

       ∂ t ρ = ∂ 2 x ρ + ρ for t > 0 and x > µ t , ρ(µ t , t) = 0, ∞ µt ρ(y, t) dy = 1 for t > 0, ρ(•, t)dλ → dω(•)
in the vague topology as t ց 0. and(ρ, µ) satisfies the equation (FBP ′ ).

(FBP ′ ) We say that (ρ, µ) is a classical solution to (FBP ′ ) above if µ t ∈ R ∀t > 0, t → µ t is continuous, ρ : R × (0, ∞) → [0, ∞), ρ ∈ C 2,1 ({(x, t) : t > 0, x > µ t }) ∩ C(R × (0, ∞)),
This result improves on a recent result of Lee [START_REF] Lee | A free boundary problem in biological selection models[END_REF], where local existence of a solution to (FBP ′ ) is shown (i.e. existence of a solution on a time interval [0, T ] for some T > 0), under the additional assumptions that ω is absolutely continuous with respect to Lebesgue measure with probability density φ ∈ C 2 c (R), and that there exists µ 0 ∈ R such that φ(µ 0 ) = 0, φ ′ (µ 0 ) = 1 and ∞ µ 0 φ(x) dx = 1. In [START_REF] De Masi | Hydrodynamics of the N -BBM process[END_REF], De Masi et al. study the hydrodynamic limit of the N -BBM and its relationship with the free boundary problem (FBP ′ ). The N -BBM is a variant of branching Brownian motion in which the number of active particles is kept constant (and equal to N ) by removing the leftmost particle each time a particle branches.

We shall now define this particle system more precisely. Suppose that φ ∈ L 1 (R) is a probability density function which satisfies (a) φ ∞ < ∞ and (b) ∞ r φ(x) dx = 1 for some r ∈ R. Let X 1 0 , . . . , X N 0 be i.i.d. with density φ. At time 0, the N -BBM consists of N particles at locations X 1 0 , . . . , X N 0 . These particles move independently according to Brownian motions, and each particle independently at rate 1 creates a new particle at its current location. Whenever a new particle is created, the leftmost particle is removed from the particle system.

Let X t = {X 1 t , . . . , X N t } denote the set of particle locations at time t. Let π (N ) t be the empirical distribution induced by the particle system at time t, i.e. for A ⊂ R, let

π (N ) t (A) = 1 N |X t ∩ A|.
De Masi et al. prove in [START_REF] De Masi | Hydrodynamics of the N -BBM process[END_REF] that for each t ≥ 0 there exists a probability density function ψ(•, t) : R → [0, ∞) such that, for any a ∈ R,

lim N →∞ π N t [a, ∞) = ∞ a
ψ(r, t) dr a.s. and in L 1 .

Moreover, they show that if (ρ, µ) is a classical solution of (FBP ′ ) with initial condition ω given by dω = φ dλ then ψ = ρ. The following result is then a direct consequence of Theorems 1 and 2 in [START_REF] De Masi | Hydrodynamics of the N -BBM process[END_REF] and our Corollary 2.1.

Corollary 2.2. Suppose φ ∈ L 1 (R) is a probability density function with φ ∞ < ∞ and ∞ r φ(x) dx = 1 for some r ∈ R. Construct an N -BBM with initial particle locations given by i.i.d. samples from φ, as defined above. Let π (N ) t denote the empirical distribution induced by the particle system at time t. Then for any t ≥ 0 and a ∈ R,

lim N →∞ π N t [a, ∞) = ∞ a ρ(r, t) dr = u(a, t) a.s. and in L 1 ,
where (u, µ) is the solution of (FBP) with initial condition v given by v(x) = ∞ x φ(y) dy, and ρ = -∂ x u.

Lee [START_REF] Lee | A free boundary problem in biological selection models[END_REF] points out that (FBP ′ ) can be reformulated as a variant of the Stefan problem; let (ρ, µ) be a solution of (FBP ′ ) and define w(x, t) := e -t ∂ x ρ(x, t). Then under some regularity assumptions, (w, µ) solves

∂ t w = ∂ 2
x w for t > 0 and x > µ t , w(µ t , t) = e -t , ∂ t µ t = -1 2 e t ∂ x w(µ t , t) for t > 0.

(Stefan)

The Stefan problem describes the phase change of a material and is one of the most popular problems in the moving boundary problem literature. Typically, it requires solving heat equations for the temperature in the two phases (e.g. solid and liquid), while the position of the front separating them, the moving boundary, is determined from an energy balance referred to as the Stefan condition. The Stefan problem has been studied in great detail since Lamé and Clapeyron formulated it in the 19th century [START_REF] Lamé | Mémoire sur la solidification par refroidissement d'un globe liquide[END_REF]. There are several reference books that the reader may consult such as the recent and up-to-date book [START_REF] Gupta | The classical Stefan problem: basic concepts, modelling, and analysis[END_REF]. The free boundary problem (FBP) was studied at a heuristic level in [START_REF] Berestycki | Exact solution and precise asymptotics of a Fisher-KPP type front[END_REF]. The main tool there was the following relation, which can be proved rigorously: Lemma 2.3. Let v be as in Theorem 1.1 with µ 0 ∈ R and such that γ := sup r : ∞ µ 0 v(x)e rx dx < ∞ ≥ 0. Let (u, µ) be the classical solution to (FBP). Then, for r < min(γ, 1), and for any

t ≥ 0, 1 + r ∞ 0 dx u(µ t + x, t)e rx = ∞ 0 ds e r(µ t+s -µt)-(1+r 2 )s < ∞. (2.1)
Although the proof is not very difficult, we omit it from the present work as it is not our main focus here; the main ideas can be found in [START_REF] Berestycki | Exact solution and precise asymptotics of a Fisher-KPP type front[END_REF][START_REF] Brunet | An exactly solvable travelling wave equation in the Fisher-KPP class[END_REF].

For instance, take a step initial condition v(x) = 1 {x≤0} . Using (2.1) with t = 0 and r = 1ǫ gives

∞ 0 ds e -ǫ 2 s+(1-ǫ)(µs-2s) = 1 ∀ǫ > 0. (2.2)
Whether (2.2) completely characterizes the function µ or not is an open question.

Feynman-Kac formulae

In this section, we state the versions of the Feynman-Kac formula which we shall use repeatedly in the rest of the paper. So as not to interrupt the flow of the main argument, the proof for Proposition 3.1 is postponed to Section 7.

We introduce the heat kernel

p t (x) = 1 √ 4πt e -x 2 4t . (3.1)
For x ∈ R, we let P x denote the probability measure under which (B t ) t≥0 is a Brownian motion with diffusivity constant √ 2 started at x. We let E x denote the corresponding expectation. The symbol * denotes convolution; for instance,

p t * v(x) = ∞ -∞ dy p t (x -y)v(y) = E x [v(B t )]
is the solution at time t to the heat equation on R with an initial condition v.

Proposition 3.1. Suppose that A ⊆ R × (0, ∞) is an open set, and that w : A → R is C 2,1
and bounded, and satisfies

∂ t w = ∂ 2 x w + Kw + S for (x, t) ∈ A, (3.2)
where K : A → R, S : A → R are continuous and bounded. Then, if one of the conditions below is met, we have the following representation for w(x, t) with (x, t) ∈ A:

w(x, t) = E x w(B τ , t -τ )e τ 0 K(Bs,t-s) ds + τ 0 dr S(B r , t -r)e r 0 K(Bs,t-s) ds , (3.3)
where τ is a stopping time for (B s ) s≥0 .

For the representation (3.3) to hold, it is sufficient to have one the following:

1. The stopping time τ is such that (B s , t -s) ∈ A for all s ≤ τ , 2.
The set A is given by A = (x, t) : t ∈ (0, T ) and x > µ t for some T > 0 and some continuous boundary t → µ t with µ t ∈ R ∪ {-∞} ∀t ∈ [0, T ], the stopping time τ is given by τ = inf s ≥ 0 : B s ≤ µ t-s ∧ t (the first time at which (B τ , t -τ ) ∈ ∂A) and, furthermore, w is defined and bounded on Ā, continuous on Ā ∩ (R × (0, ∞)) and satisfies

w(•, t) → w(•, 0) in L 1 loc as t ց 0.
Although this is a very classical result, we give a proof in Section 7 for the sake of completeness and because we could not find a exact statement with stopping times or a discontinuous initial condition in the literature. The proof that (3.3) holds under condition 1 essentially follows the proof of Theorem 4.3.2 in [START_REF] Durrett | Stochastic calculus: a practical introduction[END_REF].

Proposition 3.1 gives some useful representations for the u n defined in (1.1).

Corollary 3.2. Let v : R → [0, 1] be measurable, let n ≥ 2 and let u n (x, t) denote the solution to (1.1). Then by Proposition 3.1:

• taking K = 1 -u n-1 n
and S = 0, for τ a stopping time with τ < t:

u n (x, t) = E x u n (B τ , t -τ )e τ 0 (1-u n-1 n
(Bs,t-s))ds .

(3.4)

• taking K = 1 -u n-1
n , S = 0, and τ = t:

u n (x, t) = E x v(B t )e t 0 (1-u n-1 n
(Bs,t-s))ds .

(3.5)

• taking K = 0, S = u n -u n n and τ = t: u n (x, t) = E x v(B t ) + t 0 dr u n (B r , t -r) -u n n (B r , t -r) = p t * v(x) + t 0 dr p r * u n (x, t -r) -u n n (x, t -r) (3.6) • taking K = 1, S = -u n n and τ = t: u n (x, t) = E x v(B t )e t - t 0 dr e r u n n (B r , t -r) = e t p t * v(x) - t 0 dr e r p r * u n n (x, t -r). (3.7)
Proof. This is a direct consequence of the previous result.

We will also use the following representation for solutions of the free boundary problem (FBP):

Corollary 3.3. If v is as in Theorem 1.1 and (u, µ) is a classical solution of (FBP) with initial condition v, then for t > 0 and x ∈ R, u(x, t) = E x e τ 1 {τ <t} + e t v(B t )1 {τ =t} , ( 3.8
)

where τ = inf{s ≥ 0 : B s ≤ µ t-s } ∧ t.
Proof. This is a direct application of Proposition 3.1 under condition 2, with K = 1 and S = 0.

Finally, we use the following result to recognise solutions to partial differential equations: Lemma 3.4. Suppose that a < b, t 0 < t 1 , and that g :

[a, b] × [t 0 , t 1 ] → [0, ∞) is continuous and for x ∈ [a, b] and t ∈ [t 0 , t 1 ], g(x, t) = E x [g(B τ , t -τ )e τ ] ,
where τ = inf{s ≥ 0 :

B s ∈ {a, b}} ∧ (t -t 0 ). Then g ∈ C 2,1 ((a, b) × (t 0 , t 1 )) with ∂ t g = ∂ 2 x g + g for (x, t) ∈ (a, b) × (t 0 , t 1 ).
Proof. The proof is the same as the proof of Exercise 4.3.15 in [START_REF] Karatzas | Brownian motion and stochastic calculus[END_REF], where an outline proof is given.

Proof of Theorem 1.2

In this section, we suppose v : R → [0, 1] is measurable. Let u n denote the solution of (1.1) and define u as in (1.2). We shall use the following basic results on the smoothing effect of convolution with the heat kernel p t as introduced in (3.1).

Lemma 4.1. Suppose t > 0.

1. If x → a(x) is bounded, then x → p t * a(x) is C ∞ and (p t * a) (n) (x) = p (n) t * a(x). 2. If (x, s) → b(x, s) is such that b s := b(•, s) ∞ < ∞ for each s ∈ (0, t), and the map s → bs √ s is integrable on [0, t], then f (x) := t 0 ds ∞ -∞ dy p s (x -y)b(y, s) = t 0 ds p s * b(x, s) is C 1 and f ′ (x) = t 0 ds p ′ s * b(x, s).
Proof. The first statement holds since for every n ∈ N and t > 0, there exists a polynomial function q n,t : R → R such that |p

(n) t (x -y)| ≤ |q n,t (x -y)|e -(x-y) 2 /(4t) ∀x, y ∈ R.
Then for the second statement, we have that

f s (x) := p s * b(x, s) is smooth, with f ′ s (x) = p ′ s * b(x, s) ≤ ∞ -∞ dy p ′ s (x -y) b s = b s √ πs .
Since s → bs √ πs is integrable on [0, t], the result follows.

The following result of Uchiyama provides a useful bound on the spatial derivative of u n .

Lemma 4.2 ([17]

, Section 4). For x ∈ R and t > 0,

|∂ x u n (x, t)| ≤ 1 √ πt + √ 8 √ π . (4.1)
Proof. We briefly recall Uchiyama's proof. Using Lemma 4.1 to differentiate (3.6) with respect to x, and bounding the result (using v ∈ [0, 1] and u n -u n n ∈ [0, 1]) yields:

|∂ x u n (x, t)| ≤ ∞ -∞ p ′ t (x -y) dy + t 0 ds ∞ -∞ dy p ′ s (x -y) = 1 √ πt + 2 √ t √ π .
This bound reaches its minimum 8/π at t = 1/2. For t ≤ 1/2, the result follows immediately. For t ≥ 1/2, since u n (•, t) is also the solution at time 1/2 of (1.1) with initial condition

u n (•, t - 1/2), it follows that |∂ x u n (x, t)| ≤ 8/π ∀x ∈ R and t ≥ 1/2.
In the following two lemmas, we prove the continuity of u.

Lemma 4.3. For any t > 0, the map x → u(x, t) is Lipschitz continuous, with Lipschitz constant

1 √ πt + √ 8 √ π .
Proof. For x ∈ R and h > 0, we can write, using (4.1),

u n (x + h, t) -u n (x, t) ≤ 1 √ πt + √ 8 √ π h.
Then take the n → ∞ limit to conclude.

Lemma 4.4. The map (x, t) → u(x, t) is continuous on R × (0, ∞). Furthermore, u(•, t) → v in L 1 loc as t ց 0, and if v is continuous at x then u(x, t) → v(x) as t ց 0.
Proof. Using the bound u n -u n n ∈ [0, 1] in the expression for u n in (3.6), we have that for x ∈ R, t 0 ≥ 0 and t > 0,

u n (x, t) -p t * v(x) ∈ [0, t] and u n (x, t 0 + t) -p t * u n (x, t 0 ) ∈ [0, t],
where for the second expression we used that u n (•, t + t 0 ) is the solution at time t of (1.1) with initial condition u n (•, t 0 ). Taking the n → ∞ limit, it follows that

u(x, t) -p t * v(x) ∈ [0, t] and u(x, t 0 + t) -p t * u(x, t 0 ) ∈ [0, t].
Since the solution to the heat equation p t * v converges to v in L 1 loc as t ց 0, we have that

u(•, t) → v in L 1 loc . If v is continuous at x, then p t * v(x) → v(x)
as t ց 0, and hence u(x, t) → v(x) as t ց 0.

It remains to prove that u is continuous. By Lemma 4.3, we have

|p t * u(x, t 0 ) -u(x, t 0 )| = E x u(B t , t 0 ) -u(x, t 0 ) ≤ 1 √ πt 0 + √ 8 √ π E x [|B t -x|] = 1 √ πt 0 + √ 8 √ π 4t π .
Therefore by the triangle inequality,

|u(x, t 0 + t) -u(x, t 0 )| ≤ t + 1 √ πt 0 + √ 8 √ π 4t π .
Hence by the triangle inequality and then by Lemma 4.3, for x 1 , x 2 ∈ R, t 0 ≥ 0 and t > 0,

|u(x 1 , t 0 + t) -u(x 2 , t 0 )| ≤ |u(x 1 , t 0 + t) -u(x 1 , t 0 )| + |u(x 1 , t 0 ) -u(x 2 , t 0 )| ≤ t + 1 √ πt 0 + √ 8 √ π 4t π + |x 1 -x 2 | ,
and the result follows.

We now turn to the semigroup property.

Lemma 4.5. Suppose v : R → [0, 1], take t 0 ≥ 0 and, as throughout this section, let u n and u denote the functions defined in (1.1) and (1.2). Furthermore, for t ≥ t 0 , let u n;t 0 (•, t) denote the solution at time t -t 0 to (1.1) with the initial condition v(•) replaced by u(•, t 0 ). Then for t ≥ t 0 and x ∈ R,

lim n→∞ u n;t 0 (x, t) = lim n→∞ u n (x, t) = u(x, t).
Proof. Since u n (x, t 0 ) ≤ u(x, t 0 ) ∀x ∈ R, it follows by the comparison principle that u n (x, t) ≤ u n;t 0 (x, t) ∀x ∈ R, t ≥ t 0 . Then for t ≥ t 0 , by the Feynman-Kac formula (3.5),

u n;t 0 (x, t) -u n (x, t) = e t-t 0 E x u(B t-t 0 , t 0 )e - t-t 0 0 u n-1 n;t 0 (Bs,t-s) ds -u n (B t-t 0 , t 0 )e -t-t 0 0 u n-1 n (Bs,t-s) ds = e t-t 0 E x u(B t-t 0 , t 0 ) -u n (B t-t 0 , t 0 ) e -t-t 0 0 u n-1 n;t 0 (Bs,t-s) ds + e t-t 0 E x u n (B t-t 0 , t 0 ) e - t-t 0 0 u n-1 n;t 0 (Bs,t-s) ds -e -t-t 0 0 u n-1 n (Bs,t-s) ds ≤ e t-t 0 E x u(B t-t 0 , t 0 ) -u n (B t-t 0 , t 0 ) ,
where, in the last step, we used that u n;t 0 ≥ u n , u n ≥ 0, u ≥ u n and u n;t 0 ≥ 0. By dominated convergence, the right hand side converges to zero as n → ∞, and this completes the proof.

At this point, it is convenient to introduce the two sets

U := (x, t) ∈ R × (0, ∞) : u(x, t) = 1 and S := (x, t) ∈ R × (0, ∞) : u(x, t) < 1 . (4.2)
By the continuity of u, the set S is open.

The next proposition focuses on the set S, while Proposition 4.7 below is about the behaviour of u n in the set U .

Proposition 4.6. The map u is C 2,1 on S and satisfies

∂ t u = ∂ 2 x u + u on S. ( 4
′ , t ′ ) ∈ [a, b] × [t 0 , t 1 ], u n (x ′ , t ′ ) = E x ′ u n (B τ , t ′ -τ )e τ 0 1-u n-1 n (Bs,t ′ -s) ds , ( 4.4)
where τ = (t

′ -t 0 ) ∧ inf{s ≥ 0 : B s ∈ (a, b)} is the time at which (B τ , t ′ -τ ) hits the boundary of [a, b] × [t 0 , t 1 ].
We now take the n → ∞ limit. For a given Brownian path

(B s ) s≥0 , since (B s , t ′ -s) ∈ S for s ∈ [0, τ ], we have u n (B s , t ′ -s) → u(B s , t ′ -s) < 1 as n → ∞ for s ∈ [0, τ ] and so, since τ ≤ t ′ -t 0 , τ 0 u n-1 n (B s , t ′ -s)ds → 0 as n → ∞.
Hence by dominated convergence in (4.4),

u(x ′ , t ′ ) = E x ′ u(B τ , t ′ -τ )e τ for any (x ′ , t ′ ) ∈ [a, b] × [t 0 , t 1 ]. (4.5)
The result then follows by Lemma 3.4.

To complete the proof of Theorem 1.2, it only remains to note that if v (1) ≤ v (2) are two measurable functions, and if u (i) n is the solution to (1.1) with initial condition v (i) , then by the comparison principle u (1) n ≤ u (2) n and hence u (1) ≤ u (2) . We finish this section by proving two more results on the behaviour of u n which will be used in the proof of Proposition 1.3 in the next section, but which do not require any additional assumptions on v.

Proposition 4.7. If (x, t) is in the interior of U , then lim n→∞ u n n (x, t) = 1.
Before proving this result properly, we give a heuristic explanation. As in the proof of Proposition 4.6, choose a rectangle [a, b] × [t 0 , t 1 ] in the interior of U , and write (4.4) for a point (x ′ , t ′ ) ∈ (a, b) × (t 0 , t 1 ). We take the limit n → ∞ again. By construction, u n (x ′ , t ′ ) → 1 and u n (B τ , t ′ -τ ) → 1, so we obtain

1 = lim n→∞ E x ′ e τ 0 1-u n-1 n (Bs,t ′ -s) ds .
This equation strongly suggests the result, because if there were a region where lim sup n→∞ u n n < 1 which was visited by the paths (B s , t ′ -s) with a strictly positive probability then the limiting expectation above would be larger than 1. However, we were not able to turn this heuristic into a proper proof of Proposition 4.7, so we used a completely different method.

Proof. Take (x, t) in the interior of U . For ǫ > 0, let A = [-ǫ 0.49 , ǫ 0.49 ].

(The exponent 0.49 could be any positive number smaller than 1/2.) Choose ǫ sufficiently small that [x -ǫ 0.49 , x + ǫ 0.49 ] × [t -ǫ, t] ⊂ U . Note that u n is a monotone sequence and converges pointwise to 1 on [x -ǫ 0.49 , x + ǫ 0.49 ] × [t -ǫ, t]. Therefore, by Dini's theorem, we can choose n 0 sufficiently large that u n (x + y, t -ǫ) > 1 -ǫ 2 for all y ∈ A and all n ≥ n 0 . Let w n (y, s) denote the solution to

     ∂ s w n = ∂ 2 y w n + w n -w n n
for y ∈ R and s > 0,

w n (y, 0) = 1 - ǫ 2 1 {y∈A} for y ∈ R. (4.6)
Then, by the comparison principle, u n (x + y, t -ǫ + s) ≥ w n (y, s) for n ≥ n 0 , s ≥ 0 and y ∈ R. Heuristically, the domain A is so "large" that, for times s ≤ ǫ, the solution w n behaves locally near y = 0 as if started from a flat initial condition. This suggests that ∂ 2 y w n (0, s) is very small for s ∈ [0, ǫ]. Indeed, starting from (3.7) we have

w n (y, s) = e s p s * w n (y, 0) - s 0 dr e s-r ∞ -∞ dz p s-r (y -z)w n n (z, r).
Taking the derivative with respect to y, using Lemma 4.1, yields

∂ y w n (y, s) = e s p ′ s * w n (y, 0) - s 0 dr e s-r ∞ -∞ dz p ′ s-r (y -z)w n n (z, r).
Then integrating by parts with respect to z in the second term, we have that

∂ 2 y w n (0, s) = e s p ′′ s * w n (0, 0) -n s 0 dr e s-r ∞ -∞ dz p ′ s-r (-z)∂ z w n (z, r)w n-1 n (z, r).
Clearly, ∂ z w n (z, r) has the opposite sign to z, while p ′ s-r (-z) has the same sign as z. Hence the double integral is negative and

∂ 2 y w n (0, s) ≥ e s p ′′ s * w n (0, 0) = 1 - ǫ 2 e s 2p ′ s (ǫ 0.49 ) = -1 - ǫ 2 e s ǫ 0.49 2 √ πs 3 2
e -ǫ 0.98 4s .

The function s → s -3 2 e -ǫ 0.98 /(4s) reaches its maximum at s = ǫ 0.98 /6 and is increasing on [0, ǫ 0.98 /6). Thus, for ǫ small enough, s → s -3 2 e -ǫ 0.98 /(4s) is increasing on [0, ǫ] and so

∂ 2 y w n (0, s) ≥ -ǫ -1.01 e -ǫ -0.02 4 for s ∈ [0, ǫ].
This bound is uniform in n and goes to zero faster than ǫ. Thus, we can choose ǫ small enough that ∂ 2 y w n (0, s) > -ǫ/2 ∀s ∈ [0, ǫ]. We use this in (4.6) and obtain, by the comparison principle, w n (0, s) ≥ y n (s) for s ∈ [0, ǫ], where y n is the solution of

∂ s y n (s) = - ǫ 2 + y n (s) -y n (s) n , y n (0) = 1 - ǫ 2 .
For n sufficiently large, y n (s) is an increasing function of s. Since, for s ≥ 0, y n (s) ≥ y n (0) = 1 -ǫ/2 and y n (s) n ≤ e n(yn(s)-1) , we see, again by the comparison principle, that y n (s) ≥ z n (s) ∀s ≥ 0, where z n is the solution of

∂ s z n (s) = 1 -ǫ -e n(zn(s)-1) , z n (0) = 1 - ǫ 2 .
This last equation can be solved explicitly, giving

e -n(zn(s)-1) = 1 -e -n(1-ǫ)s 1 -ǫ + e -n (1-ǫ)s-ǫ 2 .
Hence setting s = ǫ and letting n → ∞, we obtain that for ǫ sufficiently small, lim n→∞ e -n(zn(ǫ)-1) = 1 1 -ǫ .

It follows that as n → ∞,

z n (ǫ) = 1 + log(1 -ǫ) n + o 1 n .
Therefore lim

n→∞ z n (ǫ) n = 1 -ǫ.
Since for ǫ sufficiently small and n sufficiently large we have u n (x, t) ≥ w n (0, ǫ) ≥ y n (ǫ) ≥ z n (ǫ), this implies that for ǫ > 0 sufficiently small, lim inf

n→∞ u n n (x, t) ≥ 1 -ǫ,
which yields the desired conclusion. Proof. Let

u * (x, t) := u(x, t) if u(x, t) < 1, 0 if u(x, t) = 1.
Then we have almost everywhere

u n (x, t) -u n n (x, t) → u * (x, t). ( 4
∂ x u(x, t) = p ′ t * v(x) + t 0 dr p ′ r * u * (x, t -r),
and hence, by dominated convergence, ∂ x u is continuous on R × (0, ∞), as required.

Proof of Proposition 1.3

In this section, we suppose that v : R →

[0, 1] is a non-increasing function such that v(x) → 0 as x → ∞ and v(x) → 1 as x → -∞. Let µ 0 = inf{x ∈ R : v(x) < 1} ∈ {-∞} ∪ R. Let u n
denote the solution of (1.1), and define u as in (1.2). For t > 0, let

µ t = inf {x ∈ R : u(x, t) < 1} ∪ {∞} ∈ R ∪ {∞, -∞}.
Note that since v is non-increasing, by the comparison principle we have that x → u n (x, t) is non-increasing for each n and each t ≥ 0, and therefore the same property holds for u. Hence, since u is continuous on R × (0, ∞), we have that for t > 0

u(x, t) = 1 ⇔ x ≤ µ t .
We first prove that µ t ∈ R for t > 0 and bound the increments of µ.

Proposition 5.1. µ t ∈ R for any t > 0. Furthermore, there exists a non-negative continuous increasing function ǫ → a ǫ with a 0 = 0 such that for any t > 0 and any ǫ ≥ 0,

µ t+ǫ -µ t ≥ -a ǫ . (5.1)
If µ 0 ∈ R, the above also holds at t = 0.

Proof. By (3.5), we have that for x ∈ R and t > 0,

u(x, t) ≤ e t E 0 [v(B t + x)] , (5.2)
and so, by dominated convergence, u(x, t) → 0 as x → ∞. Hence µ t < ∞ ∀t > 0.

We now turn to showing that (5.1) holds if µ t ∈ R; we shall then use this to show that µ t > -∞ for t > 0. Take v : R → R measurable with 0 ≤ v ≤ v, and, for x ∈ R and t ≥ 0, let u(x, t) = e t E x v(B t ) .

(5.3)

Let T = sup{t ≥ 0 : u(x, t) < 1 ∀x ∈ R}; (we call T the time at which u hits 1). Then

u(x, t) ≤ u(x, t) ∀x ∈ R, t ≤ T.
(5.4) Indeed, note that u(x, t) is the unique bounded solution to ∂ t u = ∂ 2 x u + u with initial condition v. By Theorem 1.2, for t < T , u(•, t) is equal to the solution of (1.1) and (1.2) with v as initial condition. Again by Theorem 1.2, it follows that u(•, t) ≤ u(•, t) for t < T . By continuity, we now have (5.4). Now fix ǫ > 0. Let v(x) = η1 {x∈[-a,a]} for some fixed η ∈ (0, 1) and a > 0 to be chosen later. For this choice of v, define u(x, s) as in (5.3). For ǫ sufficiently small, the pair (η, a) can be chosen in such a way that u hits 1 at time ǫ (we shall explain below how this is done); by symmetry, the position where u hits 1 is x = 0.

Fix t ≥ 0 such that µ t ∈ R. Our definition of v ensures that v(x -µ t + a) ≤ u(x, t) for all x ∈ R. Then, by (5.4) and the semigroup property in Theorem 1.2, u(x -µ t + a, ǫ) ≤ u(x, t + ǫ). In particular, 1 = u(0, ǫ) ≤ u(µ t -a, t + ǫ) and so µ t+ǫ ≥ µ t -a.

We now complete the proof of (5.1) by showing that it is possible, when ǫ is sufficiently small, to choose a = a ǫ := ǫ 1/3 and to find η ∈ (0, 1) such that u hits 1 at time ǫ, as required. Introduce

f (s) = u(0, s) = ηe s P 0 (|B s | < a ǫ ) = ηe s aǫ -aǫ dy √ 4πs e -y 2 4s .
Note that

P 0 (B ǫ ≥ a ǫ ) = ∞ aǫ dy √ 4πǫ e -y 2 4ǫ = ∞ aǫ/ √ ǫ dy √ 4π e -y 2 /4 ≤ e -ǫ -1/3 /4 .
Hence for ǫ sufficiently small,

e ǫ aǫ -aǫ dy √ 4πǫ e -y 2 4ǫ > 1,
and we can find η < 1 such that

u(0, ǫ) = f (ǫ) = ηe ǫ aǫ -aǫ dy √ 4πǫ e -y 2 4ǫ = 1.
It only remains to show that f (s) < 1 for s < ǫ. To do this, we simply show that f ′ (s) ≥ 0 for s < ǫ. We have

f ′ (s) = ηe s aǫ -aǫ dy √ 4πs e -y 2 4s - a ǫ s 1 √ 4πs e -a 2 ǫ 4s . ( 5.5)
Clearly, for ǫ sufficiently small and s ≤ ǫ, the first term in the parenthesis of (5.5) is arbitrarily close to 1 while the second term is arbitrarily close to 0. Hence, f ′ (s) > 0 ∀s ≤ ǫ, which concludes the proof of (5.1). Finally, we can now show that in fact µ t > -∞ for t > 0. Indeed, let v(x) = η1 {x∈[-a,a]} where a > 0 and η ∈ (0, 1) are such that u hits 1 at some time s ≤ t. (By the above argument, such a pair (η, a) can always be found. By symmetry, the position where u hits 1 is x = 0.) Now choose x 0 such that v(x -x 0 ) ≤ v(x) ∀x ∈ R (this is always possible as we assumed v(x) → 1 as x → -∞). Then by (5.4) we have u(x -x 0 , s) ≤ u(x, s) ∀x ∈ R and, in particular, 1 ≤ u(x 0 , s), which implies that µ s ≥ x 0 . We now have that µ s ∈ R for some s ≤ t and therefore, by (5.1), µ t > -∞.

Proposition 5.2. The following left-limit exists for every t > 0 and satisfies: (làg)

lim ǫց0 µ t-ǫ ≤ µ t .
Proof. Suppose that the left limit lim ǫց0 µ t-ǫ does not exist for some t > 0, and choose b and c such that lim inf

ǫց0 µ t-ǫ < b < c < lim sup ǫց0 µ t-ǫ .
Then for any ǫ > 0, there exists ǫ ′ ∈ (0, ǫ) such that µ t-ǫ ′ > c. There also exists ǫ ′′ ∈ (0, ǫ ′ ) such that µ t-ǫ ′′ < b, so that µ t-ǫ ′′ -µ t-ǫ ′ < b -c. However, by Proposition 5.1, µ t-ǫ ′′ -µ t-ǫ ′ ≥ -a ǫ , which is a contradiction if ǫ is sufficiently small that a ǫ < c -b. Hence the left limit lim ǫց0 µ t-ǫ exists. By Proposition 5.1 again, µ t-ǫ ≤ µ t + a ǫ → µ t as ǫ → 0, and so lim ǫց0 µ t-ǫ ≤ µ t .

Proposition 5.3. The map t → µ t is right-continuous (càd and hence càdlàg), i.e. for every t ≥ 0, lim

ǫց0 µ t+ǫ = µ t .
Proof. Proposition 5.1 already implies that for t ≥ 0, lim inf ǫց0 µ t+ǫ ≥ µ t . It now remains to prove that for any t ≥ 0, lim sup ǫց0 µ t+ǫ ≤ µ t . Indeed, fix t > 0 (we shall consider the case t = 0 separately). For z > 0, by the definition of µ t , we have u(µ t + z, t) < 1. Then since u is continuous on R × (0, ∞), u(µ t + z, t + ǫ) < 1 for ǫ sufficiently small, and so µ t+ǫ ≤ µ t + z. Hence lim sup ǫց0 µ t+ǫ ≤ µ t + z, and the result follows since z > 0 was arbitrary. It remains to consider the case t = 0. First suppose µ 0 ∈ R and take z > 0. Since v is non-increasing, we have that v(y)

≤ v(µ 0 + z/2) < 1 ∀y ≥ µ 0 + z/2. Since u(•, ǫ) → v in L 1 loc
as ǫ ց 0, and u(•, ǫ) is non-increasing for ǫ > 0, it follows that u(µ 0 + z, ǫ) < 1 for ǫ sufficiently small, and so µ ǫ < µ 0 + z. Hence for any z > 0, lim sup ǫց0 µ ǫ ≤ µ 0 + z. By the same argument, if µ 0 = -∞ then, for any z ∈ R, u(z, ǫ) < 1 for ǫ small enough. Therefore µ ǫ < z and so for any z ∈ R, lim sup ǫց0 µ ǫ < z.

We can finally complete the following important step: Proposition 5.4. The map t → µ t is continuous on [0, ∞).

Proof. By Propositions 5.3 and 5.2, we already have that t → µ t is càdlàg, and that for t > 0, lim ǫց0 µ t-ǫ ≤ µ t . Thus the only way in which µ could fail to be continuous would be if lim ǫց0 µ t-ǫ < µ t for some t > 0. Suppose, for some t > 0, that lim ǫց0 µ t-ǫ = a < b = µ t , and take c ∈ (a, b). Define f (s) = u(c, s) and observe that f is continuous on (0, ∞).

Since lim ǫց0 µ t-ǫ = a, we have f (t -ǫ) < 1 for all ǫ > 0 sufficiently small, but since µ t = b, we have lim s→t f (s) = f (t) = 1. Fix t 0 ∈ (0, t) such that f (s) < 1 ∀s ∈ [t 0 , t), and define (ũ(x, s), x ∈ R, s ≥ t 0 ) as the solution of the boundary value problem

       ∂ t ũ = ∂ 2 x ũ + ũ for x > c and s > t 0 , ũ(c, s) = f (s)
for s > t 0 , ũ(x, t 0 ) = u(x, t 0 ) for x ∈ R.

(5.6)

By Theorem 1.2, and since u(x, s) < 1 for s ∈ [t 0 , t) and x > c, we have that ∂ t u = ∂ 2 x u + u for x > c and s ∈ (t 0 , t). Since the solution of the boundary value problem (5.6) is unique it follows that for all s ∈ [t 0 , t) and x ≥ c we have ũ(x, s) = u(x, s). By taking s ր t we also have, by continuity, ũ(x, t) = u(x, t) for x ≥ c. But since µ t = b, we must have ũ(x, t) = u(x, t) = 1 ∀x ∈ [c, b]. Furthermore, lim x→∞ ũ(x, t) = lim x→∞ u(x, t) = 0. This is impossible because for each s > t 0 , the solution ũ(•, s) of the boundary value problem is analytic (see Theorem 10.4.1 in [START_REF] Cannon | The One-Dimensional Heat Equation[END_REF]).

The proof of Proposition 1.3 is now essentially complete. The map t → µ t is continuous on [0, ∞), whether µ 0 is finite or -∞. Therefore, defining U and S as in (4.2), we see that the topological boundary between these two domains is simply ∂U = ∂S = {(µ t , t) : t > 0}. It has measure zero, and hence by Lemma 4.8, u(•, t) is C 1 for every t > 0 and ∂ x u is continuous on R × (0, ∞).

Proof of uniqueness

In this section we prove that the classical solution to (FBP) is unique. We start with the following very simple lemma. Lemma 6.1. If (u, µ) is a classical solution of (FBP), then for t > 0,

µ t = inf{x ∈ R : u(x, t) < 1}.
Proof. Suppose, for a contradiction, that µ t < x := inf ({x ∈ R : u(x, t) < 1} ∪ {∞}) for some t > 0. Take c ∈ (µ t , x) and ǫ > 0 small enough that, by continuity, µ t+s < c ∀s ∈ [0, ǫ]. Then by Corollary 3.3, for y ∈ (c, x) and δ ∈ (0, ǫ],

u(y, t + δ) ≥ e δ P y B s ∈ [c, x] ∀s ≤ δ .
This is strictly larger than 1 for δ sufficiently small, which is a contradiction. This lemma implies that if u 1 ≤ u 2 then µ 1 ≤ µ 2 , and so the proof of the comparison property of Theorem 1.1 will be a consequence of Theorem 1.2 and the uniqueness of classical solutions of (FBP). Furthermore, it implies that if (u, µ) and (ũ, μ) are two classical solutions to (FBP) with the same initial condition v, it is sufficient to show that u = ũ to obtain that µ = μ.

For t > 0, let G t denote the Gaussian semigroup operator, so that for

f ∈ L ∞ (R) ∪ L 1 (R), G t f (x) = p t * f (x) = ∞ -∞ 1 √ 4πt e -(x-y) 2 4t
f (y) dy.

Note the following basic properties of our operators: for f, g ∈ L ∞ (R) ∪ L 1 (R), m > 0 and t > 0, we have for either norm that

C m f -C m g ≤ f -g , G t f ≤ f . (6.6)
For the supremum norm, we also have that

C m f -f ∞ ≤ max f ∞ -m, 0 . (6.7)
For w : R → [0, ∞), δ > 0 and x ∈ R,

C 1 e δ w(x) = min(e δ w(x), 1) = e δ min(w(x), e -δ ) = e δ C e -δ w(x). (6.8)

Using (6.8), we can rewrite u n,δ,-as

u n,δ,-= [e δ G δ C e -δ ] n v = [G δ C 1 e δ ] n v = G δ [C 1 e δ G δ ] n-1 C 1 e δ v. (6.9)
We can also write u n,δ,+ as

u n,δ,+ = [C 1 e δ G δ ] n v δ,+ = C 1 e δ G δ [C 1 e δ G δ ] n-1 v δ,+ . (6.10) Now let f := e δ G δ [C 1 e δ G δ ] n-1 v δ,+ -e δ G δ [C 1 e δ G δ ] n-1 v
, and let g := u n,δ,+ -u n,δ,--f . By the triangle inequality, we have

g ∞ ≤ u n,δ,+ -e δ G δ [C 1 e δ G δ ] n-1 v δ,+ ∞ + e δ G δ [C 1 e δ G δ ] n-1 v -u n,δ,- ∞ .
By our expression for u n,δ,+ in (6.10) and the properties of G δ and C 1 in (6.6) and (6.7) respectively, the first term on the right hand side is bounded above by e δ -1. A second application of the triangle inequality then yields (6.6). Replacing u n,δ,-by its expression in (6.9) gives

g ∞ ≤ e δ -1 + (e δ -1) G δ [C 1 e δ G δ ] n-1 v ∞ + G δ [C 1 e δ G δ ] n-1 v -u n,δ,- ∞ . Clearly G δ [C 1 e δ G δ ] n-1 v ∞ ≤ 1 by
g ∞ ≤ 2(e δ -1) + G δ [C 1 e δ G δ ] n-1 v -G δ [C 1 e δ G δ ] n-1 C 1 e δ v ∞ ≤ 2(e δ -1) + e δ(n-1) v -C 1 e δ v ∞ ,
where (6.6) was used repeatedly in the second inequality. (6.6), and so g ∞ ≤ 2(e δ -1) + e δ(n-1) (e δ -1) ≤ (2 + e δn )(e δ -1).

But v-C 1 e δ v ∞ = C 1 v-C 1 e δ v ∞ ≤ e δ -1 by
(6.11) By (6.6) applied repeatedly, for either norm we have f ≤ e δn v δ,+ -v .

We now need to consider two cases.

• If µ 0 = -∞, then, by our definition of v δ,+ in (6.2), we have v δ,+ -v ∞ = δ and so f ∞ ≤ e δn δ.

• If µ 0 ∈ R, then by our definition of v δ,+ in (6.1), v δ,+ -v 1 ≤ δ and so f 1 ≤ e δn δ.

In either case, if

A ≥ 1 2 then A -A |f (x)| dx ≤ 2Ae δn δ ≤ 2Ae δn (e δ -1).
By (6.11), we also have

A -A |g(x)| dx ≤ 2A(2 + e δn )(e δ -1).
By a final application of the triangle inequality to u n,δ,+ -u n,δ,-= f + g, the result follows.

(ξ s -f (s)) = 0 = 0.
Proof. By a union bound, we have that

P min s≤1 (ξ s -f (s)) = 0 ≤ P min s≤1/2 (ξ s -f (s)) = 0 + P min s≤1/2 (ξ 1-s -f (1 -s)) = 0 . Given any fixed continuous function b : [0, ∞) → R with b(0) = 0, there is at most one value of z ∈ R such that min s≤1/2 {b(s) + 2sz -f (s)} = 0.
Thus, recalling the definition of p t in (3.1),

P min s≤1/2 (ξ s -f (s)) = 0 = E P min s≤1/2 (ξ s -f (s)) = 0 ξ 1/2 = ∞ -∞ dz p 1/4 (z)P min s≤1/2 1 √ 2 ξ 2s + 2sz -f (s) = 0 = E ∞ -∞ dz p 1/4 (z)1 {min s≤1/2 { 1 √ 2 ξ 2s +2sz-f (s)}=0} = 0,
where the second equality holds since ξ s ∼ N (0, 2s(1 -s)), the third equality follows by Fubini's theorem and the last equality follows because for each realisation of (ξ s ) s∈[0,1] there is at most one value of z for which the integrand is non-zero. By the same argument,

P min s≤1/2 (ξ 1-s -f (1 -s)) = 0 = 0,
and the result follows.

Proof of Proposition 3.1. Fix (x, t) ∈ A. We begin by proving the result under condition 1. For σ ∈ [0, τ ], let

M σ = w(B σ , t -σ)e Iσ + σ 0 dr S(B r , t -r)e Ir , where I σ = σ 0 K(B s , t -s) ds.
Since w is C 2,1 on A, for σ ≤ τ , we apply Itô's formula (with no leading 1 2 in front of the ∂ 2

x term because (B s ) s≥0 has diffusivity √ 2):

dM σ = ∂ x w(B σ , t -σ)e Iσ dB σ + ∂ 2 x w(B σ , t -σ)e Iσ dσ -∂ t w(B σ , t -σ)e Iσ dσ + w(B σ , t -σ)e Iσ K(B σ , t -σ)dσ + S(B σ , t -σ)e Iσ dσ = ∂ x w(B σ , t -σ)e Iσ dB σ + [-∂ t w + ∂ 2 x w + Kw + S](B σ , t -σ)e Iσ dσ = ∂ x w(B σ , t -σ)e Iσ dB σ ,
where we used (3.2) in the last line, since (B σ , t -σ) ∈ A for σ ≤ τ . We see that (M σ ) σ≤τ is a local martingale. Therefore, since (M σ ) σ≤τ is bounded, we have that

w(x, t) = E x [M 0 ] = E x [M τ ],
which yields the result (3.3) under condition 1.

We now turn to condition 2, with A = {(x, t) : t ∈ (0, T ), x > µ t } and τ = inf s ≥ 0 : B s ≤ µ t-s ∧ t. The stopping time τ is the first time that (B τ , t -τ ) ∈ ∂A. For ǫ > 0 and δ > 0, introduce the stopping times

τ ǫ,δ = inf s ≥ 0 : B s ≤ µ t-s + δ ∧ (t -ǫ), τ ǫ = inf s ≥ 0 : B s ≤ µ t-s ∧ (t -ǫ) = τ ∧ (t -ǫ).
By (3.3) under condition 1 with stopping time τ ǫ,δ we have that for x > µ t ,

w(x, t) = E x w(B τ ǫ,δ , t -τ ǫ,δ )e τ ǫ,δ 0 K(Bs,t-s) ds + τ ǫ,δ 0 dr S(B r , t -r)e r 0 K(Bs,t-s) ds .
We now take the limit δ → 0. Since µ is continuous, τ ǫ,δ → τ ǫ as δ → 0, and since w, K and S are bounded, and w is continuous on Ā ∩ (R × (0, ∞)), we obtain, by continuity and dominated convergence, (7.1)

w(x, t) = E x w(B τǫ , t -τ ǫ )e
We now take the limit ǫ → 0 to prove (3.3) under condition 2. Note that τ ǫ ր τ as ǫ ց 0; as S and K are bounded, by dominated convergence We now turn to the first term on the right hand side of (7.1). As above, for r ≥ 0, let I r = r 0 K(B s , t -s) ds. Write E x w(B τǫ , t -τ ǫ )e Iτ ǫ = E x w(B τǫ , t -τ ǫ )e Iτ ǫ 1 {τ <t} + E x w(B t-ǫ , ǫ)e I t-ǫ 1 {τ =t} (we used that τ ǫ = t -ǫ when τ = t). Since w and K are bounded, and w is continuous on Ā ∩ (R × (0, ∞)), by continuity and dominated convergence the first term on the right hand side converges to E x w(B τ , t -τ )e Iτ 1 {τ <t} as ǫ ց 0. For the second term, write dy w(y, ǫ) φ t-ǫ (y; x, t) -φ t (y; x, t) .

E
(7.4) To show that the second line of (7.2) goes to zero as ǫ ց 0, it is then sufficient to show that φ t-ǫ (y; x, t) -φ t (y; x, t) is bounded by an integrable function and goes to 0 as ǫ ց 0.

Let (ξ t ) t∈[0,1] denote a Brownian bridge (with diffusivity Set r = t -ǫ and take the ǫ ց 0 limit. Now by the continuity of ξ, Since x > µ t , for y > µ 0 we can apply Lemma 7.1, which yields that the probability that the lower and upper bounds above are different is zero. We can conclude, by dominated convergence, that lim ǫց0 φ t-ǫ (y; x, t) = φ t (y; x, t).

1
For y < µ 0 , we have that y < µ ǫ for ǫ sufficiently small, and so for ǫ sufficiently small, φ t-ǫ (y; x, t) = 0 = φ t (y; x, t).

Since t > 0, and φ t-ǫ (y; x, t) ≤ p t-ǫ (x -y)e t K ∞ by (7.3), it is easy to see that φ t-ǫ (•; x, t) can be uniformly bounded for ǫ < t/2 by a function with Gaussian tails. Therefore, by dominated convergence we see that (7.4) (which is the second line of (7.2)) goes to 0 as ǫ ց 0. It only remains to consider the third line of (7.2). Using (7.3), we can write

E x w(B t , ǫ)e It 1 {τ =t} = ∞ -∞
dy w(y, ǫ)φ t (y; x, t). (7.5)

Since w is bounded, φ t (y; x, t) ≤ p t (x -y)e t K ∞ and w(•, ǫ) → w(•, 0) in L 1 loc as ǫ ց 0, we have that This completes the proof.

∂ 8 √ 1 √

 81 y w n (y, s) = e s p ′ s * w n (y, 0) -p s-r (y -z)n∂ z w n (z, r)w n-1 n (z, r).Note that |∂ z w n (z, r)| ≤ 1 √ πr +√ π ∀z ∈ R by Lemma 4.2, and the map r → e s-r integrable on [0, s]. Hence by Lemma 4.1, we can take the derivative with respect to y again, to obtain, at y = 0,

Lemma 4 . 8 .

 48 If the topological boundary ∂U = ∂S between U and S has measure zero, then x → u(x, t) is C 1 for every t > 0, and ∂ x u is continuous on R × (0, ∞).

τǫ 0 K

 0 (Bs,t-s) ds + τǫ 0 dr S(B r , t -r)e r 0 K(Bs,t-s) ds .

  B r , t -r)e r 0 K(Bs,t-s) ds → E x τ 0 dr S(B r , t -r)e r 0 K(Bs,t-s) dsas ǫ ց 0.

√ 2 )

 2 from 0 to 0 in time 1 and introduce, for x, y fixed,F (ξ, s, r) = √ rξ + x + s y -x r , so that (F (ξ s/r , s,r)) s≤r is a Brownian bridge from x to y in time r. Then by (7.3), φ r (y; x, t) = p r (x -y)E x e Ir 1 {τ ≥r} B r = y = p r (x -y)E x e r 0 K(Bs,t-s)ds 1 {µ t-s <Bs ∀s<r} B r = y = p r (x -y)E e r 0 K F (ξ s/r ,s,r),t-s ds 1 {µ t-s <F (ξ s/r ,s,r) ∀s<r} .

  dy w(y, ǫ)φ t (y; x, t) → ∞ -∞ dy w(y, 0)φ t (y; x, t) = E x w(B t , 0)e It 1 {τ =t}as ǫ ց 0.

  .3) Proof. Choose (x, t) ∈ S. Let a, b, t 0 and t 1 be such that x ∈ (a, b), t ∈ (t 0 , t 1 ) and [a, b]×[t 0 , t 1 ] ⊂ S. By (3.4), we have that for (x

7 Proof of the Feynman-Kac results from Section 3

  Before proving Proposition 3.1, we need the following result: Let f : [0, 1] → R∪{-∞} be continuous with f (0) < 0 and f (1) < 0. Let (ξ t ) t∈[0,1] denote a Brownian bridge (with diffusivity √ 2) from 0 to 0 in time 1. Then

	Lemma 7.1. P min s≤1

  x w(B t-ǫ , ǫ)e I t-ǫ 1 {τ =t} = E x w(B t-ǫ , ǫ)e I t-ǫ 1 {τ =t} -1 {τ ≥t-ǫ}+ E x w(B t-ǫ , ǫ)e I t-ǫ 1 {τ ≥t-ǫ} -E x w(B t , ǫ)e It 1 {τ =t} + E x w(B t , ǫ)e It 1 {τ =t} .In this equation, we set w(B t , ǫ) to an arbitrary bounded value when B t < µ ǫ .) It is clear by dominated convergence that the first line on the right hand side of (7.2) goes to 0 as ǫ ց 0.Let us now show that the second line of (7.2) goes to 0 as ǫ ց 0. Define φ r (y; x, t) as φ r (y; x, t) = E x δ(B r -y)e Ir 1 {τ ≥r} = p r (x -y)E x e Ir 1 {τ ≥r} B r = y .Thus, the second line of (7.2) can be written as E x w(B t-ǫ , ǫ)e I t-ǫ 1 {τ ≥t-ǫ} -E x w(B t , ǫ)e It 1 {τ =t} =

	∞
	-∞
	(7.2)
	((7.3)

(This is the probability, weighted by e Ir , that the path of length r started from (x, t) arrives at (y, t -r) without touching the left boundary.) By integrating over the value of B r , we have that for r ≤ t,

E x w(B r , ǫ)e Ir 1 {τ ≥r} = ∞ -∞

dy w(y, ǫ)φ r (y; x, t).

  {µ t-s <F (ξ s/t ,s,t) ∀s<t} ≤ lim inf

ǫց0

1 {µ t-s <F (ξ s/(t-ǫ) ,s,t-ǫ) ∀s<t-ǫ} ≤ lim sup ǫց0 1 {µ t-s <F (ξ s/(t-ǫ) ,s,t-ǫ) ∀s<t-ǫ} ≤ 1 {µ t-s ≤F (ξ s/t ,s,t) ∀s<t} .

Acknowledgements: JB and EB thank E. Presutti and A. De Masi for their hospitality at the GSSI, Italy where part of this work was conducted. EB was partially supported by ANR grant ANR-16-CE93-0003 (ANR MALIN). JB was partially supported by ANR grants ANR-14-CE25-0014 (ANR GRAAL) and ANR-14-CE25-0013 (ANR NONLOCAL).

For m > 0, let C m denote the cut operator given by C m f (x) = min(f (x), m).

Suppose that v : R → [0, 1] is as in Theorem 1.1, i.e. v is non-increasing, v(x) → 0 as x → ∞ and v(x) → 1 as x → -∞. For n ∈ Z ≥0 and δ > 0, introduce u n,δ,-(x) := e δ G δ C e -δ n v(x) and u n,δ,+ (x) := C 1 e δ G δ n v δ,+ (x),

where we now define v δ,+ . Recall that

)

Our proof of uniqueness relies on the Feynman-Kac representation of Corollary 3.3 and the following two results. Lemma 6.2. Suppose (u, µ) is a classical solution of (FBP) with initial condition v. Then for n ∈ Z ≥0 , δ > 0 and x ∈ R, u n,δ,-(x) ≤ u(x, nδ) ≤ u n,δ,+ (x). ( 6.3) Lemma 6.3. For any δ > 0, n ∈ Z ≥0 , and

Suppose that (u, µ) and (ũ, μ) are classical solutions of (FBP) with initial condition v. Then by Lemmas 6.2 and 6.3, for t > 0, n ∈ Z ≥0 and A ≥ 1 2 ,

Since n ∈ Z ≥0 can be taken arbitrarily large, it follows that A -A u(x, t)-ũ(x, t) dx = 0. Letting A → ∞, by continuity of u(•, t) and ũ(•, t) it follows that u(x, t) = ũ(x, t) ∀x ∈ R. Therefore (u, µ) is the unique classical solution to (FBP) with initial condition v.

It remains to prove Lemmas 6.2-6.3. We shall require the following preliminary result for the proof of Lemma 6.2.

Lemma 6.4. Suppose v

Then µ + t ∈ R ∀t ≥ 0 and t → µ + t is continuous. This is a simple result about the heat equation, which can be proved, for instance, using the same techniques as in Section 5.

Proof of Lemma 6.2. We shall show the following result: suppose that v : R → [0, 1] is as in Theorem 1.1 and that (u, µ) is a classical solution of (FBP) with initial condition v. Let

Then we shall prove that

and (6.3) follows by the definition of v δ,+ and by induction on n.

We now prove (6.4). Let τ = inf{s ≥ 0 :

By continuity of µ t and µ + t (from Lemma 6.4), we have t 0 > 0. We will show below that t 0 = ∞. Take t < t 0 and, again, let τ = inf{s ≥ 0 : B s ≤ µ t-s } ∧ t. By Proposition 3.1 we have

Then, again by Corollary 3.3,

where the second line follows since v ≤ v + and since, on {τ < t}, we have B τ = µ t-τ < µ + t-τ and so u + (B τ , t -τ ) ≥ 1. By continuity, the inequality also holds for t = t 0 and so

Suppose, for a contradiction, that t 0 < ∞. Then, by continuity, µ + t 0 = µ t 0 . Hence u(µ t 0 , t 0 ) = 1 = u + (µ t 0 , t 0 ) and ∂ x u(µ t 0 , t 0 ) = 0, and so by (6.5), ∂ x u + (µ t 0 , t 0 ) = 0.

Note that u + is smooth on R × (0, ∞) and, by the same argument as in Lemma 4.2, for t > 0,

We now have a contradiction. Therefore t 0 = ∞ and we have µ + δ > µ δ and u + (x, δ) ≥ u(x, δ) ∀x ∈ R by (6.5). This completes the proof of (6.4).

Proof of Lemma 6.3. Some of the ideas in this proof are from Section 4.3 of [START_REF] De Masi | Non local branching Brownians with annihilation and free boundary problems[END_REF].

In this proof, we use both the supremum norm ∞ and the L 1 norm 1 . When a property holds for both norms, we simply write it with .