Julie Lucas
email: julie.lucas@univ-lr.fr

Alicia Schouman

Laura Lyphout

Xavier Cousin

Christel Lefrancois

Allometric relationship between body mass and aerobic metabolism in zebrafish Danio rerio

Keywords: allometric scaling exponent, aerobic metabolism, zebrafish

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Active metabolic rate (R A ) is the maximal aerobic metabolic rate of an organism in a highly active state and standard metabolic rate (R S ) isthe minimal metabolic rate necessary for supporting maintenance activities (e.g. ventilation) measured in resting, starved and nonmaturing individual [START_REF] Fry | The effects of the environment on animal activity[END_REF][START_REF] Brett | The respiratory metabolism and swimming performance of young sockeye salmon[END_REF]. Aerobic metabolic scope (R AS ) is the difference between R A and R S and represents the capacity of an organism to provide oxygen to sustain energy demanding activities [START_REF] Fry | The effects of the environment on animal activity[END_REF]. R AS can be influenced by a set of environmental parameters such as temperature, oxygen andsalinity (Shurmann&Steffensen, 1997;Lefrançois&Claireaux, 2003;[START_REF] Johansen | Increasing ocean temperature reduces the metabolic performance and swimming ability of coral reef damselfishes[END_REF]. Ithas therefore been used in many studies to assess the influence of environmental factors on metabolic performance and energydemanding activitiesin various fish species [START_REF] Fry | The effect of environmental factors on the physiology of fish[END_REF][START_REF] Priede | Metabolic scope in fish[END_REF]Claireaux&Lefrançois, 2007).It has been demonstrated, in many species however, thatintrinsic parameters such as ontogeny and body size can also influence metabolism independently of any effects of environmental conditions (Clark & Johnston, 1999;[START_REF] Bokma | Evidence against universal metabolic allometry[END_REF][START_REF] Killen | Little left in the tank: metabolic scaling in marine teleosts and its implications for aerobic scope[END_REF][START_REF] Killen | The intraspecific scaling of metabolic rate with body mass in fishes depends on lifestyle and temperature[END_REF][START_REF] Moran | Ontogenetic scaling of fish metabolism in the mouse-toelephant mass magnitude range[END_REF] ZebrafishDaniorerio(Hamilton 1822) is a small teleost species whose biological characteristics make it suitable for experimental studies (e.g. small size, easy breeding, high fecundity; Miklosi& [START_REF] Miklosi | The Zebrafish as a Model for Behavioral Studies[END_REF][START_REF] Lawrence | The husbandry of zebrafish (Daniorerio): A review[END_REF]. Its short life cycle and rapid development also facilitate investigations regarding the effects of environmental factors on different lifestages. In the last two decades,some studies have been publishedregarding the metabolic performance ofthe D.rerioexposed to various environmental constraints [START_REF] Plaut | Swimming metabolism of wild-type and cloned zebrafishBrachydaniorerio[END_REF][START_REF] Plaut | Effects of fins size on swimming performance, swimming behaviour and routine activity of zebrafishDaniorerio[END_REF][START_REF] Marit | Acute exposure to 2,4-dinitrophenol alters zebrafish swimming performance and whole body triglyceride levels[END_REF], 2012). Despite the increasing interest in D.rerio, only the studies of Barrionuevo&Burggren (1999) and [START_REF] Burggren | Developing animals flout prominent assumptions of ecological physiology[END_REF] investigated how body mass influenced metabolism, although they did not establish the regression describing this relationship.The aim of the present study was therefore to investigate the allometricrelationship between body mass and metabolic rate in D.rerio, and derive the species-specific scaling exponent. This was achieved by measuring R A and R S atthree different life stages.

Tenbroodstock couples (wild-type Tuebingenstrain, TU) werereared together in a 10 l tank.

Spawnings were obtained from these couples each week over two month. In order to avoid any bias due to familial characteristics, at least 5 spawns were mixed. Then, 100 larvae were randomly selected and reared to be tested at three different life stages: (i) larvae, 5 days post fertilization, (ii) 60 dpf juveniles and (iii) 5 months old adults. From 2 weeks onwards, fish were kept in groups of 10 individuals in 3l aquaria with system water prepared as a mix of reverse-osmosis treated and tap water (both being filtered through sediment and activated charcoal filters) to obtain a water with constant conductivity of 300µS. Fish were reared at a constant temperature of 28 °C under artificial light with a constant photoperiod of 14h:10h (L:D) and were fed with dry food (Inicio plus, Biomar, France, http://www.biomar.com) and brine shrimp (Océan Nutrition Europe BVBA, Belgium, http://www.oceannutrition.eu) in the morning and afternoon, respectively. Fish were fasted 24 h prior to the experiments.

To assess their metabolic rate, fish were placed into circular size-adapted respirometers. For larvae, the set-up consisted of 4independent glass micro respirometer chambers (diameter d= 1.12 cm, volume V= 0.985.10 -3 l; Loligo systems, Denmark). For the two other life stages, 8 larger respirometers were run simultaneously (d= 3.75cm, V=0.061l and 7.50 cm, 0.179 l for juveniles and adults respectively).Respirometers were submerged into buffer tanks (depth x length x height: 10x20x31cm for larvae, 10x75x75xcmfor juveniles and adults) and filled with temperature-controlled, oxygenated system water as above. Flush pumps controlled the water supply in each respirometer. Each flush pump was controlled by a timer, allowing intermittent flow respirometry [START_REF] Steffensen | Some errors in respirometry of aquatic breathers: how to avoid and correct for them[END_REF] where phases of oxygen renewal alternated with phases of oxygen consumption measurements with a period of 15:15min for larvae and 30:30 min for both juvenilesand adults. In addition, the set-up was completed by amultichannel peristaltic pump was used to mix the water within each respirometer. Each respirometer was equipped with a fiber optic sensor (PreSens) connected to a multichannel oxygen measuring system (OXY 4 mini, PreSens)to record dissolved oxygen levels. Optic fibers were calibrated at 0% and 100% of air saturation at a temperature of 28°C. A factor of conversion based on oxygen solubility into water was used to convert oxygen data from percentage saturation to mgO 2 .l -1 (i.e. 100% was equivalent to 7.94mgO 2 .l -1 for a 28°C temperature and a 0 salinity).Oxygen saturation was recorded every five seconds with the program Oxyview (PreSens).

Larvae were tested in groups of 10 individuals per chamber. Juveniles and adults were tested individually; each fish was only tested once. Each experimental trial comprised two consecutive steps. First,to assessR A , fish metabolism was increased through chasing (Schurmann&Steffensen, 1997;Lefrançois&Claireaux, 2003;Jourdan-Pineauet al., 2010;Cannas et al., 2013[START_REF] Clark | Physiological benefits of being small in a changing world: responses of Coho salmon (Oncorhynchuskisutch) to an acute thermal challenge and a simulated capture event[END_REF]. Each group of larvae or individual fish was transferred from the rearing aquaria to a 6-7 ml Petri dish or one liter tank respectively, where they were chased with a stick. When exhausted (i.e. did not respond to stimulation), they were transferred in respirometers to measure oxygen consumption over 20 min. After this first measurement of oxygen consumption, a second one was performed to confirm the accuracy of the RA assessment. To do that fish were chased again inside the respirometer and their oxygen consumptionmeasured for a new period of 20 min.To estimateR S , fish were left undisturbed in the respirometerfor 48h, during which oxygen consumptionwas regularly and automatically measured. After the 48h period of measurements, the fish were removed from the respirometers and anesthetized with benzocaïne at a concentration of 50mg.l -1 . The wet body mass of each individual was determined, as well as the standard length L S of juveniles and adults: (a) larvae, 5 dpf (n=11 groups of 10 larvae; mean±S.E., mass, M=0.245.10 -3 ±0.036.10 - 3 g), (b) 60 dpf juveniles (n=14, M=0.097 ± 0.035g, L S =18.5 ± 2.2 mm) and (c) 5 months adults (n=22, M=0.326 ± 0.103g, L S =26.9 ± 1.9 mm).

A blank measurement without fish was performed before and after each trial, to quantify background respiration.Linear change is assumed in background oxygen consumptionover the 48h-experimental trial and subtracted the calculated background from the corresponding total oxygen consumptionmeasured.

The non specific aerobic metabolic rate (or oxygen consumption MO2 in mgO2.h. 1 ) was calculated according to the following formula:

MO 2 =Δ[O 2 ].V. Δt -1 equation (1) whereΔ[O 2 ] (in mgO 2 .l -1
) is the variation in oxygen concentration during the measurement periodΔt(in h) and V(in l) is the volume of the respirometer. MO 2 correspond to the R A ,themaximalMO 2 obtained after the fish being chased, or to the R S , which is the MO 2 assessed according to the method described by [START_REF] Steffensen | Oxygen consumption in four species of teleosts from Greenland: no evidence of metabolic cold adaptation[END_REF]. Briefly, to assess R S , thefrequency distribution of the MO 2 values was plotted recorded during the last 24 hours of each trial. This generally produced a bimodal distribution where the higher and the lower mode were considered to reflect routine metabolic rate, i.e. energy required by the animal for normal activity, and R S , respectively.R A and R S were assessed for each individual.

The non specific aerobic metabolic rate of organisms typically increases with body mass according to the allometric equation:

R=aM b equation (2)
whereR is the metabolic rate (R S or R A in mgO 2 .h -1 ), a is the species-specific scaling constant (or proportionality constant), M is the body mass (in g), and b is the scaling exponent. This equation is a power function, where the value of b provides information on how the variable of interest changes with body size. Equation ( 2) can be linearized with a log transformation:

log Y=log a + b log M equation (3)
The logarithm base 10 of each metabolic variable (R S or R A )was therefore plotted against the logarithm base 10 of body mass. Fitting a linear regression allowed provided the scaling exponent b and the log of the scaling coefficient (loga), i.e. the slope and the intercept, respectively. Previous studies in fishes [START_REF] Post | Metabolic ontogeny of teleost fishes[END_REF][START_REF] Killen | Little left in the tank: metabolic scaling in marine teleosts and its implications for aerobic scope[END_REF][START_REF] Moran | Ontogenetic scaling of fish metabolism in the mouse-toelephant mass magnitude range[END_REF] have argued themetabolic rate of larvae is likely to scale differently from juveniles and adults because of their incomplete development. These groups were, therefore, analyzed separately. A first linear regression was therefore derived for the three stages and a second considering only juveniles and adults. The two scaling exponents b and scaling factors a werethen compared as described by [START_REF] Zar | Biostatistical analysis[END_REF] with a modification of the t-test. Results were considered significant atP<0.05.

For R S , scaling factor a RS and scaling exponent b RS was 0.273±0.038 (± S.E.) and 0.965±0.015, respectively when the three life stages were considered. , reported byBarrionuevo&Burggren (1999) and [START_REF] Burggren | Developing animals flout prominent assumptions of ecological physiology[END_REF] and converted to mgO 2 .h -1 , were consistent with R S values measured in the current study (Fig. 1A). It appears that their data fit with standard metabolic rate regression. 0.276(±0.051)M 0.931(±0.068) for n=60 (P<0.0001).In addition, the studies of Barrionuevo&Burggren (1999) and [START_REF] Burggren | Developing animals flout prominent assumptions of ecological physiology[END_REF] measured oxygen consumption in fish presenting a resting state at the temperature of 28 °C which permitted to complete our data set regarding the standard metabolic rate at the same temperature. After conversion into mgO 2 .h -1 , these values were added on the graph (A, ■).

This is the first study examining the scaling of metabolic rate with body mass over the lifecycle of D.rerio. The results indicate that a single scaling exponent can be employed irrespective of life stgae in this species. This is in agreement with the observations made by [START_REF] Killen | Little left in the tank: metabolic scaling in marine teleosts and its implications for aerobic scope[END_REF] on three species of marine teleost (e.g. ocean poutMacrozoarcesamericanusBloch & Schneider 1801; lumpfish Cyclopteruslumpu,sL. 1758; and shorthorn sculpinMyoxocephalusscorpius, L. 1758), while the contrary was shown by [START_REF] Post | Metabolic ontogeny of teleost fishes[END_REF] for other species (e.g. the common carp Cyprinus carpi, L. 1758; rainbow trout Oncorhynchusmykis, Walbaum 1792; sea bream Pagrusmajo, Temmink& Schlegel 1843). [START_REF] White | Physiological functions that scale to body mass in fish[END_REF] studied mass-specific R S of 31 fish species and reported that several species have higher allometric exponents in early larval stages compared to their later stages. Also, in their study of yellowtail kingfish Seriolalalandi (Valenciennes 1833), [START_REF] Moran | Ontogenetic scaling of fish metabolism in the mouse-toelephant mass magnitude range[END_REF] considered that the allometric exponent changed continually during development. In fact, metabolism in larvae may be affected by processes specific to this early ontogenic stage such as higher rates of protein turnover, development of energy consuming tissue or organs, or the progressive transition from cutaneous gas exchange to branchial respiration or changes in swimming ability [START_REF] Post | Metabolic ontogeny of teleost fishes[END_REF]Killenet al., 2007;[START_REF] White | Physiological functions that scale to body mass in fish[END_REF][START_REF] Gore | Cardiac and metabolic physiology of early larval zebrafish (Daniorerio) reflects parental swimming stamina[END_REF].Nontheless, understanding of ontogenic changes in aerobic scope in fishes throughout remains limited. In D. rerio, the general pattern observed may be related to its short lifecycle, 5dpf larvae having already completed most of their organogenesis [START_REF] Kimmel | Stages of Embryonic Development of the Zebrafish[END_REF]. It could be interesting to obtain data for 21dpf, when larval metamorphosis occurs with maturation of several physiological functions and the completion of the transfer of respiratory gas exchange from predominantly cutaneous to predominantly branchial.

As expected, the scaling exponents b RA and b RS differed significantly from 1, confirming that metabolic rate scaled allometrically with body mass in this species (Student"s t-test, t 1.47 =55.88, P<0.0001 for b RA ; t 1.47 =5.224, P<0.05 for b RS ).

Although studies by [START_REF] Kleiber | Body size and metabolism[END_REF] on mammals and Hemmingsen (1960) onunicells, multicellular ectotherms and endotherms suggested that 0.75 was a universal scaling exponent, recent investigations on fishes have revealed significant heterogeneity in scaling exponents among species [START_REF] Glazier | Beyond the "3/4-power law": variation in the intraand interspecific scaling of metabolic rate in animals[END_REF][START_REF] Glazier | The 3/4-power law is not universal: evolution of isometric, ontogenetic metabolic scaling in pelagic animals[END_REF](Glazier, , 2009c;;[START_REF] White | The scaling and temperature dependence of vertebrate metabolism[END_REF][START_REF] Downs | Scaling metabolic rate with body mass and inverse body temperature: a test of the Arrhenius fractal supply model[END_REF][START_REF] White | Allometric estimation of metabolic rates in animals[END_REF].Among the teleosts, several studies have found that the scaling exponent of metabolic rate differed from 0.75 [START_REF] Post | Metabolic ontogeny of teleost fishes[END_REF]Clark & Johnston, 1999;[START_REF] Bokma | Evidence against universal metabolic allometry[END_REF]Peck, 2004), which is in agreement with the present results. It is worth noticing that a lot of scaling studies in teleosts are based on routine or "resting" metabolic rate, rather than R S or R A . This is not directly comparable with the present results, but illustrates the interspecific variation in allometric scaling exponents among fishes. For instance, [START_REF] Moran | Ontogenetic scaling of fish metabolism in the mouse-toelephant mass magnitude range[END_REF] found a scaling exponent of 0.90 forS. lalandi. [START_REF] Bokma | Evidence against universal metabolic allometry[END_REF] examined intraspecific allometry for the sea trout Salmotrutatruta in various life stages and found b RMR to be 0.86.

Indeed, lifestyle, swimming mode and environmental characteristics of the habitat are all known to modify metabolic rate, as well as the scaling exponent in fishes [START_REF] Killen | Little left in the tank: metabolic scaling in marine teleosts and its implications for aerobic scope[END_REF][START_REF] Killen | The intraspecific scaling of metabolic rate with body mass in fishes depends on lifestyle and temperature[END_REF]. Previous studies have also found that b RA was significantly different from b RS [START_REF] Brett | Metabolic rates and critical swim speeds of sockeye salmon (Oncorhynchusnerka)[END_REF][START_REF] Weiser | Developmental and metabolic constraints of the scope for activity in young rainbow trout (Salmogairdneri )[END_REF]Weibelet al., 2004;[START_REF] Killen | Little left in the tank: metabolic scaling in marine teleosts and its implications for aerobic scope[END_REF], and this was true in D.

rerio(Student"s t-test t 1.94 =4.475, P<0.05), which is also suggested by several studies.

However, b RA generallytends to be higher than b RS (Weibelet al., 2004;Killenet al., 2007) while the opposite was observed in D. rerio, where b RA <b RS. . This may, at least in part, reflect the fact that D. rerio is a domesticated species that has been reared in laboratory for many generations, whereas the study by Killenet al. (2007) was on individuals captured in wild.

In conclusion, this study estimated the scaling exponent b across three life stages of D. rerio

(5 dpf to 6 month old fish). This provides a basis for further experiments regarding the effects of environmental factors on aerobic metabolismin this species.

FIG. 1 .

 1 FIG. 1. Relationships between aerobic metabolic rate and body mass for D. rerio. On each graph, the black line represents the scaling relationship across the three life stages: 5 day-old larvae (○, L), 2 month-old juveniles (□, J) and 6 month-old adults (Δ, A). The white line shows the scaling only for juveniles and adults (J+A). The scaling factor a and scaling exponent b were estimated through the allometric equation R=aM b (equation 2).For each of these constants, the value is expressed as the average (± standard error). For standard metabolic rate (graph A), considering all stages R S(L+J+A) is equal to 0.799 (±0.038) M 0.965 (±0.085) for n=50 (P<0.0001), considering juveniles and adults, R A(J+A) is equal to 0.809 (±0.084) M0.969 (±0.110) for n=37 (P<0.0001). Regarding active metabolic rate (graph B): R A(L+J+A) 0.273(±0.024)M0.926(±0.010) for n=78 (P<0.0001), R A(J+A)

  J+A) =0.931±0.068)did not differ significantly whether larvae were considered or not (Student"s t-test, for a RA : t 1.81 =0.01270, P>0.05; for b RA : t 1.81 =0.00596, P>0.05). Data for

	a RA(J+A) =0.809±0.051) as well as	scaling exponents (b RA(L+J+A) =0.926±0.009 and
	b RA(
		These constantswere
	0.276±0.084 and 0.969±0.110, respectively, when only juveniles and adults were taken into
	consideration. These values did not differ significantly for either constant(Student"s t-test,
		799±0.024 and

t 1.82 =-0.00127, P>0.05; Student"s t-test, t 1.81 =0.00068, P>0.05, respectively). The same pattern was observed for R A . Scaling factors (a RA(L+J+A) =0.

"resting metabolic rate" of D. rerio at 28°C

The authors thank Didier Leguay and Michel Prineau, for their help during the experiment.

All experiments were carried out at Ifremer (Plateformed'Ecophysiologie des Poissons), La Rochelle station, France.This study was financially supported by the ANR project ConPhyPoP (CES 09_002) and JL received a doctoral grant of the Regional Council of Poitou-Charentes. This study was conducted under the approval of Animal Care Committee of France under the official licence of Marie-Laure Bégout (17-010).


domain: sdv sde

text:
Julie Lucas
email: julie.lucas@univ-lr.fr

Antoine Bonnieux

Laura Lyphout

Xavier Cousin

P Miramand

C Lefrancois

Trophic contamination

Keywords: metabolic rates, static respirometry, sub-lethal concentration, petroleum hydrocarbons

Present as complex mixtures in environment, pyrolytic polycyclic aromatic hydrocarbons (PY PAH) result from combustion of organic matter and enter aquatic ecosystems through atmospheric deposition [START_REF] Hylland | Polycyclic aromatic hydrocarbon (PAH) ecotoxicology in marine ecosystems[END_REF]. Due to their high liposolubility, PAHare typically adsorbed by organic matter or marine sediments,bioaccumulated by organisms at the lowest trophic levels (e.g. invertebrates; O 'Connor &Lauenstein, 2006;[START_REF] Bustamante | Bioaccumulation and metabolisation of 14C-pyrene by the Pacific oyster Crassostrea gigas exposed via seawater[END_REF]and transferred through trophic chains [START_REF] Hylland | Polycyclic aromatic hydrocarbon (PAH) ecotoxicology in marine ecosystems[END_REF]Vignet et al., 2014aVignet et al., , 2014b)).

In the context of environmental changes and risk management, assessment of the toxicity of PAH is necessary to evaluate their impacts on aquatic life. Past studies on fisheshave demonstrated that hydrocarbons have carcinogenic [START_REF] Hawkins | Carcinogenic effects of some polycyclic aromatic hydrocarbons onthe Japanese medaka and guppy in waterborne exposures[END_REF][START_REF] Myers | Relationships between hepatic neoplasms and related lesions and exposure to toxic chemicals in marine fish from the U.S. West Coast[END_REF][START_REF] Larcher | Carcinogenic but no genotoxic effects detected following chronic dietary exposure of zebrafish to PAH mixtures[END_REF], genotoxic (e.g. DNA damage; [START_REF] Holth | Differential gene expression and biomarkers in zebrafish (Danio rerio) following exposure to produced water components[END_REF][START_REF] Nogueira | Transcript profiling and DNA damage in the European eel (Anguilla anguilla L.) exposed to 7,12 dimethylbenz[a]anthracene[END_REF] and ontogenic effects [START_REF] Horng | A reproductive toxicology study of phenanthrene in medaka (Oryzias latipes)[END_REF][START_REF] Incardona | Cardiac toxicity of 5-ring polycyclic aromatic hydrocarbons is differentially dependent on the aryl hydrocarbon receptor 2 isoform during zebrafish development[END_REF][START_REF] Singh | Blood levels of polycyclic aromatic hydrocarbons in children and their association with oxidative stress indices: An Indian perspective[END_REF]. They havealso been found to affect reproduction [START_REF] Collier | Field studies of reproductive success in English sole (Parophrys vetulus): correlations with bioindicators of maternal contaminant exposure[END_REF][START_REF] Seruto | Evaluation of the relationships between biochemical endpoints of PAH exposure and physiological endpoints of reproduction in male California halibut (Paralichthys californicus) exposed to sediments from a natural oil seep Cherlynn[END_REF]Vignet et al. 2014a), growth [START_REF] Meador | Altered growth and related physiological responses in juvenile Chinook salmon (Oncorhynchus tshawytscha) from dietary exposure to polycyclic aromatic hydrocarbons (PAH)[END_REF][START_REF] Kim | Effects of dietary benzo[a]pyrene on growth and hematological parameters in juvenile rockfish, Sebastes schlegeli (Hilgendorf)[END_REF], Vignet et al., 2014a), metabolism [START_REF] Wedemeyer | Assessing the tolerance of fish and fish populations to environmental stress: the problems and methods ofmonitoring[END_REF][START_REF] Wedemeyer | Stress and acclimation[END_REF][START_REF] Davoodi | Effects of exposure to petroleum hydrocarbons upon the metabolism of the common sole Solea solea[END_REF]) andbehaviour (Gonzales-Doncelet al., 2008;Vignet et al., 2014b).

This study aimedatinvestigating responses of fish exposedto PAHthrough the assessment of aerobic metabolic scope (AS)as an indicator of the physiological state of the organism [START_REF] Fry | The effects of the environment on animal activity[END_REF]. AS represents the amount of oxygen the animal is able to provide for all activities beyond standard metabolism (e.g. locomotion, digestion, feeding; [START_REF] Fry | The effects of the environment on animal activity[END_REF][START_REF] Fry | The effect of environmental factors on the physiology of fish[END_REF]. It is defined as the difference between active metabolic rate (AMR), which is the highest metabolic rate the organism can sustain, usually during maximal activity, and the standard metabolic rate (SMR), the metabolic rate necessary to maintain vital functions and measured under resting conditions at a known ambient temperature (e.g. [START_REF] Brett | The respiratory metabolism and swimming performance of young sockeye salmon[END_REF][START_REF] Fry | The effects of the environment on animal activity[END_REF]White et al. 2006).ASis known to be modulated by pollutants (e.g. [START_REF] Sharp | Physiological basis ofdifferential sensitivity of fish embryonic stages to oil pollution[END_REF][START_REF] Hose | Oxygen consumption of grunion (Leuresthes tenuis) embryos exposed to the petroleum hydrocarbon, benzo[a]pyrene[END_REF][START_REF] Correa | Physiological responses of juvenile white mugil, Mugil curema, exposed to benzene[END_REF][START_REF] Davison | The effect of acute exposure to the water soluble fraction of diesel fuel oil on survival and metabolic rate of an Antarctic fish (Pagothenia borchgrevinki)[END_REF][START_REF] Nikinmaa | How does environmental pollution affect red cell function in fish?[END_REF][START_REF] Wilson | Metabolic costs and physiological consequences of acclimation to aluminum in juvenile rainbow trout (Oncorhynchus mykiss). 1: Acclimation specificity, resting physiology, feeding, and growth[END_REF][START_REF] Lannig | Temperature-dependent stress response in oysters, Crassostrea virginica: Pollution reduces temperature tolerance in oysters[END_REF][START_REF] Johansen | Increasing ocean temperature reduces the metabolic performance and swimming ability of coral reef damselfishes[END_REF][START_REF] Davoodi | Effects of exposure to petroleum hydrocarbons upon the metabolism of the common sole Solea solea[END_REF][START_REF] Christiansen | Sub-lethal levels of waterborne petroleum may depress routine metabolism in polar cod Boreogadus saida (Lepechin, 1774)[END_REF]. Focussing on PAH, [START_REF] Davison | The effect of acute exposure to the water soluble fraction of diesel fuel oil on survival and metabolic rate of an Antarctic fish (Pagothenia borchgrevinki)[END_REF] reported that the Antarctic fish Pagothenia borchgrevinki (Boulenger, 1902) doubled its ASafter exposure to an aqueous fraction of petroleum. [START_REF] Davoodi | Effects of exposure to petroleum hydrocarbons upon the metabolism of the common sole Solea solea[END_REF] showed a 30% decrease of AS in the common sole Solea solea (Quensel, 1806) acutely exposed to a fuel.This ASreduction could be explained by malfunctions in organs involved in oxygen transport (e.g. heart, gills) and an associated decrease of AMR (Claireaux, 2004). Another possibility to reduce AS isthe setting up of supplementary energy-demanding detoxification processes, which could increase SMR [START_REF] Lannig | Temperature-dependent stress response in oysters, Crassostrea virginica: Pollution reduces temperature tolerance in oysters[END_REF].

Thisstudy aimedto determine the impacts of environmentally relevant concentrations of PY PAHon the aerobic metabolism of zebrafish Danio rerio(F.

Hamilton, 1822)contaminated by ingestion. The main hypothesiswasthat chronic exposureto PYPAH would impair AS by increasing SMRand/or reducing AMR. The consequentpotential reduction of ASwouldindicatea decrease inthe capacity ofthe fish to supportoxygen-demanding activities beyond SMR. Metabolic variables were assessed for two durations of chronic exposure, 2 and 6 months (juvenile and adult stages, respectively).

Pairsof D. rerio (wild-type Tuebingen strain) were reared together in 10 ltanks.

Aquaria were filled with water prepared as a mixture of reverse osmosis-treated water and tap water, both filtered through sediment and activated charcoal filters. Rearing conditions were: temperature 28 ± 0.5 °C, conductivity 300 ± 50 µScm -1 , air saturation ≥ 80%, pH 7.5 ± 0.5, photoperiod of 14 h light/10 h dark. The fish were fed twice daily with commercial dry food (INICIO Plus, BioMar, www.biomar.com), occasionally supplemented with red sludge worms (Boschetto-Frozen fish food,www.achataquarium.fr). Over a period ofone month, spawn was obtained weekly from pairs following the protocol described in Lucas et al. (2014a) and Vignet et al. (2014a).Spawn was then mixed to avoid any parental influence.

At twoweeks old,fish were kept in groups of 30 individuals in 10 l aquaria.

Contamination with PY PAHwas achieved through the trophic pathway. Artificial dry food was contaminated with a mixture of PYPAH. The mixture was composed of 95% non-substituted PAHwith a majority of four-and five-ring PAH; a detailed description is given in Vignet et al. (2014a). The PAH concentration targeted for contamination of pellets was 5000 ngg -1 , based on concentrations measured in molluscs in the Seine estuary. This reference environmental concentration is hereafter referred to asX; it represents one of the four treatments tested in this study (PAH concentrations measured in diet [PAH] = 5816 ± 1433 ngg -1 ). Based on this reference, two other treatments were tested: a lower concentration of0.3X ([PAH] = 1763 ± 468 ngg -1 ) and a higher concentration of 3X ([PAH] = 18151 ± 4983 ngg -1 ). A fourth (control) treatment was added in which dry food was exposed only to dichloromethane, the solvent used to carry the PAH. Contamination through food wasachievedby feeding fishtwice a day with one of the four treatments. Quantification of hydroxylated metabolitesin larvae at15 days post fertilisation (dpf) indicated a dose-dependent increase of metabolites confirmingsuccessfulcontamination (total concentrations of hydroxylated metabolitesfor each treatment: control = 9.1 ngg -1 of tissue; 0.3X = 20 ngg -1 ,1X = 72 ngg -1 ; 3X = 275 ngg -1 ; Vignet et al., 2014a). Fish were fed with treated pellets from their first meal (5 dpf) to the ages of 2 months for juveniles and 6 months for adults (Table 1) with size-adapted food (≤ 125µm, 125-315µm, 315-500µm, ≥ 500µm).In accordance with protocols inVignet et al. (2014a), larvae were fed ad libitum and then, starting from two months old, the ration of food was 5 % to 2% of the biomass in each tank in order to maintain constant growth. For allfish, brine shrimps (Ocean Nutrition Europe BVBA, http://www.oceannutrition.eu/fr/default.aspx) weregiven as supplementaryfood once a day. Characteristics of the fish are reported Table 1.

Table I. Biometry of juveniles and adultsDanio rerio in each treatment (mean ± SE). X is the environmental reference concentration of 5.5 µg PY PAHs.g -1 of dry food. Control was food that had been exposed to dichloromethane only and did not contain PY PAHs. To assess the aerobic metabolic rate of fish, eightidentical circular size-adapted respirometers (diameter: 3.75 cm, volume: 0.061lfor juveniles and 7.50 cm, 0.179l for adults) were employed. These wereimmersedin two buffer tanks (depth x length x height: 10 x 75 x 75 cm for both juveniles and adults) filled with temperature-controlled andaeratedwater. Oxygen consumption was measured by intermittent-flowrespirometry [START_REF] Steffensen | Some errors in respirometry of aquatic breathers: how to avoid and correct for them[END_REF]where the water supply in each respirometer was provided by flush pumps controlled by a timer. This system alternated phases of flushing and oxygen renewal with phases of measurement of oxygen consumption (MO 2 ), each of which lasted 30 min. Finally, a multichannel peristaltic pump was installed to create continuous water flow and ensure water mixing inside each of the chambers. Each respirometer was equipped with anoptic fibre sensor (PreSens, www.presens.com) connected to a multichannel oxygen measuring system (OXY 4 mini, PreSens) to record the level of dissolved oxygen in the water. Oxygendata was sampledeach five seconds with the program Oxyview (PreSens).

Treatment

Fish were starved 24 h prior to respirometry. For each trial, eightfish (two ineach treatment) were testedindividually in respirometer, intwo consecutive phases. First, to increase fish metabolism and assess AMR, each fish was transferred and chased with a stick in a 1 ltank [START_REF] Schurmann | Effects of temperature, hypoxia and activity on the metabolism of juvenile Atlantic cod[END_REF][START_REF] Lefrançois | Influence of ambient oxygenation and temperature on metabolic scope and scope for heart rate in the common sole Solea solea[END_REF][START_REF] Dupont-Prinet | An investigation of metabolic prioritization in the European sea bass, Dicentrarchus labrax[END_REF][START_REF] Clark | Physiological benefits of being small in a changing world: responses of coho salmon (Oncorhynchus kisutch) to an acute thermal challenge and a simulated capture event[END_REF][START_REF] Cannas | PCB contamination does not alter aerobic metabolism and tolerance to hypoxia of juvenile sole (Solea solea)[END_REF]. When the fish was fatigued(did not respond to the stimulation), it was transferred into a respirometer. The oxygen consumption of the fish was immediately recorded for30 min to calculateAMR.Then, toconfirm the accuracy of the AMR assessment, each fish was chased again in therespirometer and its MO 2 measured again for a new period of 30 min.

The second step consisted ofaresting period of 48 h to reach and estimate SMR.

During this period fish were undisturbed and MO 2 was regularly and automatically measured. From these oxygen measurements, SMR was estimated according to the method described by [START_REF] Steffensen | Oxygen consumption in four species of teleosts from Greenland: no evidence of metabolic cold adaptation[END_REF]. Briefly, the frequency distribution of MO 2 values recorded during the last 24 h of the test was plotted. This generally produces a bimodal frequency distribution due to the routine activity of the fish. The higher mode (the first peak) is considered to reflect SMR and the lower mode (the second peak)

corresponds to the routine metabolic rate (RMR), the energy required by the fish for maintenance plus random activity. During all the experiments, oxygen concentration wasnever lower than 75% oxygen saturation in each respirometer.

After the 48 h period of resting MO 2 measurements, fish were removed from therespirometers and anesthetised using benzocaine (50 mg l -1 ). The length (standard and total length) and mass of each individual were determined (Table 1). To quantify microbial oxygen consumption in the respirometer, a blank measurement wascarried out before and after each trial. A linear change in background MO 2 over the 48 hexperimental trial was assumed and subtracted the expected value fromthe correspondingtotal MO 2 measured.

Oxygen consumption (MO 2 ),expressed in mg O 2 g -1 h -1 , was calculated according

to the formulaMO 2meas = Δ[O 2 ]VΔt -1 M meas -1 , where Δ[O 2 ] (in mg O 2 l -1
) is the changein oxygen concentration during themeasurement period Δt (in h),V (in l) is the volume of the respirometer minus the volume of the fish, andM meas (in g) is the measuredmass of thefish. An allometric relationship between oxygen consumption and body mass,allows correction of MO 2meas using the formulaMO 2cor =MO 2meas (M meas M cor -1 ) 1-b , where MO 2cor (in mg O 2 g -1 h -1 ) is the oxygen consumption related to a standard fish of 1 g (M cor ), MO 2meas

(in mg O 2 g -1 h -1 ) is the oxygen consumption estimated for experimented fish whose mass was M meas (in g) and b is the allometric scaling exponent describing the relationship between oxygen consumption and body mass of fish.In previous study, b was found to be equal to 0.926 and 0.965 in the case of AMR and SMR assessment respectively (Lucas et al., 2014a).Aerobic metabolic scope (AS)was calculated as the difference between AMR and SMR. AMR, SMR and AS were assessed once for each individual.

Statistical analysis was carried out using Graphpad Prism software. As the conditions of normality (tested using the Kolmogorov-Smirnoff test) and homoscedasticity (tested using the Bartlett test) of data were not met, a Kruskal-Wallis non-parametric test was used to test forsignificant differences in metabolic rates among treatments. If necessary, a Dunn post-hoc testwas applied to determine which treatments differed significantly.Differences were considered to be significant when P < 0.05.

There were no significant differences in AMR, among treatments for either juveniles or adults (P = 0.45 and P = 0.93 for juveniles and adults, respectively; Fig. 1A).

Similarly there were no significant differences in SMRamong treatments for each life stage (P = 0.23 and P = 0.74for juveniles and adults, respectively; Fig 1B).

Therefore,ASalso did not differsignificantly among the treatments for juveniles (P = 0.59) or adults (P = 0.89) (Fig. 1C). This is the first study assessing the aerobic metabolism of zebrafish chronically exposed to a mixture of pyrolytic PAH. Under these experimental conditions and at the two lifestages tested, trophic exposure to PY PAH did not affect SMR, AMR or AS of this species. It is worth noting that these data for aerobic metabolism (Fig. 1B) are in agreement with previous studies carried out for the same species and with a similar experimental approach (SMR = 0.19 mg O 2 g -1 h -1 in Barrionuevo &Burggren, 1999; SMR = 0.31 ± 0.11 mg O 2 g -1 h -1 and 0.35 ± 0.16 mg O 2 g -1 h -1 in juveniles and adults, respectively, inLucas et al., 2014a).

There was no significant difference in SMR among the four treatments. Even though the concentration of PAH and their metabolite compounds were not assessed in juveniles and adults in this study, previous work has shown that these concentrations tended to be proportional to the PAH content of the food received (Vignet et al., 2014a).

These results suggest that even the highest level of contamination tested (three times the average environmental concentration) was not sufficientlyextreme to induce significant variation inSMR. Moreover, the lack of effect on SMR is contrary to the initial hypothesis, which stated that an increase in SMR was expected because of supplementary energy costs induced by PAH detoxification processes. [START_REF] Lannig | Temperature-dependent stress response in oysters, Crassostrea virginica: Pollution reduces temperature tolerance in oysters[END_REF] showed that a 40-86% increase of SMRcanbe induced in oysters Crassostrea virginica (Gmelin 1791) exposed to cadmium. This increaseappears to be mostly due to the elevated costs of protein synthesis involved in detoxification or protective mechanismssuch as cellular repair or expression of stress proteins.

In the previous study of Lucas et al. (2014a), AMR ranged between 0.92 and 0.94 mg O 2 g -1 h -1 in juvenile and adult zebrafish. This is consistent with the current results for adults, while a slightly higher AMR was measured in juveniles (Fig. 1A). These results suggest that D. Rerio would not have a reduced capacity to sustain oxygen-demanding activities such as locomotion, digestion or growth (Claireaux & Lefrancois, 2007).

However, Vignet et al. (2014a) found a PAH dose-dependent reduction in growth despite the lack of effect of PAHon metabolism. This was probably mainly due to alteration of digestive capacity.

Despite numerous studies on fishes exposed to petroleum,there is no clear conclusionregarding effects on aerobic metabolism. In fact, some investigations on persistent organic pollutants reported that fishes maintain their aerobic metabolism. (2007) find that organic pollutants affected metabolic rates of chub Leuciscus cephalus(Linnaeus, 1758). In contrast, other studies havedemonstratedan increase [START_REF] Hose | Oxygen consumption of grunion (Leuresthes tenuis) embryos exposed to the petroleum hydrocarbon, benzo[a]pyrene[END_REF][START_REF] Correa | Physiological responses of juvenile white mugil, Mugil curema, exposed to benzene[END_REF][START_REF] Davison | The effect of acute exposure to the water soluble fraction of diesel fuel oil on survival and metabolic rate of an Antarctic fish (Pagothenia borchgrevinki)[END_REF] or a decrease [START_REF] Sharp | Physiological basis ofdifferential sensitivity of fish embryonic stages to oil pollution[END_REF][START_REF] Serigstad | Effects of oil exposure on oxygenconsumption of cod eggs and larvae[END_REF][START_REF] Prasad | Toxicity of crude oil to the metabolism offreshwater minor carp, Puntius sophore[END_REF][START_REF] Davoodi | Effects of exposure to petroleum hydrocarbons upon the metabolism of the common sole Solea solea[END_REF][START_REF] Christiansen | Sub-lethal levels of waterborne petroleum may depress routine metabolism in polar cod Boreogadus saida (Lepechin, 1774)[END_REF] ofASin fishes after petroleum exposure. However, fish contamination in these studies occurred by an aqueous pathway (involving the water soluble fraction of petroleum), which maycount for the contrasting results with the present study. Such exposition may indeed have causedalterations to the gills epitheliumleading to reduced oxygen diffusion into the blood (Claireaux et al., 2004;[START_REF] Davoodi | Effects of exposure to petroleum hydrocarbons upon the metabolism of the common sole Solea solea[END_REF].Therefore, theconcentration and/or type of PAH tested in this present study did not induce impairments in the mechanisms involved in metabolic regulation.

However, it is also worth noting that organisms which suffer long-term chronic environmental stress can present physiological adaptations to maintain their homeostasis [START_REF] Barton | Stress in fishes: a diversity of responses with particular reference to changes in circulating corticosteroids[END_REF]. Chronic exposure to PAH may have induced such adaptations in D. rerio.

This studyused the progeny of contaminated D. rerio (Lucas et al. 2014b). Even though no effects were observed on directly contaminatedparents, an increase of SMR was observed in larval progeny of fish exposed to very high concentrations of PY PAH(the 3X treatment, Lucas et al., 2014b). In addition, cardiac performance, heart rate and mRNA expression of genes encoding for cardiac activity were allmodified at the environmentally representative PAH concentration 1X (Lucas et al., 2014b) Based on these results, the effects on larvae of parental exposure to PAHis worthy of further study, asparental exposure may impact aerobic metabolism as well as cardiac function (Lucas et al., 2014b). These results will improve the understanding of the potential effects of pyrolytic PAH on the physiology of D. rerio in particular, and fish in general.

Lifestage Number of fish (n) Weight (g) Standard length (cm) Total length (cm)

	Control	Juveniles Adults	24 15	0.20±0.08 0.63±0.19	2.23±0.29 3.08±0.28	2.72±0.37 3.80±0.35
	0.3X	Juveniles Adults	23 12	0.21±0.12 0.68±0.11	2.18±0.27 3.26±0.20	2.73±0.20 3.94±0.20
	1X	Juveniles Adults	24 15	0.178±0.04 0.57±0.13	2.21±0.08 3.00±0.32	2.69±0,15 3.69±0.30
	3X	Juveniles Adults	24 15	0.18±0.08 0.44±0.16	2.18±0.29 2.84±0.40	2.65±0.33 3.36±0.41

  The lack of effect in the present study is for instance in accordance with[START_REF] Milinkovitch | Effect of dispersed crude oil exposure upon the aerobic metabolic scope in juvenile golden grey mullet (Liza aurata)[END_REF] who observed no modification of SMR, AMR and ASin golden grey mullet Liza aurata (Risso, 1810) after exposure to crude oil and dispersants. Nor didMcKenzie et al.

Acknowledgements

The authors are grateful to Didier Leguay and Michel Prineau for their help during the experiment. All experiments were carried out at Ifremer (Plateforme d'Ecophysiologie des Poissons), La Rochelle station, France. This study was financially supported by the ANR project ConPhyPoP (CES 09_002) and JL received a doctoral grant from the Regional Council of Poitou-Charentes. This study was conducted under the approval of the Animal Care Committee of France under the official licence to Marie-Laure Bégout (17-010). Barrionuevo, W.R.&Burggren, W.W. (1999). O 2 consumption and heart rate in developing zebrafish (Danio rerio): influence of temperature and ambient O 2 . American Journal of Physiology276, 505-513. Barrionuevo, W.R., Fernandes, M.N., Rocha, O. (2010). Aerobic and anaerobic metabolism for the zebrafish, Danio rerio, reared under normoxic and hypoxic conditions and exposed to acute hypoxia during development. Brazilian Journal of Biology70, 425-434.


Loïc Chaumont
email: loic.chaumont@univ-angers.fr

Ron A Doney
email: ron.doney@manchester.ac.uk

Loïc

ON DISTRIBUTIONS DETERMINED BY THEIR UPWARD, SPACE-TIME WIENER-HOPF FACTOR

Keywords: Mathematics Subject Classification. 60A10, 60E05 Wiener-Hopf factors, convolution powers, exponential moments, completely monotone function

According to the Wiener-Hopf factorization, the characteristic function ϕ of any probability distribution µ on R can be decomposed in a unique way as

where χ -(e iu , it) and χ + (e iu , it) are the characteristic functions of possibly defective distributions in

We prove that µ can be characterized by the sole data of the upward factor χ + (s, it), s ∈ [0, 1), t ∈ R in many cases including the cases where:

1) µ has some exponential moments; 2) the function t → µ(t, ∞) is completely monotone on (0, ∞); 3) the density of µ on [0, ∞) admits an analytic continuation on R.

We conjecture that any probability distribution is actually characterized by its upward factor. This conjecture is equivalent to the following: Any probability measure µ on R whose support is not included in (-∞, 0) is determined by its convolution powers µ * n , n ≥ 1 restricted to [0, ∞). We show that in many instances, the sole knowledge of µ and µ * 2 restricted to [0, ∞) is actually sufficient to determine µ. Then we investigate the analogous problem in the framework of infinitely divisible distributions.

Introduction

Let µ be any probability measure on R. Denote by (S n ) a random walk with step distribution µ, such that S 0 = 0, a.s. Define the first ladder times associated to (S n ) by τ -= inf{n ≥ 1 : S n < 0} , τ + = inf{n ≥ 1 : S n ≥ 0} .

Then the Wiener-Hopf factorization of the characteristic function ϕ(t) = e itx µ(dx) of µ can be written as, (1.1)

1 -sϕ(t) = [1 -χ -(s, it)][1 -χ + (s, it)] , |s| ≤ 1, t ∈ R ,
where χ -and χ + are the downward and upward space-time Wiener-Hopf factors, χ -(s, it) = E(s τ -e itSτ -1 {τ -<∞} ) and χ + (s, it) = E(s τ + e itSτ + 1 {τ + <∞} ) .

To paraphrase W. Feller [START_REF] Feller | An introduction to probability theory and its applications[END_REF], XVIII.3, the remarkable feature of the factorization (1.1) is that it represents an arbitrary characteristic function ϕ in terms of two (possibly defective) distributions, one being concentrated on the half line (-∞, 0) and the other one on the half line [0, ∞). However, this feature only exploits identity (1.1) for fixed s = 0 and reflects the fact that µ is determined by the knowledge of the distributions of both S τ -and S τ + . But one may wonder about the extra information brought by the joint distributions (τ -, S τ -) and (τ + , S τ + ). In particular, is it true in general that µ is determined by only one of these joint distributions? or equivalently, is it true that ϕ is determined by only one of its space-time Wiener-Hopf factors?

The aim of this paper is an attempt to answer the latter question. We will actually show that µ is determined by χ + (s, it) in some quite large classes of distributions including the case where µ has some positive exponential moments, or when t → µ(t, ∞) is completely monotone on (a, ∞), for some a ≥ 0, or satisfies some property which is slightly stronger than analycity. Obviously all these assumptions can be verified from the sole data of χ + (s, it). These different cases cover a sufficiently large range of distributions for us to allow ourselves to raise the following conjecture. Let M 1 be the set of probability measures on R.

Conjecture C. Any distribution µ ∈ M 1 whose support is not included in (-∞, 0) is determined by its upward space-time Wiener-Hopf factor χ + (s, it), |s| < 1, t ∈ R.

A crucial step in the proof of (1.1) is the following development of the factor χ + (s, t), for |s| < 1 and t ∈ R,

(1.2) log 1 1 -χ + (s, it) = ∞ n=1 s n n [0,∞)
e itx µ * n (dx) , see [START_REF] Feller | An introduction to probability theory and its applications[END_REF], XVIII.3, where µ * n is the nth fold convolution product of µ by itself. We will actually refer to µ * n , n ≥ 0 as the convolution powers of µ. This identity shows that the data of χ + is equivalent to the knowledge of the measures µ * n , n ≥ 1 on [0, ∞) and leads to the following equivalent conjecture.

Conjecture C'. Any distribution µ ∈ M 1 whose support is not included in (-∞, 0) is determined by its convolution powers µ * n , n ≥ 1 restricted to [0, ∞).

Each of the next sections corresponds to a class of probability distributions for which Conjecture C holds. For the first one in Section 2, we prove that distributions having some particular exponential moments satisfy conjecture C. Then in Section 3 we consider three other classes for which a much stronger result than Conjectures C and C' is true. We will see that there are actually many distributions which are determined by the sole data of µ and µ * 2 on [0, ∞). This is the case when the function t → µ(t, ∞) is smooth enough. In Subsection 3.2 we will consider the case where the function t → µ(t, ∞) is completely monotone on (a, ∞), for some a ≥ 0 and in Subsection 3.3 we will make a slightly stronger assumption than analicity on this function. We will also present the discrete counterpart of the completely monotone case in Subsection 3.4. Finally in Section 4 we will consider Conjecture C in the restricted class of infinitely divisible distributions and show that if the upper tail of the Lévy measure is completely monotone, then µ is determined by its upper Wiener-Hopf factor. Then we end this paper in Section 5 with some important remarks on the possibility of extending the classes of distributions studied.

In all this paper, we will denote by C the set of distributions satisfying Conjecture C. Let us give a proper definition of this set. Definition 1.1. Let C be the set of distributions µ ∈ M 1 which are determined by their upward Wiener-Hopf factor χ + (s, it), for |s| ≤ 1 and t ∈ R or equivalently by the data of their convolution powers µ * n , n ≥ 1 restricted to [0, ∞). More formally,

The problem we investigate here originates from a result in V. Vigon's PhD thesis [START_REF] Vigon | Simplifiez vos Lévy en titillant la factorisation de Wiener-Hopf[END_REF], see Section 4.5 therein, where a question equivalent to Conjecture C is raised in the setting of Lévy processes. Our question is actually more general since it concerns distributions which are not necessarily infinitely divisible. In particular, a positive answer to Conjecture C would imply that the law of any Lévy process (X t , t ≥ 0) is determined by one of its space-time Wiener-Hopf factors or equivalently by the marginals of the process (X + t , t ≥ 0), see Section 4.

2.

When µ admits exponential moments.

2.1.

Recovering characteristic function and moment generating function. In this paper, we will always assume that the support of the measure µ is not included in (-∞, 0). Let us observe that from the data of the upward Wiener-Hopf factor χ + (s, it), for |s| ≤ 1 and t ∈ R or equivalently from the data of the measures µ * n , n ≥ 1 restricted to [0, ∞), we know the sequences, P(S n < 0) and P(S n ≥ 0), n ≥ 0, as well as the distributions of both τ -and τ + . In particular we know whether (S n ) oscillates, drifts to -∞, or drifts to ∞. The next result shows that provided n → P(S n < 0) tends to 0 sufficiently fast along some subsequence, it is possible to recover the characteristic function ϕ of µ on some interval containing 0, from the measures µ * n restricted to [0, ∞).

Lemma 2.1. Assume that there is α > 0 such that, at least for a subsequence,

(2.3) P(S n < 0) ≤ e -αn .
Then for all t such that |ϕ(t)| > e -α , along this subsequence,

lim n→+∞ E(e itSn 1I {Sn≥0} ) 1/n = ϕ(t) .
In particular, if (2.3) holds then ϕ can be determined on some neighborhood of 0.

Proof. It suffices to write,

E(e itSn 1I {Sn≥0} ) 1/n = ϕ(t) n -E(e itSn 1I {Sn<0} ) 1/n = ϕ(t)[1 -ϕ(t) -n E(e itSn 1I {Sn<0} )] 1/n . (2.4)
Then from the assumption, for all t such that |ϕ(t)| > e -α and for all n such that P(S n < 0) < e -αn , ϕ(t

) -n E(e itSn 1I {Sn<0} ) ≤ |ϕ(t) -n P(S n < 0)| , ≤ (|ϕ(t)|e α ) -n .
Therefore, the left hand side of the above inequality tends to 0 along a subsequence and this yields the result thanks to equation (2.4).

Since, for all n ≥ 0,

P(S 1 < 0, S 2 -S 1 < 0, . . . , S n -S n-1 < 0) = P(S 1 < 0) n ≤ P(S n < 0) , (2.
3) cannot hold for all α > 0, unless P(S 1 ≥ 0) = 1. Note also that if (2.3) holds then the random walk (S n ) cannot drift to -∞. Moreover, if it holds along all subsequences, then (S n ) necessarily drifts to ∞ thanks to Spitzer's criterion which asserts that this happens if and only if

n -1 P(S n < 0) < ∞.
Let us denote the moment generating function of µ by φ(λ

) := R e λx µ(dx), λ ∈ R. Lemma 2.2. For λ < 0, 1. φ(λ) < ∞ if and only if E(e λSτ -1I {τ -<∞} ) < ∞. 2. If φ(λ) < 1, then (2.3) holds.
Proof. Assume that E(e λSτ -1I {τ -<∞} ) < ∞. Then we derive from the following inequality,

φ(λ) = E(e λS 1 1I {τ -=1} ) + E(e λS 1 1I {τ ->1} ) ≤ E(e λSτ -1I {τ -<∞} ) + P(τ -> 1) , that φ(λ) < ∞.
Conversely, assume that φ(λ) < ∞, and write for x > 0,

P(-S τ -> x, τ -< ∞) = k≥1 P(S 1 ≥ 0, . . . , S k-1 ≥ 0, -X k > x + S k-1 ) = k≥1 [0,∞) P(S 1 ≥ 0, . . . , S k-2 ≥ 0, S k-1 ∈ dy)P(-S 1 > x + y) = [0,∞) v(dy)P(-S 1 > x + y) , (2.5)
where v(dy) = k≥0 P(S 1 ≥ 0, . . . , S k-1 ≥ 0, S k ∈ dy) is the renewal measure on [0, ∞) of the (weak) upward ladder height process of (S n ), see Chap. XII.2 in [START_REF] Feller | An introduction to probability theory and its applications[END_REF]. In particular, this measure satisfies [0,∞) e αy v(dy) < ∞ for all α < 0. Then from (2.5) and the assumption, we obtain

E(e λSτ -1I {τ -<∞} ) = [0,∞) e λy v(dy) ∞ y e λz P(S 1 ∈ dz) ≤ [0,∞) e λy v(dy)φ(α) < ∞ .
Finally in order to prove assertion 2, it suffices to write,

P(S n < 0) ≤ E(e λSn 1I {Sn<0} ) ≤ φ(λ) n , so that if φ(λ) < 1, then (2.
= E((-S τ -) n | τ -< ∞) are finite for all n and if there is λ < 0 such that E(e λSτ -| τ -< ∞) = n≥0 mn n! (-λ) n < ∞.
We have proved that, the knowledge of the characteristic function t → χ -(1, it)/χ -(1, 0) on [-ε, ε] allows us to determine if E(e λSτ -1I {τ -<∞} ) < ∞ for some λ < 0, which is equivalent to φ(λ) < ∞, from Lemma 2.2. When this is the case, the knowledge of the characteristic function t → χ -(1, it)/χ -(1, 0) on [-ε, ε] is enough to determine the law of the random variable S τ -under P( • | τ -< ∞). Then from (1.1) we can determine the law µ, so that it belongs to class C . Remark 2.1. Actually, Lemma 2.1 might be exploited in a better way. Indeed, assume that µ satisfies (2.3). Then the characteristic function ϕ is known on some neighborhood of 0. If for instance ϕ is infinitely differentiable around 0, then all the moments of µ can be recovered. If moreover, µ is determined by these moments, then it belongs to class C . In particular µ does not need to have exponential moments as in Theorem 2.1. The question of finding such an example is still open.

Recall that φ is a convex function on the interval {α : φ(α) < ∞}. Moreover, since the support of µ is not included in (-∞, 0), φ is nondecreasing. If λ ∈ R is such that λ = inf{α : φ(α) < ∞}, then φ (λ) will be understood as the right derivative of φ at λ. Similarly, if λ = sup{α : φ(α) < ∞}, then φ (λ) will be the left derivative of φ at λ.

Lemma 2.3. For all λ ∈ R such that φ(λ) < ∞ and φ (λ) > 0, lim n→+∞ E(e λSn 1I {Sn≥0} ) 1/n = φ(λ) . Proof. Let (S (λ) n ) be a random walk with step distribution µ λ (dx) := e λx φ(λ) µ(dx). Since E(S (λ) 1 ) = R xe λx φ(λ) µ(dx) = φ (λ) φ(λ) > 0 , the random walk (S (λ)
n ) drifts to ∞, so that lim n→∞ P(S

(λ) n ≥ 0) = 1.
Then the result follows from the identity

P(S (λ) n ≥ 0) = E(e λSn 1I {Sn≥0} ) φ(λ) n .
The following theorem shows that distributions having some negative exponential moments less than 1 or some positive exponential moments bigger than 1 belong to class C . Theorem 2.2. The knowledge of the measures µ * n , n ≥ 1 restricted to [0, ∞) allows us to determine if φ satisfies one of the two following conditions:

(a) There exists λ < 0 such that φ(λ) < 1.

(b) There exists λ > 0 such that φ(λ) ∈ (1, ∞). When at least one of these two conditions holds, the measure µ belongs to the class C . Proof. From Lemma 2.2, if (2.3) does not hold, then (a) is not satisfied. Assume that (2.3) holds. From Theorem 2.1, we can determine if there is λ < 0 such that φ(λ) < ∞. If this is not the case, then clearly (a) does not hold. Assume that there is λ < 0 such that φ(λ) < ∞. From Theorem 2.1, the measure µ belongs to class C so that we can determine if (a) holds.

From our data, for all λ > 0 and n ≥ 1, the expression E(e λSn 1I {Sn≥0} ) is known. Assume that there is λ > 0 such that lim n→+∞ E(e λSn 1I {Sn≥0} ) 1/n > 1. Since E(e λSn 1I {Sn≥0} ) 1/n ≤ φ(λ), we have actually φ(λ) > 1. Moreover, our data clearly allows us to know if φ(λ) < ∞. Then since φ(0) = 1, by convexity of the function λ → φ(λ), it is clear that φ (λ) > 0, so that from Lemma 2.3, lim n→+∞ E(e λSn 1I {Sn≥0} ) 1/n = φ(λ). This means that from our data, we can determine if there is λ > 0 such that φ(λ) ∈ (1, ∞) and in this case lim n→+∞ E(e λSn 1I {Sn≥0} ) 1/n = φ(λ), so that φ(λ) is known. Moreover, from the continuity of φ on the set {x : φ(x) < ∞}, there is an interval I containing λ such that for all x ∈ I, φ(x) ∈ (1, ∞) and from the same reasoning as above, φ(x) is known for all x ∈ I, so that the measure µ is determined and we conclude that it belongs to the class C .

We can easily check that condition (b) of Theorem 2.2 is satisfied in the two following situations:

(b 1 ) φ(λ) < ∞ for all λ > 0. (b 2 ) µ is absolutely continuous in [0, ∞) and its density f satisfies ln(f (x)) ∼ -λ 0 x, as x → ∞, for some λ 0 ∈ (0, ∞). Indeed, in case (b 1 ), if µ = δ 0 then since φ is a nondecreasing convex function such that φ(0) = 1 and since the support of µ is not included in (-∞, 0), lim λ→∞ φ(λ) = ∞. In case (b 2 ) it is clear that lim λ→λ 0 -φ(λ) = ∞.
Checking condition (a) in Theorem 2.2, and in particular checking if µ has negative exponential moments under condition (2.3) requires more work on the behavior of the moments of S τ -, see the proof of Theorem 2.1.

Skip free distributions.

A distribution µ whose support is included in Z is said to be downward (resp. upward) skip free if µ(n) = 0 for all n ≤ -2 (resp. for all n ≥ 2). Clearly skip free distributions possess exponential moments. Moreover, upward skip free distributions belong to class C from Theorem 2.2 (b) and the note following its proof. Then in this subsection we shall see that the case of downward skip free distributions allows us to go a little beyond the cases encompassed by Theorems 2.1 and 2.2. We first need to make sure that our data allows us determine if the support of a distribution is included in Z.

Lemma 2.4. The support of the measure µ is included in Z if and only if the support of the measures µ

* n , n ≥ 1 restricted to [0, ∞) is included in Z + .
Proof. The direct implication is obvious. Then assume that the support of µ

* n , n ≥ 1 restricted to [0, ∞) is included in Z + , whereas the support of µ restricted to (-∞, 0] is not included in Z -. Then there is an interval I ⊂ (-∞, 0] \ Z -such that µ(I) > 0. Let n ∈ Z + \ {0} such that µ({n}) > 0 and h ∈ Z + \ {0} such that hn + inf I > 0. Then 0 < µ(I)µ({n}) h = P(S 1 ∈ I, S i+1 -S i = n, i = 1, . . . , h) ≤ P(S n+1 ∈ hn + I) .
This implies that µ * (n+1) (hn + I) > 0, where hn + I ⊂ [0, ∞) \ Z + , which contradicts the assumption.

Theorem 2.3. Downward skip free distributions belong to class C . Proof. Let µ ∈ M 1 whose convolution powers µ * n , n ≥ 1 restricted to [0, ∞) are known.
Then from Lemma 2.4 we can determine if the support of µ is included in Z or not. Let us assume that it is the case.

As already noticed at the beginning of this section, we can determine if (S n ) drifts to ∞ or not. Assume first that (S n ) drifts to ∞. From Theorem 2.2 (a), we can determine if there exists λ < 0 such that φ(λ) < 1. If this is not the case, then µ cannot be downward skip free. To see this, consider for instance the analytic extension of the Wiener-Hopf factorization (1.1) which is given in (9.3.1), p.99 of [START_REF] Doney | Fluctuation theory for Lévy processes[END_REF], for nonnegative arguments and in the case of upward skip free random walks. On the contrary, if there exists λ < 0 such that φ(λ) < 1, then from Theorem 2.2, µ can be determined. In particular, we know if µ is downward skip free or not. Now assume that (S n ) does not drift to ∞. From the same arguments as those we used to prove identity (2.5), we obtain for all n ≥ 1,

P(S τ + > n) = r≥1 v(r)P(S 1 > n + r) ,
where v is the renewal measure of the (strict) downward ladder height process of (S n ). In particular, this renewal measure satisfies v(r) ≤ 1 for all r ≥ 1. Moreover, (S n ) is downward skip free and does not drift to ∞, if and only if v(r) = 1, for all r ≥ 1. Since it is the only unknown in the above equation, we can determine if it is the case or not. Finally, knowing that µ is downward skip free, we immediately determine this distribution on R from its knowledge on [0, ∞).

It appears in the proof of Theorem 2.3 that downward skip free distributions which drift to ∞ actually satisfy condition (a) of Theorem 2.2. Hence the only additional case in this subsection is this of downward skip free distributions which does not drift to ∞.

We will denote by E the set of measures µ satisfying assumptions of Theorem 2.1 or those of Theorem 2.2 or those of Theorem 2.3, that is the set of measures satisfying both (2.3) and (2.6) or (a) or (b) or downward skip free distributions. It will be called the exponential class. From Theorems 2.1, 2.2 and 2.3 , we have E ⊂ C .

3. When µ is characterized by µ and µ * 2 on [0, ∞).

3.1. Preliminary results. We will show that in many cases, the sole data of µ and µ * µ on [0, ∞) actually suffices to determine µ. In this subsection, we give a theoretical condition for this to hold.

We first observe that in Conjectures C and C', there is no loss of generality in assuming that µ is absolutely continuous on [0, ∞). Proof. We already know from Section 1 that Conjectures C and C' are equivalent. Then clearly, it suffices to prove that if Conjecture C" is true, then Conjecture C' is true.

Conjecture C". Any absolutely continuous distribution

µ ∈ M 1 whose support is not included in (-∞, 0) is determined by its convolution powers µ * n , n ≥ 1 restricted to [0, ∞).
Let µ, µ 1 ∈ M 1 be any two distributions such that the measures µ * n 1 and µ * n agree on [0, ∞), for all n ≥ 1. Let g be any probability density function on [0, ∞), i.e. ∞ 0 g(x) dx = 1 and let μ, μ1 ∈ M 1 be the absolutely continuous measures whose respective densities are h(x) = R g(y -x) µ(dy) and h 1 (x) = R g(y -x) µ 1 (dy), x ∈ R. Denoting by g * n the n-th convolution product of the function g by itself, it is plain that for all x ≥ 0,

μ * n [x, ∞) = ∞ 0 µ * n [x + y, ∞)g * n (y) dy and μ * n 1 [x, ∞) = ∞ 0 µ * n 1 [x + y, ∞)g * n (y) dy .
Therefore since the measures µ * n and µ * n 1 agree on [0, ∞), for all n ≥ 1, the measures μ * n and μ * n 1 also agree on [0, ∞), for all n ≥ 1 and from the assumption that conjecture C" is true, we conclude that the measures μ and μ1 agree on R. Then we can identify both characteristic functions:

R e itx μ(dx) = R e itx µ(dx) ∞ 0 e -itx g(x) dx and R e itx μ1 (dx) = R e itx µ 1 (dx) ∞ 0 e -itx g(x) dx .
This implies that R e itx µ(dx) = R e itx µ 1 (dx), for all t ∈ R thanks to the fact that the characteristic function of g cannot vanish identically in an interval, see Theorem 2 in [START_REF] Smith | A note on characteristic functions which vanish identically in an interval[END_REF] and by continuity of characteristic functions. Then we conclude that µ = µ 1 on R, from injectivity of the Fourier transform.

We derive from Lemma 3.1 that there is no loss of generality in assuming that µ is absolutely continuous on R. We will sometimes make this assumption and denote the density of µ by f . Lemma 3.2. For any probability density function, f on R and for all t ≥ 0,

(3.7) ∞ 0 f (t + s) f (s) ds = 1 2 f * f (t) - t 0 f (t -s)f (s) ds , where f (s) = f (-s). Proof. It suffices to decompose f * f as f * f (t) = R f (t -s)f (s) ds = 0 -∞ f (t -s)f (s) ds + t 0 f (t -s)f (s) ds + ∞ t f (t -s)f (s) ds .
Then from a change of variables, we obtain,

∞ t f (t -s)f (s) ds = 0 -∞ f (t -s)f (s) ds = ∞ 0 f (t + s) f (s)
ds, which proves our identity.

The main idea of this section is to exploit identity (3.7) in order to characterize the function f on [0, ∞) (or equivalently f on (-∞, 0]) from the sole data of f and f * f on [0, ∞). More specifically, assume that f restricted to [0, ∞) fulfills the following property: for any two nonnegative Borel functions g 1 and g 2 defined on [0, ∞) such that

∞ 0 f (t + s)g 1 (s) ds < ∞, ∞ 0 f (t + s)g 2 (s) ds < ∞, for all t ≥ 0, the following implication is satisfied, (3.8) [0,∞) f (t + s)g 1 (s) ds = [0,∞) f (t + s)g 2 (s) ds , for all t ≥ 0 ⇒ g 1 ≡ g 2 , a.e.
Then clearly, the map t → [0,∞) f (t + s) f (s) ds characterizes f on [0, ∞) and therefore from (3.7), µ is determined on R by the sole data of µ and µ * 2 on [0, ∞). Remark 3.1. There are density functions f which do not satisfy (3.8). For instance with f (s) = 1 2 e -s , the operator t → ∞ 0 f (t + s) f (s) ds = 1 2 e -t ∞ 0 e -s f (s) ds provides a very poor information on f and certainly cannot characterize this function on [0, ∞). Also if µ has a bounded support in [0, ∞), then clearly the operator t → ∞ 0 f (t + s) f (s) ds cannot characterize f outside this support.

Note also that from (3.7), the knowledge of t → ∞ 0 f (t+s) f (s) ds and f (t), for t ≥ 0 is equivalent to this of the functions f and f * 2 on [0, ∞). Therefore, in the above examples, f is not even determined by the data of f and f * 2 on [0, ∞).

The following proposition gives a sufficient condition for (3.8) to hold. Proposition 3.1. Assume that µ is absolutely continuous on R with density f . Let us introduce the following set of functions defined on [0, ∞),

H := n k=1 α k f (t k + •) : n ≥ 1, α k ∈ R, t k ≥ 0 . If the restriction of f to [0, ∞) belongs to L ∞ ([0, ∞)) and if H is dense in L ∞ ([0, ∞)),
then for any g ∈ L 1 ([0, ∞)), the following implication is satisfied,

(3.9) ∞ 0 f (t + s)g(s) ds = 0, for all t ≥ 0 ⇒ g ≡ 0, a.e.
When (3.9) holds, the measure µ is determined on R by µ and µ * 2 on [0, ∞).

Proof.

If ∞ 0 f (t + s)g(s) ds = 0, for all t ≥ 0, then clearly, since H is dense in L ∞ ([0, ∞)), ∞ 0 h(s)g(s) ds = 0 for all h ∈ L ∞ ([0, ∞)
) and this implies that g ≡ 0, a.e. Assume now that the restriction of the measures µ and µ * 2 are known on [0, ∞). Recall the notation f from Lemma 3.2 and observe that the right hand side of identity (3.7) is known for all t ≥ 0. From (3.9) this determines f on [0, ∞) and the measure µ is determined.

Unfortunately we do not know any example of function satisfying the condition of Proposition 3.1 and finding a simple criterion on f for it to satisfy this condition remains an open problem. More specifically, we may wonder if the converse of Proposition 3.1 holds, that is if assertion (3.9) implies that H is dense in L ∞ ([0, ∞)). The latter problem can be compared with Wiener's approximation theorem which asserts that for a function f in L 1 (R) the set H (thought as a set of functions defined on R) is dense in L 1 (R) if and only if the Fourier transform of f does not vanish, see [START_REF] Rudin | Functional analysis[END_REF].

In Subsection 3.2, we give a class of density functions such that (3.8) holds and in Subsection 3.3, we give a class of density functions which are bounded on [0, ∞) and such that (3.9) holds.

3.2.

The completely monotone class. In this subsection, we assume that µ is absolutely continuous with respect to the Lebesgue measure on R and we denote by f its density.

We will show that if f restricted to (a, ∞), for some a ≥ 0 is a completely monotone function satisfying some mild additional assumption, then µ is characterized from µ and µ * 2 on [0, ∞). Let us first recall that from Bernstein Theorem, the function f is completely monotone on (a, ∞), for a ≥ 0, if and only if there is a positive Borel measure ν on (0, ∞) such that for all t > a,

(3.10) f (t) = ∞ 0 e -ut ν(du) .
Theorem 3.1. Assume that there is a ≥ 0 such that the restriction of f to (a, ∞) is completely monotone and that the support of the measure ν in (3.10) contains a sequence (a n ) such that n a -1 n = +∞. Then (3.8) holds and the measure µ is characterized by the restriction of µ and µ * 2 to [0, ∞). In particular, µ belongs to class C . Proof. Let g be any nonnegative Borel function defined on [0, ∞) such that ∞ 0 f (t + s)g(s) ds < ∞, for all t > a. Then from Fubini's Theorem, for all t > a,

∞ 0 f (t + s)g(s) ds = ∞ 0 ∞ 0 e -u(t+s) ν(du)g(s) ds = ∞ 0 e -ut ∞ 0 e -us g(s) ds ν(du) .
This expression is the Laplace transform of the measure θ(du) := ∞ 0 e -us g(s) ds ν(du). The knowledge of this Laplace transform for all t > a characterizes the measure θ(du) so that a version of the density function u → ∞ 0 e -us g(s) ds is known on a Borel set B ⊂ (0, ∞) such that ν(B) = 1. Since this density function is continuous, it is known on B and hence it is known everywhere on the support of ν. Therefore, from the assumption on ν, we can find a sequence (a n ) such that n a -1 n = +∞ and such that the Laplace transform ∞ 0 e -ans g(s) ds of the function g at a n is known for each n. From a result in [START_REF] Feller | On Müntz' theorem and completely monotone functions[END_REF], this is enough to determine the function g and (3.8) holds, see also [START_REF] Feller | An introduction to probability theory and its applications[END_REF], p.430.

Then we derive from (3.8) and Lemma 3.2 that the function f is determined on [0, ∞) from the restriction on [0, ∞) of µ and µ * 2 . Therefore the measure µ is determined.

We will denote by M the set of absolutely continuous measures µ whose density f satisfies the assumption of Theorem 3.1. This class will be called the completely monotone class. Theorem 3.1 shows that M ⊂ C . Remark 3.2. Note that since f is a density function, the measure ν in (3.10) should also satisfy

(3.11) ∞ 0 f (t) dt = ∞ 0 e -au u ν(du) ≤ 1 . Remark 3.3.
Clearly class E is not included in class M . Moreover, it is easy to find an example of a measure in class M which does not belong to class E . First we readily check that whenever the support S of ν is such that S ∩ (0, ε) = ∅, for all ε > 0, then µ has no positive exponential moments, so that it cannot belong to class E . Then let us take for instance the measure ν(du) on [0, ∞) with density u 2 1I {u∈[0,1]} . This measure satisfies the assumption of Theorem 3.1, for a = 0.

Remark 3.4. Theorem 3.1 excludes completely monotone functions of the type f (t) = n k=1 α k e -β k t , t > a, for some α k , β k > 0 and some finite n since in this case the measure ν(du) = n k=1 α k δ β k (du) does not satisfy the condition required by this theorem. It also excludes functions f whose support is bounded since completely monotone functions on (a, ∞) are analytic on (a, ∞). This remark is consistent with Remark 3.1.

The analytic class.

Let us assume again that µ has density f on R. We will now exploit the same kind of arguments as in the previous subsection by assuming that f is the Fourier transform of some complex valued function. Theorem 3.2. Assume that there is a complex valued function k such that for all t ≥ 0, (3.12)

f (t) = R e iut k(u) du .
Assume moreover that 1. the moments M n = R |u| n |k(u)| du, n ≥ 0 are finite and satisfy

lim sup n M n n! < ∞ ,
2. the function k does not vanish on any interval of R. Then f is bounded on [0, ∞) and (3.9) holds. In particular, the measure µ is characterized by the restrictions of µ and µ * 2 to [0, ∞) and µ belongs to class C . Moreover f admits an analytic continuation on R.

Proof. The fact that f is bounded follows directly from (3.12) and 1. Let g ∈ L 1 ([0, ∞)), then from (3.12) and Fubini's theorem, we can write for all t ≥ 0,

∞ 0 f (t + s)g(s) ds = ∞ 0 R e iu(t+s) k(u) du g(s) ds = R e iut ∞ 0
e ius g(s) ds k(u) du .

Assume that this expression vanishes for all t ≥ 0 and set ϕ(u) = ∞ 0 e ius g(s) ds. This means that the Fourier transform

Ψ(t) := R e iut ϕ(u) k(u) du ,
of the function u → ϕ(u)k(u), u ∈ R vanishes for all t ≥ 0. Then let us show that under our assumptions, the function Ψ is analytic on the whole real axis. First note that since |ϕ(u)| ≤ 1, u ∈ R and since all the moments M n are finite, then Ψ is infinitely differentiable on R and (3.13)

Ψ (n) (t) = R (iu) n e iut ϕ(u) k(u) du , t ∈ R .
Then notice that for all t, u, x ∈ R,

e iux e itu -1 + itu 1! -• • • - (itu) n-1 (n -1)! ≤ |tu| n n! .
We derive from this inequality and (3.13) that

(3.14) Ψ(x + t) -Ψ(x) - t 1! Ψ (x) -• • • - t n-1 (n -1)! ψ (n-1) (x) ≤ M n n! |t| n .
Set c = lim sup n M n /n!, then from Stirling's formula, for |t| < 1/(3c), the righthand side of (3.14) tends to 0, as n → +∞, hence the Taylor series of Ψ converges in some interval around x, for all x ∈ R. It follows that Ψ is analytic on R. As a consequence, Ψ is determined by its expression on the positive half line. Hence the Fourier transform of the continuous function u → ϕ(u)k(u) vanishes on R, which means that this function vanishes a.e. on R. Since k does not vanish on any interval of R and ϕ is continuous, it implies that ϕ(u) = 0, for all u ∈ R and we conclude that g(t) = 0, for almost every t ∈ [0, ∞). We have proved that (3.9) holds and from the second part of Proposition 3.1, µ is characterized by the restriction of µ and µ * 2 on [0, ∞).

We have proved above that Ψ(t) = R e iut ϕ(u) k(u) du is analytic on R. It follows from the same arguments that the continuation of f on R which is defined in a natural way by f (t) = R e iut k(u) du, t ∈ R is analytic, which proves the last assertion of the theorem. We will denote by A the class of distributions which satisfy the assumptions of Theorem 3.2. It will be called the analytic class. From Theorem 3.2, A ⊂ C .

It is very easy to construct examples of distributions in class A , simply by choosing any symmetric function k which satisfies assumptions 1. and 2. in Theorem 3.2. Let us consider for instance k(u) = e -|u| , for u ∈ R. Then

f (t) = R e iut e -|u| du = 1 2(1 + t 2 )
, t ≥ 0 .

Any extension on R of this function into a density function determines a distribution of

A which does not belong to classes E and M . Conversely, none of the classes E and M is included in A . It is straightforward for E . Then let us consider

f (t) = 1 4 1 0 e -ut u 1/2 du + ∞ 1 e -ut u -1 du , t > 0 . The measure ν(du) = u 1/2 1I [0,1] (u) + u -1 1I [1,∞) (u)
du satisfies the condition of Theorem 3.1 so that µ ∈ M (here we choose a = 0). However, f is not bounded on (0, ∞), hence it does not admit an analytic continuation on R, so that f does not belong to class A , from Theorem 3.2. (Note also that since the support of ν intersects any interval (0, ε), ε > 0, the measure µ has no positive exponential moments, see Remark 3.3.)

3.4.

A class of discrete distributions. In this section, we present the discrete counterpart of class M . More specifically, we will consider distributions whose support is included in Z.

First we need the following equivalent of Lemma 3.2 for discrete distributions. Its proof is straightforward so we omit it. Lemma 3.3. Let (q n ) n∈Z be any probability on Z. Define q * 2 n = k∈Z q n-k q k . Then for all n ≥ 1

(3.15) 0 k=-∞ q n-k q k = 1 2 q * 2 n - n-1 k=1 q n-k q k ,
where we set n-1 k=1 q n-k q k = 0 if n = 1. A sequence (a k ) k≥0 of nonnegative real numbers is called completely monotone if for all k ≥ 0 and n ≥ 1, ∆ n a k := ∆ n-1 a k -∆ n-1 a k+1 ≥ 0 , where ∆ 0 a = a. A result from Hausdorff asserts that (a k ) k≥0 is completely monotone if and only if there is a probability measure ν on [0, 1] such that for all k ≥ 0,

(3.16) a k = 1 0 t k ν(dt) .
Let us set µ({n}) = µ n , for n ∈ Z. Then by assuming that (µ n ) n≥0 is completely monotone, we obtain a new class of distributions satisfying conjecture C as shows the following theorem.

Theorem 3.3. Assume that (µ n ) n≥0 is completely monotone and that the support of the measure ν in the representation (3.16) contains a sequence (c n ) such that (-ln c n ) -1 = ∞. Then the data of µ and µ * 2 restricted to [0, ∞) allows us to determine if the support of µ is included in Z. Assume that this is the case, then the measure µ is determined by the restriction of µ and µ * 2 to [0, ∞). In particular µ belongs to class C .

Proof. First let us observe that we can derive from the data of of µ and µ * 2 restricted to [0, ∞) that the support of µ is included in Z. Indeed, assume that the support of µ restricted to (-∞, 0] is not included in Z -. Then there is an interval I ⊂ (-∞, 0] \ Z - such that µ(I) > 0. From (3.16), µ({n}) > 0 for all n ≥ 1.

Let n ∈ Z + \ {0} such that n + inf I > 0, then 0 < µ(I)µ({n}) = P(S 1 ∈ I, S 2 -S 1 = n) ≤ P(S 2 ∈ n + I) .
This implies that µ * 2 (n + I) > 0, where n + I ⊂ [0, ∞) \ Z + , which contradicts the assumption.

From the Haussdorff representation recalled above, there is a unique finite measure ν on [0,1] such that µ k = 1 0 t k ν(dt). Using this representation and Fubini's theorem, we can write for all n ≥ 0,

∞ k=0 µ -k µ n+k = 1 0 t n ∞ k=0 µ -k t k ν(dt) .
From (3.15) in Lemma 3.3 applied to (µ k ), we derive that this expression is determined from the knowledge of µ and µ * 2 on [0, ∞). This means that we know the moments of the measure

∞ k=0 µ -k t k • ν(dt).
This measure is finite and its support is included in [0, 1], hence it is determined by its moments. Then we know the generating function t → ∞ k=0 µ -k t k of the sequence (µ -k ) k≥0 on the support of ν, since this function is continuous. From the assumption, we know this generating function on a sequence (c n ) such that (-ln c n ) -1 = ∞. This is enough to determine the sequence (µ -k ) k≥0 , from [START_REF] Feller | On Müntz' theorem and completely monotone functions[END_REF].

We conclude that the measures µ and µ * 2 restricted to [0, ∞) allow us to determine µ on Z.

The set of measures satisfying the assumptions of Theorem 2.4 will be called the discrete monotone class and will be denoted by M d . Theorem 2.4 shows that M d ⊂ C . Moreover, it is clear that none of the classes E , M and A is included in M d and that these classes do not contain M d .

When µ is infinitely divisible

The aim of this section is to present a problem equivalent to conjecture C in the framework of infinitely divisible distributions. When µ is infinitely divisible, the Wiener-Hopf factorization can be understood in two different ways: we can either factorize the characteristic function ϕ as in (1.1), or we can factorize the characteristic exponent ψ, which is defined by ϕ(t) = e ψ(t) , t ∈ R .

Then let us recall the Wiener-Hopf factorization in the latter context. Let (X t , t ≥ 0) be a real Lévy process issued from 0 under the probability P and such that X 1 has law µ under this probability, that is E(e iuXt ) = e -tψ(u) , for all t ≥ 0. The characteristic exponent of µ is given explicitly according to the Lévy-Khintchine formula by

ψ(u) = iau + σ 2 2 u 2 + R\{0} (1 -e iux + iux1 {|x|≤1} ) Π(dx) ,
where a ∈ R, σ ≥ 0 and Π is a measure on R \ {0}, such that (x 2 ∧ 1) Π(dx) < ∞. Then the Wiener-Hopf factorization of ψ has the following form:

(4.17)

s + ψ(u) = κ + (s, -iu)κ -(s, iu) , u ∈ R , s ≥ 0 ,
where κ + and κ -are the Laplace exponents of the upward and downward ladder processes β) . These exponents are given explicitly for α, β ≥ 0 by the identities,

(τ + , H + ) and (τ -, H -) of X, that is E(e -ατ +/- t -iβH +/- t ) = e -tκ +/-(α,
κ -(α, β) = k -exp ∞ 0 (-∞,0) (e -t -e -αt-βx ) 1 t P(X t ∈ dx) dt (4.18) κ + (α, β) = k + exp ∞ 0 [0,∞) (e -t -e -αt-βx ) 1 t P(X t ∈ dx) dt , (4.19)
where k -and k + are positive constants depending on the normalization of the local times at the infimum and at the supremum of X. The joint law of (τ + 1 , H + 1 ) is the continuous time counterpart of the joint law (τ + , S τ + ) defined in Section 1, in the setup of random walks. We refer to Chap. VI of [START_REF] Bertoin | Lévy Processes[END_REF], Chap. IV of [START_REF] Kyprianou | Fluctuations of Lévy processes with applications[END_REF] or Chap. of [START_REF] Doney | Fluctuation theory for Lévy processes[END_REF] for complete definitions of these notions. Note that our formulation of the Wiener-Hopf factorization (4.17) includes compound Poisson processes since expression (4.19) takes account of a possible mass at 0 for the measure P(X t ∈ dx). This slight extension can be derived from p. 24 and 25 of [START_REF] Vigon | Simplifiez vos Lévy en titillant la factorisation de Wiener-Hopf[END_REF], see also the end of Section 6.4, p.183 in [START_REF] Kyprianou | Fluctuations of Lévy processes with applications[END_REF].

Set Π(t) = Π(t, ∞), t > 0 and let µ t be the law of X t .

Lemma 4.1. The data of κ + is equivalent to this of µ t on [0, ∞), for all t ≥ 0. Moreover, the knowledge of κ + allows us to determine the function Π(t), t > 0.

Proof. From identity (4.20)

1 t P(X t ∈ dx) dt = ∞ 0 P(τ u ∈ dt, H u ∈ dx) du u , x ≥ 0, t > 0 ,
which can be found in Section 5.2 of [START_REF] Doney | Fluctuation theory for Lévy processes[END_REF], we see that the law of X t in [0, ∞), for all t ≥ 0 is determined by the law of (τ, H) and hence by κ + . (Note that equation (4.20) is also valid for compound Poisson processes.) Conversely, it follows directly from formula (4.19) that κ + is determined by the data of the measure µ t on [0, ∞), for all t ≥ 0.

The second assertion is a consequence of the first one and Exercise 1 of chap.I in [START_REF] Bertoin | Lévy Processes[END_REF], which asserts that the family of measures 1 t P(X t ∈ dx) converges vaguely toward Π, as t → 0.

The above lemma enables us to make the connection between the two Wiener-Hopf factorizations (1.1) and (4.17). Let us state it more specifically in the following proposition. Proof. The result is straightforward from Lemma 4.1. Indeed, knowing κ + we can determine µ n = µ * n restricted to [0, ∞), for all n ≥ 1 and from Section 1 that this data is equivalent to that of χ + . Definition 4.1. We will denote by C i the class of infinitely divisible distributions µ which are determined by the data of their upward Wiener-Hopf factor κ + (s, t), for s, t ≥ 0 or equivalently by the data of the measures µ t , t > 0 restricted to [0, ∞).

Let us denote by I the set of infinitely divisible distributions. Then it is straightforward that I ∩ C ⊂ C i . In particular if Conjecture C is true, then C i = I . It was proved in Chapter 4 of [START_REF] Vigon | Simplifiez vos Lévy en titillant la factorisation de Wiener-Hopf[END_REF] that infinitely divisible distributions having some exponential moments belong to class C i , which is a consequence of our results. The latter work uses a different technique based on the analytical continuation of the Wiener-Hopf factors κ + and κ -.

Let k -, δ -, γ -and k + , δ + , γ + be the killing rate, the drift and the Lévy measure of the subordinators H -and H + , respectively and let us set γ+ (x) = γ + (x, ∞) and γ-(x) = γ -(x, ∞). Let also U -be the renewal measure of the downward ladder height process

H -, that is U -(dx) = ∞ 0 P(H - t ∈ dx). Theorem 4.1.
Assume that the function t ∈ (a, ∞) → Π(t) is completely monotone, for some a ≥ 0, that is there exists a Borel measure ν on (0, ∞) such that for all t > a, Assume moreover that the support of ν contains a sequence (a n ) such that n a -1 n = +∞. Then the measure µ belongs to the class C i .

Proof. The proof relies on Vigon's équation amicale inversée, see [START_REF] Vigon | Simplifiez vos Lévy en titillant la factorisation de Wiener-Hopf[END_REF], p.71, or (5.3.4) p.44 in [START_REF] Doney | Fluctuation theory for Lévy processes[END_REF] which can be written as U -(dy)Π(x + y) , x > 0 .

From Lemma 4.1, given κ + , we know both γ+ (x), for x > 0 and Π(t), for t > 0. Then we will show that under our assumption, equation (4.22) allows us to determine the renewal measure U -(dy), so that the law of X will be entirely determined, thanks to the relation:

Û-(z) = R + e -yz U -(dy) = 1 κ -(0, z)
, z > 0 , (4.23) and the Wiener-Hopf factorization (4.17).

From (4.21), (4.22) and Fubini's Theorem, we can write for all x > 0,

γ+ (x) = [0,∞) U -(dy) ∞ 0 e -(x+y)z ν(dz) = ∞ 0 e -xz Û-(z) ν(dz) . (4.24)
Then the left hand side of equation (4.24) determines the measure Û-(z) ν(dz). Since z → Û-(z) is a continuous function, then it is determined on the support of ν. From our assumption on this support and [START_REF] Feller | On Müntz' theorem and completely monotone functions[END_REF] we derive that Û-(and hence κ -(0, z) = -log E(e -zH -1 )) for z > 0, is determined.

Note that an analogous result to Theorem 4.1 holds when µ has support in Z and the sequence Π(n), n ≥ 1 satisfies the same assumptions as (µ n ) n≥0 in Theorem 3.3. One may also wonder if an assumption such as (3.12) for Π(t) would lead to a similar result to Theorem 3.2. However, in order to use the same argument as in the proof of this theorem together with equation (4.22), we need Û-(z) to be bounded, which is not the case in general.

Remark 4.1. As already observed above, the class C i contains at least all probability measures in the set I ∩ (E ∪ M ∪ A ∪ M d ) but Theorem 4.1 shows that there are other distributions in C i . Indeed, it is easy to construct an example of a compound Poisson process (X t , t ≥ 0) with intensity 1, whose Lévy measure Π satisfies conditions of Theorem 4.1 but such that the law µ(dx) = e -1 n≥0 Π * n (dx)/n! of X 1 does not belong to any of the classes E , M , A and M d .

An infinitely divisible distribution is said to be downward skip free (respectively upward skip free) if the support of the measure Π is included in (-∞, 0] (respectively in [0, ∞)).

Here is a counterpart of Theorem 2.3. Theorem 4.2. Skip free infinitely divisible distributions belong to the class C i .

Proof. The proof relies on Vigon's équation amicale, p.71 in [START_REF] Vigon | Simplifiez vos Lévy en titillant la factorisation de Wiener-Hopf[END_REF]. See also equation (5.3.3), p.44 in [START_REF] Doney | Fluctuation theory for Lévy processes[END_REF]. If δ -> 0, then from [START_REF] Vigon | Simplifiez vos Lévy en titillant la factorisation de Wiener-Hopf[END_REF], the Lévy measure γ + is absolutely continuous and we will denote by γ + (x) its density. Then Vigon's équation amicale can be written as

Π(x) = ∞ 0 γ + (x + du)γ -(u) + δ -γ + (x) + k -γ+ (x) , x > 0 .
It is plain that in the right hand side, the term ∞ 0 γ + (x + du)γ -(u) is identically 0 if and only if the Lévy process X is spectrally positive, that is µ is downward skip free. Moreover, (4.17) for u = 0 entails that the knowledge of κ + implies that of κ -(s, 0), for all s ≥ 0. In particular, we know the killing rate of the subordinator (τ - t , t ≥ 0), and this killing rate is the same as this of (H - t , t ≥ 0), that is k -. Then we conclude that X is spectrally positive if and only if there is a constant δ -such that

Π(x) = δ -γ + (x) + k -γ+ (x) , x > 0 ,
and this can be determined, since from our data, we know k -, Π(x) and γ+ (x), for x > 0.

More classes of distributions

In the previous sections, we have highlighted the subclasses E , M , M d and A of C and proved that these sets of distributions are distinct from each other. More specifically, none of them is included into another one. Then the aim of this section is to show that some of these classes can be substantially enlarged through simple arguments.

Actually for most of the subclasses investigated in this paper, we imposed conditions bearing only on µ restricted to [0, ∞), but one is also allowed to make assumptions on µ * n restricted to [0, ∞). In order to move in this direction, let us mention the following straightforward extension of results of Section 3. Proposition 5.1. Let µ ∈ M 1 be absolutely continuous with density f . If there is n ≥ 1 such that the density function f * n satisfies the same conditions as f in Theorems 3.1 or in Theorem 3.2, then µ is determined by µ * n and µ * 2n restricted to [0, ∞). In particular µ belongs to class C .

It is plain that an analogous extension of Theorem 3.3 is satisfied. Then here is a more powerful result allowing us to extend our classes of distributions. 1 restricted to [0, ∞) and known coincide. Since μ ∈ C , the measures μ and μ1 are equal. Finally, since the characteristic function of ν does not vanish on any interval of R, see Theorem 2 in [START_REF] Smith | A note on characteristic functions which vanish identically in an interval[END_REF], we obtain the identity µ = µ 1 from injectivity of the Fourier transform. Theorem 5.1 entails in particular that Conjecture C' is equivalent to the following one: Any distribution µ ∈ M 1 whose support is not included in (-∞, 0) is determined by its convolution powers µ * n , n ≥ 1 restricted to [a, ∞), for some a ≥ 0. Indeed, it suffices to choose ν = δ -a in Theorem 5.1. Finding more general examples illustrating this result is an open problem. In order to do so, one needs for instance to find the characteristic function ϕ of a random variable which belongs to class C and the characteristic function ϕ Y of a nonnegative random variable Y such that the ratio ϕ(t)/ϕ Y (-t) is the characteristic function of some random variable X. Then since the law of X -Y belongs to class C , so does the law of X from Theorem 5.1.

Note that neither Proposition 5.1 nor Theorem 5.1 allows us to enlarge class E . In order to do so in the same spirit as in Theorem 5.1, one needs to find an invertible transformation T (µ) ∈ M 1 of a distribution µ ∈ M 1 \ E , such that T (µ) * n , n ≥ 1 restricted to [0, ∞) would be known and such that T (µ) belongs to class E .

Let us end this paper with an example of a distribution which satisfies conjecture C, although it does not belong to any of the classes studied here. Assume that the support of µ is included in Z and recall that according to Lemma 2.4, this assumption can be checked from the data of the measures µ * n , n ≥ 0 restricted to [0, ∞). Assume moreover that there are positive integers a and b such that    µ(n) > 0, for all n ≥ a + b and µ(n) = 0, for all n = 0, . . . , a + b -1, µ * 2 (n) = 0, for all n = 0, . . . , a.

(5.25)

Then we can determine µ on Z -, so that µ ∈ C . Let us first show that µ(n) = 0, for all n ≤ -b. Assume that there is n ≤ -b such that µ(n) > 0. Then let k = 0, . . . , a such that k -n ≥ a + b. By definition of the convolution product 0 ≤ µ(k -n)µ(n) ≤ µ * 2 (k), but from our assumptions µ(k -n)µ(n) > 0 and µ * 2 (k) = 0, which is contradictory, hence µ(n) = 0, for all n ≤ -b. On the other hand, assumptions (5.25) entail that for all k = a + 1, . . . , a + b -1,

µ * 2 (k) =

Lemma 3 . 1 .

 31 Conjectures C, C' and C" are equivalent.

Proposition 4 . 1 .

 41 The Wiener-Hopf factor κ + allows us to determine the Wiener-Hopf factor χ + .

e

  -ut ν(du) .

Theorem 5 . 1 .

 51 Let µ ∈ M 1 . If there is ν ∈ M 1 whose support is included in (-∞, 0] and such that µ * ν ∈ C , then µ ∈ C . Proof. Let µ, µ 1 ∈ M 1 such that for each n ≥ 1, the measures µ * n and µ * n 1 restricted to [0, ∞) coincide. Set μ = µ * ν and μ1 = µ 1 * ν. Then from commutativity of the convolution product, μ * n = µ * n * ν * n and μ * n 1 = µ * n 1 * ν * n . Since the support of ν * n is included in (-∞, 0] and µ * n and µ * n 1 restricted to [0, ∞) are known and coincide, the measures μ * n and μ * n

2 kµ 2

 22 -i)µ(i) ,that is µ * 2 (a+1) = µ(-b+1)µ(a+b), µ * 2 (a+2) = µ(-b+2)µ(a+b)+µ(-b+1)µ(a+b+1),... Therefore, this system allows us to determine µ(n), for n = -b + 1, -b + 2, . . . , -1 and the conclusion follows.Let us consider for instancea = 1, b = 3 and µ(-2) = µ(-1) = 1 -c 2 and µ(n) = 1 n 3 , n ≥ 4, where c = n≥4 n -3 . Clearly, such a distribution does not belong to any of the classes A , M or M d . Then let us check that it does not belong to class E . The mean of µ satisfies k≥-< 0 , so that (2.3) does not hold. Moreover µ has no positive exponential moments. Therefore conditions of Theorems 2.1 and 2.2 are not satisfied and since µ is not downward skip free, we obtain the conclusion. Finally it cannot be proved that µ belongs class C by applying Proposition 5.1 or Theorem 5.1. However, from the above arguments, µ does belong to class C .

  3) is satisfied. is the characteristic function of the random variable S τ -under P( • | τ -< ∞). Since this random variable is negative, the knowledge of its characteristic function on [-ε, ε] allows us to determine if the moments m n

	Theorem 2.1. Assume that we know the measures µ * n restricted to [0, ∞). Then we can
	determine if (2.3) holds. Moreover if (2.3) holds, then we can determine if
	(2.6)	there is λ < 0 such that φ(λ) < ∞.
	When both (2.3) and (2.6) hold, the measure µ belongs to the class C .
	Proof. The first assertion is trivial since 1 -µ * n [0, ∞) = P(S n < 0). Assume that (2.3)
	holds. Then from Lemma 2.1, there is ε > 0 such that the characteristic function ϕ(t)
	is known for t ∈ [-ε, ε]. Recall from Section 1 that the knowledge of the measures µ * n ,
	n ≥ 1 restricted to [0, ∞) is equivalent to the knowledge of χ + (s, it), for all |s| ≤ 1 and
	t ∈ R.
	Therefore, from (1.1), we know χ -(1, it), for t ∈ [-ε, ε]. Note that t → χ -(1, it)/χ -(1, 0)

Acknowledgement We would like to thank Rodolphe Garbit, Philippe Jaming et Jean-Jacques Loeb for motivating discussions on this subject.