J S Grange
email: julie.grange@locean-ipsl.upmc.fr

The three steps of the carbonate biogenic dissolution process by microborers in coral reefs (New Caledonia)

Keywords: Microbioerosion dynamics, biogenic dissolution of calcium carbonate, euendoliths, succession of microboring communities, coral reefs, New Caledonia

Biogenic dissolution of carbonates by microborers is one of the main destructive forces in coral reefs and is predicted to be enhanced by eutrophication and ocean acidification by 2100. The chlorophyte Ostreobium sp., the main agent of this process, has been reported to be one of the most responsive of all microboring species to those environmental factors. However, very little is known about its recruitment, how it develops over successions of microboring communities and how that influences rates of biogenic dissolution. Thus, an experiment with dead coral blocks exposed to colonization by microborers was carried out on a reef in New Caledonia over a year period. Each month, a few blocks were collected to study microboring communities and the associated rates of biogenic dissolution. Our results showed a drastic shift in community species composition between the 4 th and 5 th months of exposure, i.e. pioneer communities dominated by large chlorophytes such as Phaeophila sp. were replaced by mature communities dominated by Ostreobium sp.. Prior the 4 th month of exposure, large chlorophytes were responsible of low rates of biogenic dissolution while during the community shift, rates increased exponentially (x10). After 6 months of exposure, rates slowed down and reached a 'plateau' with a mean of 0.93 kg of CaCO 3 dissolved per m² of reef after 12 months of exposure. Here we show that (a) Ostreobium sp. settled down in new dead substrates as soon as the 3 rd month of exposure but dominated communities only after 5 months of exposure, and (b) microbioerosion dynamics comprise 3 distinct steps which fully depend on community development stage and grazing pressure.

Introduction

Coral reefs are highly diverse and complex marine ecosystems, providing many services including commercial fishing, tourism and coastal protection against storms [START_REF] Knowlton | Coral reef biodiversity[END_REF]. Their maintenance results from the delicate balance between constructive forces, mainly calcification of corals and encrusting coralline algae, and destructive forces, mostly bioerosion processes [START_REF] Scoffin | Recent, temperate, sub-photic, carbonate sedimentation: Rockall bank, Northeast Atlantic[END_REF], Tribollet & Golubic 2011). Unfortunately, this equilibrium is increasingly threatened by anthropogenic and environmental factors such as eutrophication, overfishing, rising sea surface temperature and ocean acidification [START_REF] Hoegh-Guldberg | Coral reefs under rapid climate change and ocean acidification[END_REF][START_REF] Pandolfi | Projecting coral reef futures under global warming and ocean acidification[END_REF], Huand 2012). To date, 20% of coral reefs have already disappeared and 20% show signs of imminent extinction [START_REF] Wilkinson | Coral Reef Monitoring Network and Reef and Rainforest Research Centre[END_REF]. Although both constructive and destructive forces are affected by anthropogenic and climatic stresses [START_REF] Anthony | Ocean acidification causes bleaching and productivity loss in coral reef builders[END_REF][START_REF] Tribollet | Effects of elevated pCO2 on dissolution of coral carbonates by microbial euendoliths[END_REF][START_REF] Chauvin | Is the response of coral calcification to seawater acidification related to nutrient loading?[END_REF][START_REF] Carreiro-Silva | Phosphorus and nitrogen effects on microbial euendolithic communities and their bioerosion rates[END_REF][START_REF] Andersson | Ocean acidification and coral reefs: Effects on breakdown, dissolution, and net ecosystem calcification[END_REF][START_REF] Jessen | In situ effects of simulated overfishing and eutrophication on settlement of benthic coral reef invertebrates in the Central Red Sea[END_REF][START_REF] Wisshak | Sponge bioerosion accelerated by ocean acidification across species and latitudes?[END_REF], much less attention has been given to bioerosion than reef accretion.

Reef bioerosion results from the mechanical and/or chemical activity of living organisms [START_REF] Neumann | Observations on coastal erosion in Bermuda and measurements of boring rate of sponge Cliona lampa[END_REF] comprising grazers (e.g. urchins and parrotfishes), macroborers (e.g. bivalves and sponges) and microborers or euendoliths (cyanobacteria, algae and fungi) [START_REF] Glynn | Bioerosion and coral reef growth: a dynamic balance[END_REF], Tribollet & Golubic 2011). Grazers abrade substrate surfaces to feed on epilithic and euendolithic algae [START_REF] Pari | Bioerosion of experimental substrates on high islands and on atoll lagoons (French Polynesia) after two years of exposure[END_REF][START_REF] Tribollet | Cross-shelf differences in the pattern and pace of bioerosion of experimental carbonate substrates exposed for 3 years on the northern Great Barrier Reef, Australia[END_REF] while macro-and micro-borers penetrate actively by mechanical and/or chemical means into substrates to find a shelter [START_REF] Lazar | Bioerosion of coral reefs -A chemical approach[END_REF][START_REF] Zundelevich | Chemical versus mechanical bioerosion of coral reefs by boring sponges -lessons from Pione cf. vastifica[END_REF][START_REF] Garcia-Pichel | Microbial excavation of solid carbonates powered by P-type ATPase-mediated transcellular Ca 2+ transport[END_REF], Tribollet & Golubic 2011). Previous studies showed that microbioerosion in synergy with grazing is the main process driving reef bioerosion of dead carbonates when exposed several months to three years to bioeroding agents [START_REF] Chazottes | Bioerosion rates on coral reefs: interactions between macroborers, microborers and grazers (Moorea, French Polynesia)[END_REF][START_REF] Tribollet | Cross-shelf differences in the pattern and pace of bioerosion of experimental carbonate substrates exposed for 3 years on the northern Great Barrier Reef, Australia[END_REF]. Despite their microscopic size, euendoliths are probably the main agents of reef bioerosion as they are able to colonize each available micrometer of carbonate surface area, -from carbonate sand to skeletons of dead and live calcifiers such as corals, mollusks and crustose coralline algae-, and are able to dissolve as much as 1.1 kg of CaCO 3 per linear m² of reef per year, which converts to 30 mmol CaCO 3 .m -².d -1 (value obtained from the highest rate of microbioerosion quantified in coral reefs and recalculated using coral microdensity and porosity; see [START_REF] Tribollet | Cross-shelf differences in the pattern and pace of bioerosion of experimental carbonate substrates exposed for 3 years on the northern Great Barrier Reef, Australia[END_REF]. [START_REF] Chazottes | Bioerosion rates on coral reefs: interactions between macroborers, microborers and grazers (Moorea, French Polynesia)[END_REF] and Tribollet (2008a) highlighted that the highest rates of carbonate dissolution result mostly from the activity of the chlorophytes of the genus Ostreobium. Recently, experiments realized under controlled conditions or semi-controlled conditions over short periods of time showed that ocean acidification and eutrophication enhance biogenic dissolution by stimulating the activity of chlorophytes such as Ostreobium sp., suggesting an acceleration of reef framework degradation in the near future [START_REF] Chazottes | The effects of eutrophication-related alterations to coral reef communities on agents and rates of bioerosion (Reunion Island, Indian Ocean)[END_REF][START_REF] Tribollet | Effects of elevated pCO2 on dissolution of coral carbonates by microbial euendoliths[END_REF][START_REF] Carreiro-Silva | Phosphorus and nitrogen effects on microbial euendolithic communities and their bioerosion rates[END_REF][START_REF] Reyes-Nivia | Ocean acidification and warming scenarios increase microbioerosion of coral skeletons[END_REF][START_REF] Tribollet | Production of seawater alcalinity by bioeroding microflora increases with ocean acidification[END_REF]. These experiments strongly suggested the dependence of the response of biogenic dissolution on species composition of microboring communities (see also Tribollet 2008a). To better understand dynamics of the biogenic dissolution process under various environmental conditions, it is therefore essential to determine simultaneously species composition of microboring communities, their successions over time and their associated rates of biogenic dissolution under natural conditions. Interestingly these aspects have been rarely studied limiting possibilities of modeling dynamics of the biogenic dissolution process for better prediction of coral reef future.

Very few studies have indeed investigated changes in euendolithic assemblages at different time scales and the influence of such changes on rates of carbonate biogenic dissolution. Le [START_REF] Campion-Alsumard | Experimental study of the colonization of calcite fragments by marine endolithic Cyanophyceae[END_REF] was first to report successions of euendolithic communities in exposed calcite spar in the Mediterranean Sea over a period of one month. She showed that colonization of calcite by euendoliths starts after 8 days of exposure and that the early colonizers are cyanobacteria such as Mastigocoleus testarum. Unfortunately, she did not quantify rates of carbonate dissolution. Later, [START_REF] Kiene | Factors controlling the distribution of microborers in Bahamian Reef environments[END_REF], [START_REF] Gektidis | Development of microbial euendolithic communities: The influence of light and time[END_REF] and [START_REF] Vogel | Experimental studies on microbial bioerosion at Lee Stocking Island, Bahamas and One Tree Island, Great Barrier Reef, Australia: implications for paleoecological reconstructions[END_REF] reported successions of euendolithic communities in various reef carbonate substrates after [START_REF]Food selection in different, socially determined habitats[END_REF]3,6,12,24 months and very scarce information on rates of biogenic dissolution associated with those communities. They confirmed that pioneer euendolithic communities, also called 'immature communities', comprise short lived cyanobacteria such as M. testarum and the chlorophyte Phaeophila sp., and that the long-lived euendolithic chlorophyte Ostreobium sp. appears later to dominate communities after 1 or 2 years of exposure ('mature communities'). [START_REF] Chazottes | Bioerosion rates on coral reefs: interactions between macroborers, microborers and grazers (Moorea, French Polynesia)[END_REF], [START_REF] Tribollet | Endolithic microflora are major primary producers in dead carbonate substrates of Hawaiian coral reefs[END_REF] and Tribollet (2008a) specified later that euendolithic communities in coral reefs become mature at least after 6-12 months of exposure. However, the exact period of settlement of Ostreobium and when it becomes dominant remain unknown. Among all those studies, only [START_REF] Chazottes | Bioerosion rates on coral reefs: interactions between macroborers, microborers and grazers (Moorea, French Polynesia)[END_REF] and Tribollet (2008a) studied and reported simultaneously species composition of euendolithic communities and rates of reef biogenic dissolution over time, i.e. after 2, 6, 12, 24 months and 12 and 36 months, respectively.

They reported that biogenic dissolution does not vary linearly nor proportionally over time, however those authors did not present the full dynamics of the process. Thus, to better understand dynamics of biogenic dissolution of carbonates by euendoliths, we carried out an experiment with a high temporal resolution on a reef in New Caledonia. Experimental coral blocks were exposed to colonization by euendoliths and were recovered every month over a year period. The experiment did not last longer to avoid colonization by macroboring organisms (see review by Tribollet & Golubic 2011) but was long enough to report interactions between euendoliths and grazers. The specific goals of the present study were to determine [START_REF]Food selection in different, socially determined habitats[END_REF] when Ostreobium sp. settles on dead carbonate substrates newly available, (2) when exactly it dominates communities, (3) how euendolithic communities succeed to each other over a year period with a monthly resolution, and (4) how does that influence biogenic dissolution. In addition we quantified grazing pressure and how it influenced biogenic dissolution. Then, the first model of the biogenic dissolution process dynamics was developed.

Material and Methods

Site and experimental design description

This study was carried out on the coral reef of the Island 'Ile aux Canards' in New Caledonia (lat. 22°31'S, long.

166°43'E, Fig. 1a,b) located in the lagoon, a few hundred meters away from Nouméa city (South province of the Island Grande Terre). This site was selected because of its easy access by boat from the laboratory (Center of the Institut de Recherche pour le Développement), it presents a well developed reef which is relatively healthy although it is impacted by anthropogenic activities such as tourism. At this site, the tide amplitude is low (~1.5 m) and seawater is weakly turbid over the year due to the hydrodynamism occurring around the island. Local currents resuspend regularly carbonate sediments and organic matter. The mean seawater temperature during the hot season is 26 ± 2 C° (November-April) and 23 ± 1 C° during the cold season (May-October). Seawater salinity varies between 35.1 and 35.8 over a year period (monthly measurements with CTD sensors on the reef during the experiment).

To determine successions of euendolithic communities and to quantify associated biogenic dissolution and grazing rates, 60 experimental coral blocks were fixed on two steel grids (1 m x 1 m) affixed on the reef at 3 m depth on the 15 th of December 2010, i.e. during the austral summer season (Fig. 1c). Experimental coral blocks (~1.5 cm x 2 cm x 2 cm) were cut from the interior of a live colony of the massive coral Porites sp. using a band saw. Porites skeletons have been used in several previous bioerosion studies thus allowing for comparison of results [START_REF] Kiene | Bioerosion experiments at Lizard-Island, Great Barrier Reef[END_REF][START_REF] Chazottes | Bioerosion rates on coral reefs: interactions between macroborers, microborers and grazers (Moorea, French Polynesia)[END_REF][START_REF] Pari | Bioerosion of experimental substrates on high islands and on atoll lagoons (French Polynesia) after two years of exposure[END_REF][START_REF] Edinger | Normal coral growth rates on dying reefs: Are coral growth rates good indicators of reef health?[END_REF][START_REF] Schonberg | Small-scale distribution of great barrier reef bioeroding sponges in shallow water[END_REF][START_REF] Chazottes | The effects of eutrophication-related alterations to coral reef communities on agents and rates of bioerosion (Reunion Island, Indian Ocean)[END_REF][START_REF] Tribollet | Cross-shelf differences in the pattern and pace of bioerosion of experimental carbonate substrates exposed for 3 years on the northern Great Barrier Reef, Australia[END_REF], Tribollet 2008a). Micro-and bulk density of the coral skeleton were measured using the buoyant weight technique as described by [START_REF] Bucher | Skeletal micro-density, porosity and bulk density of acroporid corals[END_REF]. Between January and December 2011, 3 coral blocks were collected every month and were preserved in a buffered 4% solution of formaldehyde in seawater.

Sample analyses

Colonized blocks were cut in half perpendicular to the upper side. The first halves were used to estimate the bioeroded surface area by euendoliths (in cm 2 ) on the upper side, by taking pictures under a Scanning Electronic Microscope (SEM, Zeiss EVO LS15, Alizés Platform, IRD, Bondy, France). Twenty pictures were taken per coral block (n = 60 per month) at panoramic magnification x1500, and were analyzed using the software ImageJ.

The second halves of blocks were used to prepare thin sections according to the technique described by [START_REF] Golubic | Scanning electron microscopy of endolithic algae and fungi using a multipurpose casting-embedding technique[END_REF][START_REF] Tribollet | Large-scale spatial variability in bioerosion of experimental coral substrates on the Great Barrier Reef (Australia): importance of microborers[END_REF]. These thin sections were observed under a light microscope (Nikon Eclipse LV100) to allow the determination of (1) the species composition of euendolithic communities, (2) the relative abundance (RA) of living filaments inside substrates (n > 120 measurements per month) expressed in % (± the confidence interval), and (3) the mean depth of penetration (P 80 ) in cm at which 80%-90% of the living euendolithic filaments could be observed inside coral blocks (n > 120 measurements per month; see technique description in [START_REF] Tribollet | Effects of elevated pCO2 on dissolution of coral carbonates by microbial euendoliths[END_REF]). To complete the identification of euendoliths, small pieces of sample were decalcified (HCl < 10%) allowing the observations of a few euendolithic filaments at a time. Identification was based on morphological criteria with the help of Stjepko Golubic, and according to [START_REF] Bornet | Sur quelques plantes vivant dans le test calcaire des mollusques[END_REF] and [START_REF] Wisshak | Temperate bioerosion: ichnodiversity and biodiversity from intertidal to bathyal depths (Azores)[END_REF]. Only species with relative abundance higher than 5% are presented in this paper.

The quantity of carbonate dissolved per month, called here 'biogenic dissolution' (BD), was calculated using the following equation (adapted from [START_REF] Tribollet | Effects of elevated pCO2 on dissolution of coral carbonates by microbial euendoliths[END_REF]):

where 'BD' is expressed in g of CaCO 3 dissolved per linear square meter of reef after a certain time of exposure (e.g. one month of exposure for blocks colonized by euendoliths during one month), 'depth of penetration' is P 80 in cm, 'bioeroded surface area' is the bioeroded surface area of block surfaces exposed to colonization by euendoliths multiplied by (100 -(% of porosity of coral skeleton) in cm 2 (here we considered only 5 sides per block as the bottom side was exposed to a different light regime and porosity was 53%), 'density' is the coral skeleton microdensity (2.58 ± 0.06 g.cm -3 ) as filaments of euendoliths grow into the microstructure of the skeleton, not in its void spaces, and 'surface area of block' is the surface area of the 5 sides of each block exposed to colonization (the bottom side was discarded). We assumed here that the measured bioeroded surface area and P 80 on the upper sides of blocks were representative of the 5 sides as blocks were small enough to be exposed to the same light intensity.

By knowing the initial dimensions of coral blocks, it was also possible to estimate the percentage of carbonate removed by grazers on each sample, by analyzing pictures of thin sections taken by a high resolution Nikon camera (Nikon Camera Digital D5100 with AF-S Micro Nikkor 60mm objective) using the software imageJ (n> 12 measurements per month). Grazing rates were then calculated using the following equation:

where 'grazing' is expressed in g of CaCO 3 eroded per linear square meter of reef after a certain time of exposure (i.e. xx months), 'volume of block' is expressed in cm 3 (initial length x height x width), 'density' is the bulk density of the coral skeleton in g.cm -3 (1.86 ± 0.1 g.cm -3 ) as grazers abrade simultaneously voids and skeleton, and 'surface area of block' is similar to that used for BD, in cm 2 .

Statistical analysis

Statistical analyzes were performed under the R statistical framework (R. Development Core Team 2014). All data sets were analyzed with non-parametric tests because homogeneity of variance and normality distribution were not met (Bartlett test and Shapiro test, respectively) even after data transformation [START_REF] Hollander | Threatened reef corals of the World[END_REF]).

The Kruskal-Wallis test (with α = 5 %) was used to determine if measured variables (i.e. relative abundance and depth of penetration of euendolithic filaments, biogenic dissolution, and grazing) varied significantly among the different durations of exposure (from 1 month to 12 months). When the Kruskal-Wallis test was significant a pairwise post-hoc analysis of Mann and Whitney was realized using a correction of Bonferroni [START_REF] Hollander | Threatened reef corals of the World[END_REF]). To fit biogenic dissolution values and to obtain a rate of the biogenic dissolution after 12 months of exposure, we used a logistic growth model in R statistical framework with the nls2 library [START_REF] Grothendieck | nls2: Non-linear regression with brute force[END_REF]):

where 'A' is the maximum asymptotic value, 'X' is the time where A/2 is observed , 'S' is the maximum slope and 't' is the time of exposure. To see possible correlation between grazing and biogenic dissolution, the correlation of Spearman was used [START_REF] Hollander | Threatened reef corals of the World[END_REF]. Finally to determine the main variables driving biogenic dissolution dynamics over a year period, a hierarchical cluster analysis was performed based on dissimilarity matrix using Bray-Curtis distances [START_REF] Murtagh | Ward's hierarchical agglomerative clustering method: which algorithms implement Ward's criterion[END_REF]. A non-metric multidimensional scaling analysis (nMDS) was performed using this matrix as well as a hierarchical cluster analysis to graphically represent results [START_REF] Faith | Compositional dissimilarity as a robust measure of ecological distance[END_REF].

Results

Euendolithic species

Euendoliths observed in the experimental coral blocks comprised (i) chlorophytes including Phaeophila sp. (Fig. 2a), Eugomontia sp., (Fig. 2b) and Ostreobium sp. (Fig. 2c), (ii) two cyanobacteria comprising Mastigocoleus testarum (Fig. 2d) and Plectonema terebrans (Fig. 2e), and (iii) unidentified fungi (Fig. 2f). Due to relative morphological similarities between Eugomontia sp. and Phaeophila sp. and difficulties sometimes to distinguish these two species from unidentified large chlorophytes, all these euendoliths were grouped in one category named 'unidentified chlorophytes'. Ostreobium sp. was easily identified as this chlorophyte presents siphoneous and polymorphic filaments. Plectonema terebrans and fungi were also grouped in one category named 'thin filaments' as the diameter of their filaments did not exceed 1-2 µm and cells in some filaments of P. terebrans were difficult to distinguish. Moreover perpendicular branches and reproductive organs of fungi were usually absent rendering difficult their differentiation from P. terebrans filaments.

Successions of communities: abundance and distribution of euendoliths

After one month of exposure, euendolithic communities were mostly composed of unidentified chlorophytes even though thin filaments and the cyanobacterium M. testarum were observed (mean RA of 83% for unidentified chlorophytes, Fig. 3a). Filaments of M. testarum and P. terebrans as well as fungi were observed in all blocks during the overall duration of the experiment but their mean relative abundance never exceeded 20% (Fig. 3c,d). The cyanobacterium Hyella sp. was also observed in one month-old blocks but was very rare (RA < 0.5%). In 2 and 3 months-old blocks, the unidentified chlorophytes remained dominant with a mean RA of 52% (Fig. 3a). About 2% can be attributable to M. testarum, 0.5% to thin filaments and the rest to O. quekettii. The unidentified chlorophytes never penetrated deeper than 0.5 mm into coral skeletons (Fig. 3a). Similarly filaments of M. testarum were abundant beneath the surface of substrates although a few filaments could be observed down to 2 mm (their relative abundance at this depth was less than 2%; Fig. 3d). Between 4 and 6 months, the relative abundance of unidentified chlorophytes and O. quekettii was inversely correlated (Spearman test, r = -0.81, p<0.01), with RA of unidentified chlorophytes reaching less than 1% after 6 months of exposure (Fig. 3a) and RA of O. quekettii 42% (Fig. 3b). After more than 6 months of exposure, O. quekettii dominated euendolithic communities. Its relative abundance was maximal (> 70%) after 11 months of exposure (Fig. 3b).

The rest of the communities comprised thin filaments (13%) and rare filaments of M. testarum (4%). O. quekettii and thin filaments could penetrate deep into coral skeletons reaching sometimes more than 4 mm (Fig. 3b,c). O. quekettii was the most abundant species between 0.5 mm and 4 mm below the substrate surface with a mean RA of 43% (Fig. 3b).

Surface area bioeroded and depth of penetration (P 80 )

After one month of exposure, the mean surface area bioeroded by euendoliths was about 17.6% ± 2.1%. During the first four months of exposure, the surface area bioeroded increased significantly (Mann and Whitney, p<0.01; Table 1) and reached a mean value of 40%. After 4 months of exposure, the surface area bioeroded was 38% on average, except between 7 and 9 months of exposure (lowest mean quantified, 28%).

The mean depth of penetration of living euendolithic filaments (P 80 ) increased slowly during the first four months of exposure (Table 1). Between the 4 th and 7 th months of exposure, P 80 increased exponentially and reached a maximum mean value of 3.19 ± 0.19 mm after 7 months of exposure. Finally, after 8 months of exposure, depths of penetration tend to stabilize with a mean value of 2.55 ± 0.20 mm.

Biogenic dissolution rates

The quantity of dissolved calcium carbonate by euendoliths increased significantly over time (Mann and Whitney, p<0.001) but non-linearly (Fig. 4a). The dynamics of the biogenic dissolution process presented indeed 3 steps. First, between 1 and 3 months of exposure, biogenic dissolution increased slowly reaching 66 ± 0.8 g of dissolved per square meter of reef after 3 months of exposure (Fig. 4a). Within this period of 2 months, the mean rate of dissolved CaCO 3 was 30 g.m -2 .mo -1 . Second, between 4 and 5 months of exposure, biogenic dissolution increased exponentially reaching a mean of 739 ± 11 g.m -2 after 5 months of exposure. During this short period of time, the mean rate of biogenic dissolution was 564 g.m -2 .mo -1 . Finally after 6 months of exposure, the process slowed down dramatically. Biogenic dissolution reached a 'plateau' with a mean value varying between 742 and 1134 g.m -2 after 8 months and 11 months, respectively (Fig. 4a); the mean rate of biogenic dissolution was 23 g.m -2 .mo -1 for the period comprised between 7 and 12 months of exposure. This process, as described above, is typically a logistic growth model in three steps. A preliminary model was then developed to highlight the process kinetics and to estimate the mean rate of biogenic dissolution for a 12 months period (Fig. 5). The expression of a logistic model growth was used in R to fit our biogenic dissolution dataset. The asymptotic value or the maximum value of biogenic dissolution was 0.91 ± 0.04 kg.m -2 ; half of the biogenic dissolution was realized after 4.5 ± 0.2 months and the maximum of slope was 0.37 ± 0.15 kg.m -2 .mo -1 . The 3 steps-process was confirmed by the non-metric multidimensional scaling (nMDS) with the hierarchical cluster analysis (Fig. 6).

The low value of stress indicates that the representation is reliable: the distance between studied months (circle) correlate well with the dissimilarity of the three clusters (ellipses). In addition, the nMDS showed how the different measured biological parameters influenced each step. The early step of biogenic dissolution appeared to be driven by the abundance of unidentified chlorophytes while the second step seemed to be related to the shift in community species composition. In contrast, the final step was clearly driven by the abundance of the chlorophyte Ostreobium quekettii and the intensity of grazing.

Grazing pressure

Based on observations of block surfaces under a binocular microscope, grazing traces appeared to be due mostly to parrotfishes, confirming a few in situ observations during block collection. Between 1 and 7 months of exposure, grazing increased almost linearly (from 35 ± 11 to 365 ± 51 g of CaCO 3 eroded per square meter of reef, respectively; Fig. 4b). Except for the two peaks at 8 and 10 months of exposure (grazing rate > 600 g.m -2 ), after 7 months of exposure grazing rate values oscillated around the mean obtained after 12 months of exposure, i.e. 339 ± 81 g.m -2 (Fig. 4b). Biogenic dissolution and grazing were strongly negatively correlated after 7 months of exposure (Spearman correlation, r = -0.92, p <0.01).

Discussion

The present study of euendolithic community successions with a monthly resolution reveals the three main steps in the colonization of newly available dead carbonate substrates: (i) an early community development stage between 0 and 3 months where large chlorophytes such as Phaeophila sp. are dominant, (ii) an intermediate stage between 3 and 6 months where the chlorophyte Ostreobium sp. the main agent of biogenic dissolution, starts to settle down and to dominate communities, and (iii) a final stage, after more than 6 months of exposure, where Ostreobium sp. dominates largely microboring communities. The present work confirms that colonization by euendoliths of newly available carbonate substrates is rapid (a few days or weeks) and is dominated by large pioneer chlorophytes, especially the green alga Phaeophila sp., and cyanobacteria such as Mastigocoleus testarum (see [START_REF] Chazottes | Bioerosion rates on coral reefs: interactions between macroborers, microborers and grazers (Moorea, French Polynesia)[END_REF][START_REF] Kiene | Factors controlling the distribution of microborers in Bahamian Reef environments[END_REF], Le Campion-Alsumard et al. 1995[START_REF] Gektidis | Development of microbial euendolithic communities: The influence of light and time[END_REF][START_REF] Vogel | Experimental studies on microbial bioerosion at Lee Stocking Island, Bahamas and One Tree Island, Great Barrier Reef, Australia: implications for paleoecological reconstructions[END_REF][START_REF] Tribollet | Endolithic microflora are major primary producers in dead carbonate substrates of Hawaiian coral reefs[END_REF][START_REF] Tribollet | The boring microflora in modern coral reef ecosystems: a review of its roles[END_REF]). Those communities are characteristic of 'immature communities' [START_REF] Gektidis | Development of microbial euendolithic communities: The influence of light and time[END_REF]. Here, the relative abundance of large unidentified chlorophytes, comprising mostly Phaeophila filaments, was higher than 60% during the first 4 months of exposure which is similar to the findings of [START_REF] Kiene | Factors controlling the distribution of microborers in Bahamian Reef environments[END_REF] and [START_REF] Vogel | Experimental studies on microbial bioerosion at Lee Stocking Island, Bahamas and One Tree Island, Great Barrier Reef, Australia: implications for paleoecological reconstructions[END_REF] in limestone in Bahamian and Australian reefs. Although [START_REF] Chazottes | Bioerosion rates on coral reefs: interactions between macroborers, microborers and grazers (Moorea, French Polynesia)[END_REF] also found Phaeophila in abundance in one-month old coral blocks in French Polynesia, the relative abundance of this species in communities decreased rapidly after the second month of exposure. This could result from the intense grazing quantified on blocks in French Polynesia which certainly limited the development of epiliths and Phaeophila, and postponed Ostreobium settlement (seen after 6 months of exposure). In the present study, significant traces of grazing were only observed on blocks after a few months of exposure suggesting that epilithic and euendolithic communities developed slowly on the studied reef in New Caledonia compared to those growing in French Polynesian reefs. Grazers are indeed not attracted by substrates until epilithic (mostly turfs) and euendolithic communities on which they feed, start to be well developed [START_REF] Bruggemann | Foraging by the stoplight Parrotfish Sparisoma viride[END_REF]. Interestingly in the present study, Ostreobium sp. started settling on new dead coral skeletons as soon as the 3 rd month of exposure (but represented less than 5% of euendolithic communities). In mollusc shells [START_REF] Carreiro-Silva | The role of inorganic nutrients and herbivory in controlling microbioerosion of carbonate substratum[END_REF] also observed Ostreobium filaments after 49 days. To the contrary of what was suggested in previous studies, this green euendolithic alga appears thus as an early colonist of dead carbonate substrates in shallow reefs. Most studies have indeed reported the presence of Ostreobium sp. in dead substrates comprising limestone, calcite spar, mollusc shells and coral skeletons, only after more than 6-12 months of exposure [START_REF] Chazottes | Bioerosion rates on coral reefs: interactions between macroborers, microborers and grazers (Moorea, French Polynesia)[END_REF][START_REF] Kiene | Factors controlling the distribution of microborers in Bahamian Reef environments[END_REF], Le Campion-Alsumard et al. 1995[START_REF] Gektidis | Development of microbial euendolithic communities: The influence of light and time[END_REF][START_REF] Vogel | Experimental studies on microbial bioerosion at Lee Stocking Island, Bahamas and One Tree Island, Great Barrier Reef, Australia: implications for paleoecological reconstructions[END_REF][START_REF] Tribollet | Endolithic microflora are major primary producers in dead carbonate substrates of Hawaiian coral reefs[END_REF], Tribollet 2008a).

All pioneer euendolithic species were located within the first few hundred micrometers beneath the substrate surface, with the exception of a few filaments of M. testarum seen down to 2 mm below the surface. These species are indeed known as photophile euendoliths which do not penetrate deep into substrates as they require high light intensities to insure photosynthesis (see review [START_REF] Tribollet | The boring microflora in modern coral reef ecosystems: a review of its roles[END_REF]. In contrast, Ostreobium is a sciaphile or oligophotic euendolith, i.e. that it is adapted to low light intensities [START_REF] Fork | Light harvesting in green-alga Ostreobium sp., a coral symbiont adapted to extreme shade[END_REF][START_REF] Tribollet | Endolithic microflora are major primary producers in dead carbonate substrates of Hawaiian coral reefs[END_REF], and thus, can penetrate deep inside substrates (max depth > 4 mm after 6 months of exposure) and presents a large bathymetric distribution (found down to 300 m depth; [START_REF] Lukas | Depth distribution and form among common microboring algae from the Florida continental shelf[END_REF]; [START_REF] Kiene | Factors controlling the distribution of microborers in Bahamian Reef environments[END_REF]; [START_REF] Radtke | Microbial assemblages of the bioerosional "notch" along tropical limestone coasts[END_REF]. It can also be easily photoinhibited by sudden important light intensities like in the case of a rapid bleaching event in live corals [START_REF] Fine | Tolerance of endolithic algae to elevated temperature and light in the coral Montipora monasteriata from the southern Great Barrier Reef[END_REF]. Its light regime requirements may thus explain why it was not observed on one month-old blocks. Like suggested by [START_REF] Vogel | Experimental studies on microbial bioerosion at Lee Stocking Island, Bahamas and One Tree Island, Great Barrier Reef, Australia: implications for paleoecological reconstructions[END_REF], we strongly emphasize that Ostreobium can only settle on substrates once they are colonized by epilithic organisms (biofilms, and then turfs at the early stages) and pioneer euendoliths, which provide shade (see also [START_REF] Radtke | Microbial assemblages of the bioerosional "notch" along tropical limestone coasts[END_REF]. As soon as those conditions were met in the present experiment, Ostreobium started rapidly to dominate communities. The shift in euendolithic community species composition in favour of Ostreobium occurred between the 4 th and 5 th months of exposure. Although such shift was suggested previously by [START_REF] Chazottes | Bioerosion rates on coral reefs: interactions between macroborers, microborers and grazers (Moorea, French Polynesia)[END_REF] and [START_REF] Kiene | Factors controlling the distribution of microborers in Bahamian Reef environments[END_REF], its period of occurrence was not highlighted due to the temporal resolution selected in each experiment (2-3 months and then, 6 months or more). Here, after 6 months of exposure, the relative abundance of Ostreobium reached more than 45% while that of large unidentified chlorophytes (mostly Phaeophila) dropped down to 1%. The outcompetition between Ostreobium and Phaeophila was first reported by Le [START_REF] Campion-Alsumard | Microbial endoliths in skeletons of live and dead corals -porites lobata (Moorea, French Polynesia)[END_REF] in shallow waters where light intensity is high. The other species observed in the present study after 6 months of exposure were essentially the cyanobacterium P. terebrans and fungi. The presence of these species in addition to Ostreobium characterizes 'mature communities' [START_REF] Chazottes | Bioerosion rates on coral reefs: interactions between macroborers, microborers and grazers (Moorea, French Polynesia)[END_REF][START_REF] Kiene | Factors controlling the distribution of microborers in Bahamian Reef environments[END_REF][START_REF] Gektidis | Development of microbial euendolithic communities: The influence of light and time[END_REF][START_REF] Vogel | Experimental studies on microbial bioerosion at Lee Stocking Island, Bahamas and One Tree Island, Great Barrier Reef, Australia: implications for paleoecological reconstructions[END_REF], Tribollet 2008a). Such community species composition seems to be maintained after 12 months of exposure [START_REF] Chazottes | Bioerosion rates on coral reefs: interactions between macroborers, microborers and grazers (Moorea, French Polynesia)[END_REF][START_REF] Kiene | Factors controlling the distribution of microborers in Bahamian Reef environments[END_REF], Tribollet 2008a) unless a major event occurs. An extreme grazing pressure for instance, can modify dramatically community assemblages by removing most of settled euendolithic filaments in substrates allowing more light to penetrate inside the carbonate structure. The remaining filaments can then grow inside substrates until they reach their new depth of compensation (photosynthesis = respiration) while new pioneer euendoliths colonize substrate surfaces [START_REF] Schneider | Biokarst on limestone coasts, morphogenesis and sediments production[END_REF][START_REF] Chazottes | Bioerosion rates on coral reefs: interactions between macroborers, microborers and grazers (Moorea, French Polynesia)[END_REF]. In general, grazers do not remove all euendolithic filaments inside coral skeleton allowing Ostreobium to remain the dominant species in substrates exposed more than 6 months to colonization, but are sufficient to offer new surfaces available to colonization by cyanobacteria such as Hyella sp. and M. testarum. This is why these pioneer species are often observed within mature communities, but in relatively low abundance (less than 0.5% for Hyella sp. and 20% for M. Testarum here; see also [START_REF] Chazottes | Bioerosion rates on coral reefs: interactions between macroborers, microborers and grazers (Moorea, French Polynesia)[END_REF][START_REF] Tribollet | Endolithic microflora are major primary producers in dead carbonate substrates of Hawaiian coral reefs[END_REF], and Tribollet 2008a).

With a monthly resolution, we highlight how much the dynamics of euendolithic communities and grazing pressure controlled the process of biogenic dissolution over a year period. First of all, pioneer communities within the first 3 months of exposure drove a slow process of biogenic dissolution with a mean rate of 30 g.m - ².mo -1 although the amount of dissolved CaCO 3 after 3 months of exposure was 66 g.m -2 , which is similar to what found [START_REF] Chazottes | Bioerosion rates on coral reefs: interactions between macroborers, microborers and grazers (Moorea, French Polynesia)[END_REF][START_REF] Peyrot-Clausade | Initial bioerosion and bioaccretion on experimental substrates in high island and atoll lagoons (French Polynesia)[END_REF] in 2 months-old blocks in French Polynesia (~73 g.m -² after recalculation with a microdensity for Porites coral skeleton of 2.58 g.cm -3 and a porosity of 53%). This was due to the low depth of penetration of euendolithic filaments at this early stage of community development (< 400 µm), and not to the colonization of surfaces which was already important (> 30% like in [START_REF] Chazottes | Bioerosion rates on coral reefs: interactions between macroborers, microborers and grazers (Moorea, French Polynesia)[END_REF] and the large diameter of pioneer euendolithic filaments (> 10 µm). Carbonate biogenic dissolution became a very efficient process during the community shift with a mean rate of 564 g.m - 2 .mo -1 (rate obtained the 4 th and 5 th month of exposure). The rate of the biogenic dissolution increased exponentially during this short period of time (one month) and was multiplied by a factor 10. This resulted from the 3D network development of Ostreobium filaments within substrates (increase of filament abundance and depth of penetration) as described above. Once Ostreobium's population became stable after 6 months of exposure, -i.e. Ostreobium filaments reached their depth of compensation (about 4 mm which is similar to that found by Tribollet 2008a)-, the mean rate of biogenic dissolution slowed down and reached a 'plateau' (23 g.m - 2 .mo -1 ). Interestingly the estimated average rate of biogenic dissolution provided by the model after 12 months of exposure (0.91 kg m -2 .y -1 ) was of the same order to rates estimated on offshore reefs on the Great Barrier Reef after one year of exposure, although those reefs where more influenced by oceanic conditions than the present studied site (0.5 to 1.1 kg m -2 .y -1 after recalculation with microdensity and porosity of Porites skeletons, [START_REF] Tribollet | Cross-shelf differences in the pattern and pace of bioerosion of experimental carbonate substrates exposed for 3 years on the northern Great Barrier Reef, Australia[END_REF]. Those rates are comparable as the same techniques were used for their quantification (SEM picture analysis and thin sections; see Tribollet 2008a). Tribollet (2008a)'s methodology, although it prevents from confounding initial microbioerosion like in [START_REF] Chazottes | Bioerosion rates on coral reefs: interactions between macroborers, microborers and grazers (Moorea, French Polynesia)[END_REF] and avoids subjective estimations like in [START_REF] Vogel | Experimental studies on microbial bioerosion at Lee Stocking Island, Bahamas and One Tree Island, Great Barrier Reef, Australia: implications for paleoecological reconstructions[END_REF] and [START_REF] Carreiro-Silva | The role of inorganic nutrients and herbivory in controlling microbioerosion of carbonate substratum[END_REF], allows only the quantification of the amount of CaCO 3 dissolved by euendoliths alive at the time of collection. It is like an instantaneous 'snap shot' of a community activity at a given time. It cannot provide the cumulative effects of euendolithic communities on carbonate substrates over a given time period and does not reflect the potential long term impact of abiotic factors. The survey of biogenic dissolution rates at different periods of time like in the present study allows however, determining the possible impacts of biotic and abiotic factors on the process. Here, biogenic dissolution variation within the last 7 months of the experiment resulted from grazing activity as a strong inverse correlation between the two processes was found during this period. Grazing was especially important after 8 and 10 months of exposure reducing biogenic dissolution by about 30% (the 'residual biogenic dissolution' was thus measured; see definition in [START_REF] Chazottes | Bioerosion rates on coral reefs: interactions between macroborers, microborers and grazers (Moorea, French Polynesia)[END_REF]. This synergy between grazers and euendoliths has first been well described by [START_REF] Schneider | Biokarst on limestone coasts, morphogenesis and sediments production[END_REF] in the Adriatic Sea and later by [START_REF] Chazottes | Bioerosion rates on coral reefs: interactions between macroborers, microborers and grazers (Moorea, French Polynesia)[END_REF] in coral reefs. It has been reported as the main bioerosion process in dead substrates after more than 12 months by [START_REF] Tribollet | Cross-shelf differences in the pattern and pace of bioerosion of experimental carbonate substrates exposed for 3 years on the northern Great Barrier Reef, Australia[END_REF]. Here, it also appears as the main bioerosion process on dead carbonates exposed 6 to 12 months to colonization.

In conclusion, the present experiment revealed the 3 steps in the process of biogenic dissolution over a year period confirming that it is a neither linear, nor proportional process over time. It also suggested that rates estimated after 12 months of exposure should be considered as conservative rates (probably minimum values). This is the first time that such pattern is highlighted under natural conditions with a moderate grazing pressure.

Due to the experimental design, temporal resolution, and biotic conditions (intense grazing), [START_REF] Chazottes | Bioerosion rates on coral reefs: interactions between macroborers, microborers and grazers (Moorea, French Polynesia)[END_REF] could only report part of the kinetics of the process. The present study also highlights that the biogenic dissolution process is particularly efficient during the community shift between the 4 th and the 5 th month of exposure and after a grazing episode (similar slopes). It took indeed about one month for the remaining Ostreobium filaments to reach again their depth of compensation as the amount of calcium carbonate dissolved per square meter after a high grazing episode (e.g. 8 months) was of the same order to that estimated prior to grazing (Fig. 7). This fast growth in depth inside substrates may be the result of the avoidance of an increased light intensity by Ostreobium filaments as epiliths and euendoliths near the surface have been scrapped off by grazers. This reinforces the idea that the chlorophyte Ostreobium is a very efficient bioeroding microorganism in coral reef systems. It should be noticed here however, that euendolithic community dynamics over 12 months were studied starting in the summer season (December). Biofilms and epilithic covers may develop slowly in winter due to reduce light intensity and seawater temperature, and thus may slow down euendolith settlement (especially that of Ostreobium), successions and therefore, biogenic dissolution. This should be investigated over several years to highlight possible seasonal and inter-annual variations. Such data would definitely improve our capacity to better predict the response of biogenic dissolution to climatic and anthropogenic factors, and therefore to better evaluate coral reef future by 2100.

FIGURES

Fig. 1 Fig. 3

 13 Fig. 1 The area of study. a -Position of New Caledonia in the Southwest Pacific Ocean; b -Study site at "Ile aux Canards" in New Caledonia; c -Two experimental exposure grids with 60 experimental coral blocks fixed on the reef at 3 m-depth Fig. 2 Euendolithic filaments or borings observed on colored thin sections or after decalcification under light microscopy, or under SEM. A -Phaeophila sp. filament (Chlorophyta, on thin section); b -Eugomontia sp. filament with a few cells (Chlorophyta, after decalcification); c -Ostreobium sp. filaments (Chlorophyta, on thin section) with the white arrow indicating the typical shape of this alga, and the black arrow highlighting filaments of fungi; d -Mastigocoleus testarum filaments (Cyanobacteria, after decalcification) with heterocystous cells (black arrow); e -Plectonema terebrans filaments (Cyanobacteria, after decalcification); f -Borings of unidentified euendolithic fungi (under SEM). Scale bare = 20 µm Fig. 3 Mean relative abundance (RA) of euendolithic species as a function of time (in months) and their depth of penetration in substrates (in mm). a -Unidentified chlorophytes (i.e. Phaeophila sp. + Eugomontia sp. + unidentified species); b -The chlorophyte Ostreobium sp., c; Thin filaments (i.e. Plectonema terebrans + fungi); d -The cyanobacterium Mastigocoleus testarum

Fig. 4 Fig. 5 Fig. 7

 457 Fig. 4 Bioerosion by euendoliths (a) and grazers (b) as a function of time. Rates are expressed in kg of CaCO 3 dissolved per linear m 2 of reef (Mean ± standard error) Fig. 5 Preliminary logistic growth model of the process of biogenic dissolution in kg of CaCO 3 dissolved per linear m² of reef as a function of time. Solid line relates to fitted values, and dotted lines show the confidence interval of fitted values Fig. 6 Non-metric multidimensional scaling (nMDS) realized after clustering the relative abundance of euendolithic species, biogenic dissolution (BD), and grazing over the duration of the experiment Fig. 7 Graphic representation of temporal dynamics of the process of biogenic dissolution of carbonates by euendoliths and interaction with grazers

Acknowledgements

We would like to deeply thank John Butscher and the diving team of IRD located in New Caledonia (Center of IRD, Nouméa), especially Armelle Renaud and Bertrand Bourgeois, for helping us on the field. We thank Cécile Dupouy for in situ CTD measurements. We thank Sandrine Caquineau for helping using the Scanning Electronic Microscope on the ALIZES facility (IRD-UPMC), which was funded by the Région Ile-de-France. We also thank Stjepko Golubic for discussions and help with euendolith identification. Finally, we thank the Institut National des Sciences de l'Univers (INSU) -EC2CO (Microbien and PNEC programs), the Institut de Recherche pour le Développement, and the Grand Observatoire de l'Environnement et de la Diversité Terrestre et Marine du Pacifique Sud (GOPS) for funding this project.

TABLES


). Each month, 5 blocks were sampled: 3 blocks for this study and 2 additional blocks for the study of the impact of microborers on the geochemical composition of coral skeletons (see Poster #2254 C.Brahmi et al.)

Introduction

Coral reefs maintenance results from the balance between constructive and destructive forces, those being mainly due to the agents of bioerosion. Microborers -cyanobacteria, microalgae and fungi-are the main agents of biogenic dissolution in dead substrates such as coral skeletons. However, very little is known on the early stages of their community successions and associated biogenic dissolution rates. In this study we present To characterize the biogenic dissolution per month due to microborers, 4 parameters were determined on blocks:

(1) Species composition of microboring communities,

(2) Succession of microborers, (3) Filament depth of penetration in blocks (4) the bioeroded surface area at the surface of blocks.

Here, biogenic dissolution is expressed in kg of CaCO 3 dissolved per square meter of coral reef and per unit of time.

All data were analysed using R software. -M. testarum and thin filaments (fungi and P. terebrans) are present through the duration of the experiment but are not abundants (<20%, Fig. 4).

-O. quekettii start settling on substrates after 3 months of exposure and dominates after 6 months (Fig. 4), characterizing mature communities.

-Chlorophytes other than O. quekettii and cyanobacteria penetrate no more than 2 mm into substrates (Fig. 5). Those organisms are known to request light (photophile species).

-O. quekettii and thin filaments (mainly fungi) penetrate up to 4mm into substrates (Fig. 5). O. quekettii is a sciaphile species, adapted to low light intensities. -After 12 months of colonization, 4.3 kg of CaCO 3 are dissolved per square meter of reef (gross dissolution rates, Fig. 6).

-The dynamic of biogenic dissolution is non-linear (Fig. 6) and seems to be strongly correlated with the successions of microborer communities (Fig. 4, 6 and7) -The dynamic of biogenic dissolution comprised 3 stages (Fig. 6 and7):

Stage 1: Low dissolution rates Settlement and dominance of Chlorophytes other than O. quekettii that remains at the surface (Fig. 4 and5).

Stage 2: High dissolution rates Settlement of O. quekettii (Fig. 4) that penetrate down to 4 mm (Fig. 5). Increase of M. testarum abundance (Fig. 4).

Stage 3: Stabilization of dissolution rates O. quekettii dominates microborer communities (Fig. 4) and drives the biogenic dissolution process (Fig. 7).

Stage 1 Stage 2

Stage 3

Community Composition

Observed microborers comprised (Fig. 3):

-3 species of chlorophytes (Eugomontia sp., Phaeophila sp., Ostreobium quekettii and unidentified species), This is the first study on the successions of microborer communities and the associated biogenic dissolution with a montly resolution (Grange et al. in prep.).

- -3 stages in this process are noticed over the 12 months period : (1) A slow increase between 0-3 months, (2) an exponential increase between 4 and 7 months, and then (3) a «plateau» after 7 months.

-Parameters such as grazing and environmental conditions (T°C, S...) could explain the variability after 7months. Community successions

Conclusions

The pattern of early biogenic dissolution of carbonates by microborers

-

  The early stages of community successions over a year period -When Ostreobium quekettii, the main agent of carbonate dissolution settles down and becomes the dominant species -The dynamics of the biogenic dissolution process with a monthly resolution and the correlation with the microborer communities

Figure 1 :

 1 Figure 1: Map of the study site in New Caledonia.Figure 2: A-Coral block after 6 months of microbioersion; B-The 2 grids at 3-m depth

  Figure 1: Map of the study site in New Caledonia.Figure 2: A-Coral block after 6 months of microbioersion; B-The 2 grids at 3-m depth

Figure 4 :

 4 Figure 4: Relative abundance of microborers in pourcent according to the time of exposure in months.

Figure 5 :Figure 6 :

 56 Figure 5: Relative abundance of microborers in pourcent according to the distance from the block surface. See legend in the Fig.4

Figure 7 :

 7 Figure 7: Principal Component Analysis

-

Figure 3 :

 3 Figure 3: A-Phaeophila sp.; B-Ostreobium quekettii; C-Hyella sp.; D-Mastigocoleus testarum; E-Fungi or Plectonema terebrans. Scale bar = 40µm


Richard Eon
email: richard.eon@univ-rennes1.fr

Mihai Gradinaru
email: mihai.gradinaru@univ-rennes1.fr

Gaussian asymptotics for a non-linear Langevin type equation driven by an α-stable Lévy noise

Keywords: stable Lévy noise, non-linear Langevin type equation, Lévy driven stochastic differential equation, Brownian motion, exponential ergodic processes, Lyapunov function, convergence in probability, functional central limit theorem for martingales. MSC2010 Subject Classification Primary 60F17, secondary 60G52, 60J75, 60J65, 60H10, 60G44

.

Introduction

In many physical, engineering or financial mathematics models based on random perturbations, the usual construction is performed by using the standard white noise and studying the resulting diffusion process. The Gaussian feature and the continuity of paths of the Brownian motion are essential when choosing the tools used for this kind of situation. During the last fifteen years, the study of some particular phenomena, as discontinuous behaviour of paths or self-similarity in time scale, focuses on another type of random perturbation, mainly an α-stable Lévy noise. The resulting processes are called in physical literature anomalous (fractional) diffusions or Lévy flights.

In this paper, we consider the one-dimensional and non-linear Langevin type equation driven by an α-stable Lévy process. Let us denote by x ε t the one-dimensional process describing the position of a particle at time t ≥ 0, having the speed v ε t

x ε t = x 0 + t 0 v ε s ds, t ≥ 0, (1.1)
and such that v ε t is a small α-stable Lévy process in a potential U(x) := 2 β+1 |x| β+1 ,

dv ε t = εdℓ t - 1 2 U ′ (v ε t )dt, v ε 0 = v 0 , (1.2)
in other words v ε t verifies the following integral equation

v ε t = v 0 + εℓ t - t 0 sgn(v ε s )|v ε s | β ds, t ≥ 0. (1.3)
Here β > -1 and {ℓ t : t ≥ 0} is an α-stable Lévy process, α ∈ (0, 2]. If α ∈ (0, 2), the Lévy process is a pure jump process with càdlàg paths and the jump measure is given by ν(dz) = |z| -1-α 1 R\{0} (z)dz.

The 2-stable Lévy process is the standard Brownian motion {b t : t ≥ 0} which is continuous. In all cases, the process has the property of self-similarity which means that the processes {ℓ t : t ≥ 0} and {c -1 /α ℓ ct : t ≥ 0} have the same law, for any c > 0.

The case of a harmonic potential (β = 1, linear equation), when the speed is a Ornstein-Uhlenbeck process, was already considered by Hintze and Pavlyukevich [START_REF] Hintze | Small noise asymptotics and first passage times integrated Ornstein-Uhlenbeck process driven by α-stable Lévy process[END_REF]. The dynamic of the integrated Ornstein-Uhlenbeck process appears in some financial mathematics (volatility) models (see for instance Barndorff-Nielsen and Shephard [START_REF] Barndorff-Nielsen | Integrated OU processes and non-Gaussian OU-based stochastic volatility models[END_REF]) or in models in physics of plasma (see for instance Chechkin, Gonchar and Szydlowski [START_REF] Chechkin | Fractional kinetics for relaxation and superdiffusion in a magnetic field[END_REF]). In the paper by Hintze and Pavlyukevich, the authors study the asymptotic behaviour of the integrated Ornstein-Uhlenbeck and prove that this process converges weakly, as ε → 0, to the underlying α-stable Lévy process. In particular, when the driving process is a Brownian motion (α = 2), the asymptotic behaviour is Gaussian. In [START_REF] Hintze | Small noise asymptotics and first passage times integrated Ornstein-Uhlenbeck process driven by α-stable Lévy process[END_REF], asymptotics of the first exit time from an interval are deduced. Several physical papers pointed out that new interesting phenomena appear when one considers super-harmonic potentials (see for instance Metzler, Chechkin, Klafter [START_REF] Metzler | Levy statistics and anomalous transport: Levy flights and subdiffusion[END_REF]).

Our goal is to answer to the same question in the situation of a super-harmonic potential (non-linear equation): what is the asymptotic behaviour of the position process x ε t , as ε → 0 ? On the one hand, the non-linear case introduces new technical difficulties, mainly since the solution is no longer explicit. Indeed, this fact was essential to prove weak convergence in the linear case. On the other hand, different conditions on the two parameters α and β will generate different asymptotics for the position process. The intuition suggests that the big jumps should be compensated by a strong negative drift (for instance if β > 1) and small jumps should have some regularising effect. In the present paper, we answer to the question by showing that for α and β in some unbounded domain, the position process x ε t will behave as a Brownian motion when ε goes to 0. In other words, we get Gaussian asymptotic behaviour even if α is smaller than 2, provided that β is not very small, more precisely if β + α 2 > 2. When α and β are in somehow "small" the previous heuristic fails. To get convergence toward a stable process, one needs to change the approach and other technical difficulties appear (this case will be presented in a forthcoming paper, see [START_REF] Gradinaru | Small noise asymptotics of solutions of the Langevin equation with non-linear damping subject α-stable Lévy forcing[END_REF]).

To state the main result of the present paper, we will perform some scaling transformations. Without loss of generality, we can assume that the initial position is the origin x 0 = 0. Moreover we will assume that the initial speed vanishes v 0 = 0, contrary to the linear case. By using the self-similarity, it is clear that the process {L ε t := εℓ ε -α t : t ≥ 0} is also an α-stable process. Let us denote, for t ≥ 0,

X ε t := x ε ε -α t and V ε t := v ε ε -α t (1.4)
satisfying, respectively,

X ε t = 1 ε α t 0 V ε s ds and V ε t = L ε t - 1 ε α t 0 sgn(V ε s )|V ε s | β ds. (1.5) Set Ľε t := L ε t ε α 2 /(α+β-1) ε α /(α+β-1) = ℓ t ε -α(β-1) /(α+β-1) ε (β-1) /(α+β-1)
and

V ε t := V ε t ε α 2 /(α+β-1) ε α /(α+β-1) , (1.6)
provided that α + β -1 > 0. Again by self-similarity, Ľε is distributed as an α-stable Lévy process and we have

X ε t = ε α(2-β) α+β-1 tε -α 2 /(α+β-1) 0 V ε s ds and V ε t = Ľε t - t 0 sgn( V ε s )| V ε s | β ds. (1.7)
Let us note that if α = 2, all previous computations hold true with ℓ, L or Ľ replaced respectively by b, B or B a standard Brownian motion. Our main result is the following:

Theorem 1.1. Assume that 0 < α ≤ 2 and β + α 2 > 2.
There exists a positive constant κ α,β such that the process ε

α(β+ α /2-2) α+β-1 x ε ε -α t : t ≥ 0 = ε α(β+ α /2-2) α+β-1 X ε t : t ≥ 0 (1.8)
converges in distribution toward a Brownian motion process with variance κ α,β , as ε → 0. Moreover, if α = 2, the result is true even for -1 < β ≤ 1.

Remark 1.2.

1. If the driving noise is the Brownian motion α = 2, the convergence in the theorem holds in the space of continuous functions C([0, ∞)) endowed by the uniform topology. If the driving noise is α-stable with α ∈ (0, 2), the convergence in Theorem 1.1 holds in the Skorokhod space of càdlàg functions D([0, ∞)) endowed by J 1 (or simple) Skorokhod topology. Our situation is simpler than in [START_REF] Hintze | Small noise asymptotics and first passage times integrated Ornstein-Uhlenbeck process driven by α-stable Lévy process[END_REF] since the limit is a continuous paths process.

2. If the driving noise is the Brownian motion α = 2, the normalizing factor is ε 2(β-1) /(β+1) and it behaves differently following with the position of β with respect to 1 (if β = 1, the position process X ε converges in distribution toward a standard Brownian motion, see also Remark 2.4 below).

3. The case when β + α 2 = 2 should be considered as a critical for some phase transition from Gaussian to stable case. It should be reasonable that there is some continuity but the proof seems more delicate since natural integrability conditions are not fulfilled.

4. The constant κ α,β has an integral representation (see (2.9) and (3.28) ) and it is more explicit when the driving noise is the Brownian motion (α = 2).

5. Again, as an application, one can find asymptotics of the first exit time from an interval : Corollary 2.1, p. 269, in [START_REF] Hintze | Small noise asymptotics and first passage times integrated Ornstein-Uhlenbeck process driven by α-stable Lévy process[END_REF] applies.

Let us explain the method of proof and the organisation of the paper. To simplify the notations, all along the paper we will denote

θ = θ α,β := α α + β -1 ∈ (0, 1). (1.9)
It is a simple observation that

ε θ(β+ α 2 -2) X ε t = ε αθ 2 tε -αθ 0 V ε s ds , hence Theorem 1.
1 is in fact a second order type ergodic theorem. By using stochastic calculus, we will show that the latter quantity is a sum of a square integrable martingale and a term which tends in probability toward 0 as ε → 0. The result is then obtained by using the functional central limit theorem for martingales and the continuous-mapping theorem.

In the next section, we consider the case when the driving noise is the Brownian motion: in this case computations are performed by using Itô's calculus and are more explicit. For instance, the constant κ 2,β can be written in terms of the scale function and the speed measure. In Section 3, we follow the same structure of the proof for a pure jump driving noise. Computations are more technical and new ideas are needed: for instance, we need to find and use a Lyapunov function which allows to perform the same reasoning by using Lévy-Itô's calculus. We collect in the Appendix the technical proofs.

Brownian motion driving noise

Recall that in this case, {b t : t ≥ 0} is a standard one-dimensional Brownian motion, β > -1 and we set

Bε t := B ε t ε 4 /(β+1) ε 2 /(β+1) = b t ε 2(1-β) /β+1 ε (β-1) /(β+1) , and
V ε t := V ε t ε 4 /(β+1) ε 2 /(β+1) . (2.1)
Recall also that

X ε t = ε 2(2-β) (β+1) tε -4 /(β+1) 0 V ε s ds and V ε t = Bε t - t 0 sgn( V ε s )| V ε s | β ds . (2.2)
Bε is distributed as a standard Brownian motion so, to simplify the notation, we will suppress the index ε, as well as for V ε .

2.1 The speed process V ε x 0 e -c β (y) dy and m β (dx) := 2e c β (x) dx, where c β (x) := -

2 β + 1 |x| β+1 . (2.3) Since ∞ 0 m β ([0, x]
)e -c β (x) dx = ∞, by Theorem 52.1, p. 297 in [START_REF] Rogers | Diffusions, Markov Processes, and Martingales[END_REF], the pathwise uniqueness holds to (2.2 2 ). Finally, there exists a pathwise unique strong solution V to the equation (2.2 2 ).

Convergence in probability

The main result of this section is the following Proposition 2.1. As ε → 0, {V ε t : t ≥ 0} converges to 0 in probability uniformly on each compact time interval.

By (2.1 2 ), the relation between V ε and V is V ε t = ε 2 /(β+1)
Vt ε -4 /(β+1) . To prove Proposition 2.1, we need a preliminary result: Lemma 2.2.

Fix p ≥ 2.

There exists a positive constant C p,β such that, for any t ≥ 0,

E | Vt | p ≤ C p,β t.
(2.4)

2. Fix p ≥ 4 and T > 0. There exists a positive constant C ′ p,β such that

E sup 0≤t≤T | Vt ε -4 /(β+1) | p ≤ C ′ p,β T 2 ε -8 /(β+1) . (2.5)
Proof of Proposition 2.1. Taking p > 4 in Lemma 2.2, we deduce that for any T > 0, as ε → 0, sup 0≤t≤T |V ε t | converges to 0 in L p (Ω), and the conclusion follows.

Proof of Lemma 2.2. By using Itô's formula and the equation (2.2 2 ), we can write

| Vt | p = p t 0 sgn( Vs )| Vs | p-1 d Bs + p t 0 (1/2)(p -1)| Vs | p-2 -| Vs | p-1+β ds Since β > -1, there exists a constant C p,β > 0 such that p (1/2)(p -1)|x| p-2 -|x| p-1+β ≤ C p,β , ∀x ∈ R.
We deduce that

| Vt | p ≤ C p,β t + p t 0 sgn( Vs )| Vs | p-1 d Bs (2.6)
We show that

t 0 sgn( Vs )| Vs | p-1 d Bs is a martingale. Fix T > 0, for all t ≤ T , since (a + b) 2 ≤ 2(a 2 + b 2 ) and |x| 2p-2 ≤ 1 + |x| 2p
, by using the Burkholder-Davis-Gundy inequality, we can see that there exists a positive constant

C ′ 1 such that E sup 0≤u≤t | Vu | p 2 ≤ 2C 2 p,β T 2 + 2p 2 E sup 0≤u≤t u 0 sgn( Vs )| Vs | p-1 d Bs 2 ≤ 2C 2 p,β T 2 + 2p 2 C ′ 1 t 0 E | Vs | 2p-2 ds ≤ 2p 2 C ′ 1 T + 2C 2 p,β T 2 + 2p 2 C ′ 1 t 0 E | Vs | 2p ds ≤ 2p 2 C ′ 1 T + 2C 2 p,β T 2 + 2p 2 C ′ 1 t 0 E sup 0≤u≤s | Vu | p 2 ds.
By Gronwall's lemma, we get, for all t ≤ T ,

E sup 0≤u≤t | Vu | p 2 ≤ (2p 2 C ′ 1 T + 2C 2 p,β T 2 )e 2p 2 C ′ 2 T . Hence t 0 sgn( Vs )| Vs | p-1 d
Bs is a martingale and we get (2.4) by taking expectation in (2.6). It is now possible to improve the inequality (2.4). Indeed, it can be used to see that

E sup 0≤t≤T | Vt ε -4 /(β+1) | p = E sup 0≤t≤T | Vt ε -4 /(β+1) | p /2 2 ≤ p 2 2 E sup 0≤t≤T t ε -4 /(β+1) 0 | Vs | p /2-1 d Bs 2 + 2 C 2 p /2,β T 2 ε -8 /(β+1) ≤ p 2 2 C ′ 1 T ε -4 /(β+1) 0 E | Vs | p-2 ds + 2C 2 p /2,β T 2 ε -8 /(β+1) ≤ p 2 4 C ′ 1 C p-2,β T 2 ε -8 /(β+1) + 2C 2 p /2,β T 2 ε -8 /(β+1) . Therefore (2.5) follows taking C ′ p,β := p 2 4 C ′ 1 C p-2,β + 2C 2 p /2,β .

Ergodicity

Recall that we introduced the scale function and the speed measure in (2.3). Since s β (∞) = ∞ and m β (R) < ∞, the diffusion V is regular (see for instance (45.2) and (46.10) pp. 272-275 in [START_REF] Rogers | Diffusions, Markov Processes, and Martingales[END_REF]) and is a recurrent and ergodic process with the invariant measure m β (see for instance Theorem 53.1, p. 300 in [START_REF] Rogers | Diffusions, Markov Processes, and Martingales[END_REF]). Therefore, for all

f ∈ L 1 (m β ), lim T →∞ 1 T T 0 f ( Vs )ds = 1 m β (R) R f (x)m β (dx), almost surely. (2.7)

The position process X ε

We recall that the infinitesimal generator of V is given by L

2,β = 1 2 d 2 dx 2 -sgn(x)|x| β d dx . Introduce g β (x) := x 0 +∞ y -2ze c β (z) dz e -c β (y) dy, x ∈ R, (2.8)
and note that (L 2,β g β )(x) = x, for all x ∈ R. Set

κ 2,β := 1 m β (R) R g ′ β (x) 2 m β (dx) = - 2 m β (R) R xg ′ β (x)m β (dx) (2.9)
(the latter equality is obtained by integrating by parts). We can give now the proof of the main result.

Proof of Theorem 1.1 for the case α = 2. By applying Itô's formula, we can see that

g β ( Vt ) = t 0 g ′ β ( Vs )d Bs + t 0 (L 2,β g β )( Vs )ds = t 0 g ′ β ( Vs )d Bs + t 0
Vs ds, and therefore

ε 2(β-1) /(β+1) X ε t = -ε 2 /(β+1) t ε -4 /(β+1) 0 g ′ β ( Vs )d Bs + ε 2 /(β+1) g β ( Vt ε -4 /(β+1) ) .
The continuous local martingale

M ε t := -ε 2 /(β+1) t ε -4 /(β+1) 0 g ′ β ( Vs )d Bs
has the quadratic variation

M ε t = ε 4 /(β+1) t ε -4 /(β+1) 0 g ′ β ( Vs ) 2 ds .
As a consequence of (2.7), for all t, M ε t → κ 2,β t a.s., as ε → 0, where κ 2,β is given by (2.9), and it is the constant in the statement of Theorem 1.1. Indeed, using Whitt's theorem (see Theorem 2.1(ii), p. 270 in [START_REF] Whitt | Proofs of the martingale FCLT[END_REF]), we deduce that M ε converges in distribution (as a process) toward κ 1 /2 2,β B. We will prove that the second term in the right hand side converges in probability uniformly on compact sets to 0. At this level, we need a technical result: Lemma 2.3. There exist two positive constants µ β , ν β such that for all x ∈ R,

|g β (x)| ≤ µ β |x| (2-β)∨1 + ν β .
(2.10)

We postpone the proof of the lemma to the Appendix and finish the proof of Theorem 1.1 in the case α = 2. By using the classical inequality

(a + b) 2m ≤ 2 2m-1 (a 2m + b 2m ), (m ≥ 1 integer), we obtain |ε 2 /(β+1) g β ( Vt ε -4 /(β+1) )| 2m ≤ 2 2m-1 µ 2m β ε (4m) /(β+1) | Vt ε -4 /(β+1) | 2m((2-β)∨1) + 2 2m-1 ν 2m β ε (4m) /(β+1) .
By choosing the integer m ≥ 1 such that p := 2m((2 -β) ∨ 1) > 4, we can use Lemma 2.2 and we get for all T > 0, lim

ε→0 E sup 0≤t≤T ε 4m /(β+1) g 2m β ( Vt ε -4 /(β+1) ) = 0.
We finish the proof of the theorem by employing the joint convergence theorem and the simple continuousmapping theorem (Theorem 11.4.5 p. 379 and Theorem 3.4.1, p. 85 in [START_REF] Whitt | Stochastic-process limits: an introduction to stochastic process limits and their applications to queues[END_REF]) on the space of continuous functions C([0, ∞)) endowed with the uniform topology.

α-stable driving noise

Recall that Ľε is distributed as a α-stable Lévy process (see (1.6 1 )) so, to simplify the notation, we will suppress the index ε, as well as for V ε (see (1.7 2 )).

3.1

The speed process V ε

Existence and uniqueness

If β > 1, the drift coefficient in (1.7 2 ) is a locally Lipschitz function and it is well known (see, for instance, Theorem 6.2.11, p. 376 in [START_REF] Applebaum | Lévy Processes and Stochastic Calculus[END_REF]) that there exists a locally pathwise unique strong solution V for equation (1.7 2 ) defined up to an explosion random time τ . Moreover it can be proved that τ = ∞ a.s. hence V is a global solution. For the sake of completeness, we give the proof of the latter statement (see also [START_REF] Samorodnitsky | Tails of solutions of certain nonlinear stochastic differential equations driven by heavy tailed Levy motions[END_REF], p. 73) by following some ideas in [START_REF] Fournier | On pathwise uniqueness for stochastic differential equations driven by stable Lévy processes[END_REF], pp. 156-157.

Lemma 3.1. For any α ∈ (0, 2), any δ ∈ (0, α) and any T > 0, E sup

t∈[0,T ] | Vt | δ < ∞.
Proof. By Itô-Lévy's decomposition, there exists a Poisson process N and its compensated Ñ such that

Ľt = t 0 |z|≤1 z Ñ (ds, dz) + t 0 |z|>1
zN (ds, dz)

and so the equation satisfied by V , starting from any x ∈ R, is

Vt = x + t 0 |z|≤1 z Ñ (ds, dz) + t 0 |z|>1 zN (ds, dz) - t 0 sgn( Vs )| Vs | β ds. (3.1)
Consider another equation where we skip the (third) big jumps term

Y t = x + t 0 |z|≤1 z Ñ (ds, dz) - t 0 sgn(Y s )|Y s | β ds, (3.2)
and apply Itô-Lévy's formula. We obtain

Y 2 t = x 2 + M t + t 0 |z|≤1 [(Y s + z) 2 -Y 2 s -2zY s ]ν(dz)ds -2 t 0 |Y s | β+1 ds = x 2 + Mt + t |z|≤1 z 2 ν(dz) -2 t 0 |Y s | β+1 ds, (3.3)
where the local martingale term is given by

Mt := t 0 |z|≤1 [(Y s + z) 2 -Y 2 s ] Ñ (ds, dz).
The constants depending only on α and β will be denoted c α or k α,β and could change from line to line in this proof. Let us write the third term in (3.3) as c α t and note that lim |y|→∞ (c α -2|y| β+1 ) = -∞.

We deduce that there exists a positive constant k α,β such that, for all t ≥ 0,

Y 2 t ≤ x 2 + k α,β t + Mt . (3.4)
By Kunita's inequality (see for instance [START_REF] Applebaum | Lévy Processes and Stochastic Calculus[END_REF], p. 265) and by our convention on constants,

E sup 0≤s≤t Y 2 s ≤ x 2 + k α,β t + c α t 0 |z|≤1 E (Y s + z) 2 -Y 2 s 2 ν(dz)ds ≤ x 2 + k α,β t + c α t 0 E[Y 2 s ]ds ≤ x 2 + k α,β t + c α t 0 E sup 0≤u≤s Y 2 u ds. (3.5)
Applying Gronwall's inequality, we get

E sup 0≤u≤t Y 2 u ≤ (x 2 + k α,β t)e cαt . (3.6)
Hence M is a (true) square integrable martingale and, taking expectation in (3.4), we obtain

E[Y 2 t ] ≤ x 2 + k α,β t. (3.7)
Re-injecting this in (3.5), we get that, for any T > 0, there exists a positive constant C α,β,T depending also on T , such that

E sup t∈[0,T ] Y 2 t ≤ C α,β,T (1 + x 2 ). (3.8)
We proceed with the study of (3.1). Denote by 0 < T 1 < T 2 < . . . the jumping times of N restricted to {|z| > 1}, and by (Z n ) the jumps which are i.i.d. random variables with distribution λ -1 1 {|z|>1} ν(dz), where λ := {|z|>1} ν(dz). Therefore t 0 |z|>1 zN (ds, dz) = n∈N Z n 1 {Tn≤t} and (3.1) coincides with (3.2) on each time interval (T n , T n+1 ). Since V is a solution of (3.2) on [0, T 1 ), by using (3.8),

E sup t∈[0,T1∧T ) V 2 t G ≤ C α,β,T (1 + x 2 ), with G := σ(T 1 , T 2 , . . . ).
By using the Jensen inequality and the classical inequality

(a + b) δ ≤ c δ (a δ + b δ ), we obtain E sup t∈[0,T1∧T ) V δ t G ≤ C α,β,δ,T (1 + |x| δ ). Furthermore, VT1 = VT1-+ Z 1 , hence | VT1 | δ ≤ c δ | VT1-| δ + |Z 1 | δ . Since δ < α, E(|Z 1 | δ ) < ∞.
Consequently, we have

E sup t∈[0,T1∧T ] V δ t G ≤ C α,β,δ,T (1 + |x| δ ).
Using the same inequality on (T n , T n+1 ), but starting from VTn , we can show that, for any n ≥ 0,

u n := E sup t∈[Tn∧T,Tn+1∧T ] V δ t G ≤ C ′ T,δ (1 + E[| VTn | δ |G]) (with T 0 = 0).
Then the sequence

(u n ) n≥0 satisfies u 0 ≤ C ′ T,δ (1 + |x| δ ) and u n+1 ≤ C ′ T,δ (1 + u n ), implying that there exists C T,δ,x > 1 such that u n ≤ C n+1
T,δ,x . We deduce that

E sup t∈[0,Tn∧T ] V δ t G ≤ u 0 + • • • + u n-1 ≤ C n+1 T,δ,x C T,δ,x -1 .
Finally,

E sup t∈[0,T ] V δ t ≤ n≥0 E 1 Tn<T <Tn+1 E sup t∈[0,Tn∧T ] V δ t G ≤ 1 C T,δ,x -1 n≥0 C n+2 T,δ,x (λT ) n n! e -λT < ∞.

Ergodicity

The ergodic feature of the process V is a consequence of Proposition 0.1, p. 604 in [START_REF] Kulik | Exponential ergodicity of the solutions to SDE's with a jump noise[END_REF]. Indeed, provided that β > 1, the drift coefficient b(x) = -sgn(x)|x| β and the jump measure ν(dz) = |z| -1-α 1 R\{0} dz clearly satisfy the conditions in the cited result. Hence V is an exponential ergodic (and Harris recurrent) process having an unique invariant distribution, denoted by m α,β , which satisfies

m α,β ([x, +∞)) ∼ |x|→∞ +∞ |x| ν([u, +∞)) -b(x) du = C |x| α+β-1
(3.9)

as follows from Theorem 4.1, p. 92 in [START_REF] Samorodnitsky | Tails of solutions of certain nonlinear stochastic differential equations driven by heavy tailed Levy motions[END_REF]. Clearly, the identity function, id ∈ L 1 (m α,β ) under the hypothesis of Theorem 1.1, β + α 2 -2 > 0. By the classical ergodic theorem, for all f ∈ L 1 (m α,β ), lim

T →∞ 1 T T 0 f ( Vs )ds = R f (x)m α,β (dx), a.s. ( 3.10)
Recall that we are interested on the behaviour as ε → 0 of

ε θ(β+ α 2 -2) x ε ε -α t = ε αθ 2 tε -αθ 0 Vs ds, (3.11)
where θ is given by (1.9). In other words, we are studying a large time behaviour of a functional of V , hence it is quite natural to perform the study in steady state. In fact, we can prove the following lemma (see also [START_REF] Bhattacharya | On the functional central limit theorem and the law of iterated logarithm for Markov processes[END_REF], Theorem 2.6, p. 194):

Lemma 3.2. Suppose that β + α 2 -2 > 0. Assume that the process ε αθ /2 tε -αθ 0 Vs ds : t ≥ 0 converges, as ε → 0, in distribution toward a Brownian motion, provided that V is starting with m α,β as an initial distribution. Then the same process converges in distribution toward a Brownian motion when V0 = 0.

Proof. In this proof we will denote the process in (3.11) by Z ε,0 (t), and for ∆ ≥ 0,

Z ε,∆ (t) := ε αθ 2 tε -αθ +∆ ∆ Vs ds.
First, let us prove that Z ε,∆ (•) converges in distribution, as ∆ → ∞ and ε → 0, toward a Brownian motion, when V0 = 0. Denoting by µ ∆ the distribution of V∆ , for each bounded continuous real function ψ on C([0, +∞)), by the Markov property, we have

E ψ(Z ε,∆ (•)) | V0 = 0 = E ψ(Z ε,0 (•)) | V0 ∼ µ ∆ .
We can write, for all ε > 0,

E ψ(Z ε,0 (•)) | V0 ∼ µ ∆ -E ψ(Z ε,0 (•)) | V0 ∼ m α,β = R E ψ(Z ε,0 (•)) | V0 = y µ ∆ (dy)-m α,β (dy) ≤ ||ψ|| ∞ R p(∆, 0, dy) -m α,β (dy) ≤ ||ψ|| ∞ ||p(∆, 0, dy) -m α,β (dy)|| TV ,
where p(t, x, dy) = P x ( Vt ∈ dy) is the transition kernel of V (and therefore p(∆, 0, dy) = µ ∆ (dy)) , and • TV is the norm in total variation. Since V is (exponentially) ergodic, we get that

lim ∆→∞ E ψ(Z ε,0 (•)) | V0 ∼ µ ∆ -E ψ(Z ε,0 (•)) | V0 ∼ m α,β = 0, uniformly in ε.
Second, by choosing ∆ = ∆(ε) = ε -αθ /4 we obtain

sup t≥0 Z ε,∆(ε) (t) -ε αθ 2 tε -αθ +∆(ε) 0 Vs ds ≤ ε αθ 2 ∆(ε) 0 | Vs |ds = ε αθ 4 1 ∆(ε) ∆(ε) 0 | Vs |ds.
The right hand side term of the latter inequality tends to 0 almost surely, by using the ergodicity (3.10). Therefore ε αθ /2 •ε -αθ +∆(ε) 0 Vs ds converges in distribution, as ε → 0, toward a Brownian motion when V0 = 0. Clearly, lim ε→0 (t -∆(ε)ε αθ ) = t, and applying Lemma p. 151 in [START_REF] Billingsley | Convergence of Probability Measures[END_REF] (a consequence of the continuous mapping theorem for the composition function), we can conclude.

In the sequel, we will always assume that V is starting with m α,β as an initial distribution. Let us recall that the infinitesimal generator of V is given by

(L α,β g)(x) = -sgn(x)|x| β g ′ (x) + R g(x + y) -g(x) -yg ′ (x)1 |y|≤1 ν(dy) , (3.12)
with the domain D L α,β . Also denote (T t ) t≥0 the semi-group associated to the operator L α,β (or to the process V ). We collect in the following lemma some useful properties of the process V .

Lemma 3.3.

1. The domain D L α,β contains the space of bounded twice differentiable functions C 2 b (R). 2. For all p ≥ 1, T t is a contraction semi-group on L p (m α,β ) and for each f ∈ L p (m α,β ),

lim t→0 T t f -f L p (m α,β ) = 0. (3.13)
Proof.

To prove the first point, we fix f ∈ C 2 b (R) and we show that

(L α,β f )(x) < ∞. First, -sgn(x)|x| β f ′ (x) is well defined for all x ∈ R. Since f ∈ C 2 b (R), ∀y ∈ [-1, 1], f (x + y) -f (x) -yf ′ (x) ≤ y 2 sup z∈[x-1,x+1] |f ′′ (z)| < ∞,
and we find

|y|≤1 f (x + y) -f (x) -yf ′ (x) ν(dy) ≤ sup z∈[x-1,x+1] |f ′′ (z)| |y|≤1 y 2 ν(dy) < ∞.
Since f is bounded, we have

|y|>1 f (x + y) -f (x) ν(dy) ≤ 2||f || ∞ |y|>1 ν(dy) < ∞ , hence f ∈ D L α,β .
We proceed with the proof of the second point. Fix f ∈ L p (m α,β ) and we show first that

T t f L p (m α,β ) ≤ f L p (m α,β ) . Since T t f p L p (m α,β ) = R |T t f (x)| p m α,β (dx) = R |E x (f ( Vt ))| p m α,β (dx),
by the Jensen inequality (p ≥ 1), we get

T t f p p ≤ R E x (|f ( Vt )| p )m α,β (dx) = E m α,β (|f ( Vt )| p ) = ||f || p L p (m α,β ) .
Finally, we prove (3.13). Since

C 2 b (R) is dense in L p (m α,β ), there exists f η ∈ C 2 b (R) such that ||f -f η || L p (m α,β ) ≤ η /3.
Since T t is a contraction semi-group and m α,β is a probability measure, we get

T t f -f L p (m α,β ) ≤ 2 f -f η L p (m α,β ) + T t f η -f η ∞ ≤ (2η) /3 + T t f η -f η ∞ .
Since T t is a Feller semi-group (see for instance, [START_REF] Applebaum | Lévy Processes and Stochastic Calculus[END_REF], p. 151), for t small enough, we have T t f η -f η ∞ ≤ η /3 and we deduce (3.13). The proof is complete.

Convergence in probability

The main result of this section concerns the behaviour of the speed process which is described by using a Lyapunov function.

Proposition 3.4. Suppose that β + α 2 > 2 and let p and γ such that Then, as ε → 0, {ε αθ /2 h p,γ (ε -θ V ε t ) : t ≥ 0} converges to 0 in probability uniformly on each compact time interval. More precisely, there exists q > 2 such that, for any fixed T > 0,

p > 1, pγ > 2, 2 -β < γ < α 2 . ( 3
lim ε→0 E sup t∈[0,T ] ε αθ 2 h p,γ ε -θ V ε t q = E sup t∈[0,T ] ε αθ 2 h p,γ Vt ε -αθ q = 0. (3.16)
In order to prove this result, we need the following lemma whose proof is postponed to the Appendix.

Lemma 3.5.

1. If pγ > 2, h p,γ is a twice differentiable function and there exists a positive constant k such that for all

(x, y) ∈ R 2 , -if |x| < 1 then |h p,γ (x + y) -h p,γ (x)| ≤ k(|y|1 {|y|≤1} + |y| γ 1 {|y|>1} ); -if |x| ≥ 1 then |h p,γ (x + y) -h p,γ (x)| ≤ k(|y||x| γ-1 1 {|y|≤i(x)} + |y| γ 1 {i(x)<|y|} ),
where i(x) := (2|x| pγ + 1) 1 /pγ -|x|.

2. Assume that pγ > 2 and 2 -β < γ < α. There exist a continuous function f p,α,β,γ , a compact set K and a constant d (depending only on p, α, β, γ) such that

∀x ∈ R, f p,α,β,γ (x) ≥ 1 + |x|, f p,α,β,γ (x) ∼ |x|→∞ γ|x| γ+β-1 , (3.17) and (L α,β h p,γ )(x) ≤ -f p,α,β,γ (x) + d1 K . (3.18)
Proof of Proposition 3.4. By (1.6 2 ), we can write

ε αθ 2 h p,γ V ε t ε θ = ε αθ 2 h p,γ Vt ε -αθ (3.19)
and the first equality in (3.16) is clear. Since 2 -β < α 2 and β > 1, we can fix q such that 2 p ∨ (2 -β) < γ < 2γ < qγ < α and 2 < q < β-1 α + 2. By noting that h p,γ (x) q = h p q ,qγ (x), we can write

E sup t∈[0,T ] ε αθ 2 h p,γ Vt ε -αθ q = ε q αθ 2 E sup t∈[0,T ] h p q ,qγ Vt ε -αθ .
Employing Itô's formula with h p q ,qγ , we get

h p q ,qγ ( Vt ) -h p q ,qγ ( V0 ) = R t + t 0 (L α,β h p q ,qγ )( Vs )ds, (3.20)
where

R t := t 0 R
h p q ,qγ ( Vs + y) -h p q ,qγ ( Vs ) Ñ (dy, ds).

By Lemma 3.5 applied to the function h p q ,qγ , we see that there exists c > 0 such that, for all t ∈ [0, T ],

t 0 (L α,β h p q ,qγ )( Vs )ds ≤ ct.
Moreover, let us note that h p q ,qγ is continuous and that h p q ,qγ (x) ∼ |x| qγ , as |x| → ∞. Hence, by the choice of q, we have h p q ,qγ ∈ L 1 (m α,β ). Replacing in (3.20), we obtain

ε q αθ 2 E sup t∈[0,T ] h p q ,qγ Vt ε -αθ ≤ ε q αθ 2 h p q ,qγ L 1 (m α,β ) + ε (q-2) αθ 2 cT + ε q αθ 2 E sup t∈[0,T ] R t ε -αθ .
Since q > 2, the first and the second term converge toward 0. For the last term, we use Kunita's first inequality (see for instance [START_REF] Applebaum | Lévy Processes and Stochastic Calculus[END_REF], p. 265): since V0 ∼ m α,β , then for all t, Vt ∼ m α,β and there exists a positive constant C such that

E sup t∈[0,T ] R t ε -αθ ≤ E sup t∈[0,T ] R 2 t ε -αθ 1 /2 ≤ C √ T ε -αθ 2 R 2
h p q ,qγ (x + y) -h p q ,qγ (x) 2 ν(dy)m α,β (dx).

It is sufficient to show that

R 2 h p q ,qγ (x + y) -h p q ,qγ (x) 2 ν(dy)m α,β (dx) < ∞. (3.21)
This fact is obtained by using Lemma 3.5. If |x| ≥ 1,

h p q ,qγ (x + y) -h p q ,qγ (x) 2 ≤ k 2 (|y| 2 |x| 2qγ-2 1 {|y|≤i(x)} + |y| 2qγ 1 {i(x)<|y|} ), hence R h p q ,qγ (x + y) -h p q ,qγ (x) 2 ν(dy) = O(|x| 2qγ-α ), as |x| → +∞,
and, since q < β-1 α + 2, we get (3.21). If |x| < 1,

h p q ,qγ (x + y) -h p q ,qγ (x) 2 ≤ k 2 (|y| 2 1 {|y|≤1} + |y| 2qγ 1 {|y|>1} )
and R 2 h p q ,qγ (x + y) -h p q ,qγ (x) 2 ν(dy) is finite independently of x. Since m α,β is a probability measure, (3.21) is verified again. The proof is complete except for Lemma 3.5.

The position process X ε

We are ready to prove our main result concerning the behaviour of the position process. Recall that, thanks to Lemma 3.2, we assume that V is starting with m α,β as an initial distribution.

Proof of Theorem 1.1 for the case α ∈ (0, 2). Thanks to (3.17), Theorem 3.2, p. 924 in [START_REF] Glynn | A Liapounov bound for solutions of the Poisson equation[END_REF] applies and we deduce that the Poisson equation Lg = id admits a solution ĝ satisfying |ĝ| ≤ c(h p,γ + 1), with c a positive constant. Applying Itô-Levy's formula with ĝ, we get

ĝ( Vt ) -ĝ( V0 ) = t 0 Vs ds + M t , (3.22)
where

M t := t 0 R [ĝ(z + Vs ) -ĝ( Vs )] Ñ (ds, dz). (3.23)
Step 1) We prove that M given by the latter formula is a square integrable true martingale. On one hand we have

E[ĝ( Vt ) 2 ] = E[ĝ(V 0 ) 2 ] = R ĝ(x) 2 m α,β (dx) < ∞.
Indeed, recall that h 2 p,γ is continuous and it behaves as |x| 2γ in the neighbourhood of the infinity. Recalling that γ was chosen such that 4 p ∨ (4 -2β) < 2γ < α, by using (3.9), we see that

R h p,γ (x) 2 m α,β (dx) < ∞.
On the other hand, we can write Applying again Theorem 3.2, p. 924 in [START_REF] Glynn | A Liapounov bound for solutions of the Poisson equation[END_REF], we deduce that the Poisson equation L α,β g = |id| admits a solution g satisfying |g| ≤ c ′ (h p,γ + 1) with c ′ a positive constant. Moreover

s 0 T u |id|( V0 )du = T s g( V0 ) -g( V0 ).
Replacing in the latter inequality

E t 0 Vs ds 2 ≤ 2 t 0 E | V0 ||T s g( V0 ) -g( V0 )| ds = 2 t 0 ds R |x||T s g(x) -g(x)|m α,β (dx).
At this level, we need to apply the Hölder inequality to conclude that

E t 0 Vs ds 2 < ∞. (3.24) First, if β < 2 then we choose γ close enough to 2 -β such that g ∈ L (3-β) /(2-β) (m α,β ). Since 3-β 2-β > 1, using the second part of Lemma 3.3, we get T s g -g L (3-β) /(2-β) (m α,β ) ≤ 2 g L (3-β) /(2-β) (m α,β ) .
By the Hölder inequality and the fact that |id| ∈ L 3-β (m α,β ), we get (3.24). Second, if β ≥ 2, we choose γ < 1 close enough to 0 such that |id| ∈ L 1 /(1-γ) (m α,β ). Since g ∈ L 1 /γ (m α,β ), using again Lemma 3.3, we get

T t g -g L 1 /γ (m α,β ) ≤ 2 g L 1 /γ (m α,β ) . Since |id| ∈ L 1 /(1-γ) (m α,β
), we can apply the Hölder inequality and get (3.24) again. We conclude that M given by (3.23) is a square integrable true martingale. Moreover, we can compute its quadratic variation

M t = t 0 R [ĝ(y + Vs ) -ĝ( Vs )] 2 ν(dy)ds, (3.25)
hence

E[ M t ] = t R 2 [ĝ(x + y) -ĝ(x)] 2 ν(dy)m α,β (dx) < ∞. (3.26)
Step 2) Performing a simple time change in (3.22), we see that the process in (1.8) can be written

ε θ(β+ α 2 -2) X ε t = ε αθ 2 ĝ Vt ε -αθ -ĝ( V0 ) -ε αθ 2 M t ε -αθ . (3.27)
In this step, we show that the martingale term on the right hand side of the latter equality converges to a Brownian motion by using Whitt's theorem (see Theorem 2.1 (ii) in [START_REF] Whitt | Proofs of the martingale FCLT[END_REF], pp. 270-271). We need to verify the hypotheses of this result. In order, since the function

x → R [ĝ(x + y) -ĝ(x)] 2 ν(dy) ∈ L 1 (m α,β ),
by using (3.25) and the ergodic theorem (3.10), we deduce that

lim ε→0 ε αθ 2 M • ε -αθ t = lim ε→0 ε αθ 2 tε -αθ 0 R [ĝ(y + Vs ) -ĝ( Vs )] 2 ν(dy)ds = t R 2
[ĝ(x + y) -ĝ(x)] 2 ν(dy)m α,β (dx).

The condition [START_REF] Fournier | On pathwise uniqueness for stochastic differential equations driven by stable Lévy processes[END_REF] in [START_REF] Whitt | Proofs of the martingale FCLT[END_REF], p. 271 is fulfilled. Again by (3.25), we see that M has no jump, hence the condition (4) in [START_REF] Whitt | Proofs of the martingale FCLT[END_REF], p. 270 is trivial. Let us note also that, by (3.22), the jumps of the martingale M t are J(M t ) := ĝ( Vt ) -ĝ( Vt-). Therefore we deduce that the jumps of the martingale term on the right hand side of (3.27) are

J ε αθ 2 M t ε -αθ := ε αθ 2 ĝ Vε -αθ t -ĝ Vε -αθ t-≤ c ε αθ 2 h p,γ Vε -αθ t + h p,γ Vε -αθ t-+ 2 ≤ 2c ε αθ 2 sup t∈[0,T ] h p,γ ε -θ V ε t + 1 ,
by using the fact that |ĝ| ≤ c(h p,γ + 1) and (3.19). By Proposition 3.4,

lim ε→0 E sup t∈[0,T ] J ε αθ 2 M t ε -αθ 2 = 0.
Therefore we can apply Whitt's theorem to deduce that ε (αθ) /2 M t ε -αθ : t ≥ 0 converges in distribution (as a process) toward κ (3.28)

The constant κ α,β is positive by noting that ν is absolutely continuous with respect to the Lebesgue measure, that m α,β has a non-empty support, and that ĝ could not be a constant function, since Lĝ = id.

Step 3) By using that |ĝ| ≤ c(h p,γ + 1), we get

ĝ Vt ε -αθ -ĝ( V0 ) 2 ≤ 4c 2 h p,γ Vt ε -αθ 2 + h p,γ ( V0 ) 2 + 2
hence, using Proposition 3.4,

lim ε→0 E ε αθ sup t∈[0,T ] ĝ Vt ε -αθ -ĝ( V0 ) 2 = 0
hence ε (αθ) /2 ĝ Vt ε -αθ -ĝ( V0 ) : t ≥ 0 converges in probability toward 0, uniformly on compact sets.

Step 4) Our processes are valued in the Skorokhod space of càdlàg functions D([0, ∞)) endowed with J 1 (or simple) Skorokhod topology (see [START_REF] Whitt | Stochastic-process limits: an introduction to stochastic process limits and their applications to queues[END_REF], §3.3). It is not difficult to see that a sequence which converges in probability toward 0, uniformly on compact sets, is also convergent in probability for J 1 metric, hence in distribution in J 1 topology. Recall that in the Skorokhod space, the addition is not a continuous map (see for instance [START_REF] Whitt | Stochastic-process limits: an introduction to stochastic process limits and their applications to queues[END_REF], p. 84). In our case, the limits of the terms on the right hand side of equality (3.27) are, respectively 0 and a Brownian motion which have continuous paths. By using the joint convergence theorem (Theorem 11.4.5, p. 379 in [START_REF] Whitt | Stochastic-process limits: an introduction to stochastic process limits and their applications to queues[END_REF]) and the continuous-mapping theorem (Theorem 3.4.3, p. 86 in [START_REF] Whitt | Stochastic-process limits: an introduction to stochastic process limits and their applications to queues[END_REF]), we obtain the conclusion of Theorem 1.1. The first term on the right hand side of (3.30) can be written :

E ĝ Vε αθ -ĝ( V0 ) 2 = 2 ĝ(x) 2 m α,β (dx) -2E ĝ V0 )ĝ( Vε αθ = 2 ĝ(x) 2 m α,β (dx) -2E ĝ( V0 )E ĝ Vε αθ | V0 = 2 ĝ(x) 2 m α,β (dx) -2E ĝ( V0 ) T ε αθ ĝ ( V0 ) = 2 ĝ(x) 2 m α,β (dx) -2E ĝ( V0 ) ĝ( V0 ) + ε αθ 0 (T s id)( V0 )ds = -2E ĝ( V0 ) ε αθ 0 (T s id)( V0 )ds = -2 ĝ(x) m α,β (dx) ε αθ 0 (T s id)(x)ds = -2 ε αθ xĝ(x)m α,β (dx) -2 ĝ(x) m α,β (dx) ε αθ 0 (T s id) -id (x)ds.
By using the Hölder inequality, we prove that,

E ĝ Vε αθ -ĝ( V0 ) 2 ∼ -2 ε αθ xĝ(x)m α,β (dx), as ε → 0. (3.31) Indeed, if 2 -α 2 < β < 2, ĝ ∈ L 3-β /2-β (m α,β ) and lim s→0 (T s id) -id L 3-β (m α,β ) = 0, and if β ≥ 2, ĝ ∈ L 1 γ (m α,β ) and lim s→0 (T s id) -id L 1 /(1-γ) (m α,β ) = 0.
By using (3.24) and Fubini's theorem, the second term on the right hand side of (3.30) can be written

E ε αθ 0 Vs ds 2 = ε αθ 0 ds s 0 E Vs Vu du = ε αθ 0 ds s 0 E Vs-u V0 du = ε αθ 0 ds s 0 E V0 (T s-u id)( V0 ) du = ε αθ 0 du E V0 ε αθ u (T s-u id)( V0 ) ds = ε αθ 0 du E V0 T ε αθ -u ĝ ( V0 ) -ĝ( V0 ) = ε αθ 0 du x T ε αθ -u ĝ -ĝ (x)m α,β (dx).
Once again by the Hölder inequality, we prove that

E ε αθ 0 Vs ds 2 = o(ε αθ ), as ε → 0. (3.32) Indeed, if 2 -α 2 < β < 2 then id ∈ L 3-β (m α,β ), we can see that lim ε→0 sup 0≤u≤ε αθ (T ε αθ -u ĝ) -ĝ L 3-β /2-β (m α,β ) = 0. Similarly, if β ≥ 2 then id ∈ L 1 /(1-γ) (m α,β ), we see that lim ε→0 sup 0≤u≤ε αθ T ε αθ -u ĝ) -ĝ L 1 γ (m α,β ) = 0.
Finally, the third term in (3.30) is analysed by using the Cauchy-Schwartz inequality and the behaviour of the other terms. We get that and the result is proved.

-2 E ĝ Vε αθ -ĝ( V0 ) ε αθ 0 Vs ds = o(ε αθ ), as ε → 0. ( 3

Appendix

Proof of Lemma 2.3. Note that g β is an odd function. Introduce

ϕ β (x) = - +∞ x 2ye c β (y) dy. By the continuity of g β on [0, 1], it is sufficient to prove (2.3) for x > 1. Assume β ∈ [1, ∞), then, since x > 1, ϕ β (x) = +∞ x z 1-β -2z β e -2 β+1 z β+1 dz ≥ +∞ x -2z β e -2 β+1 z β+1 dz = -e -2 β+1 x β+1 , hence x 1 e 2 β+1 y β+1 ϕ β (y)dy ≥ 1 -x,
and (2.3) is true in this case. If β ∈ [0, 1), by integration by parts,

ϕ β (x) = +∞ x z 1-β -2z β e -2 β+1 z β+1 dz = -x 1-β e -2 β+1 x β+1 + 1 -β 2 +∞ x z -2β -2z β e -2 β+1 z β+1 dz ≥ -x 1-β e -2 β+1 x β+1 - 1 -β 2 x -2β e -2 β+1 x β+1 , hence, x 1 e 2 β+1 y β+1 ϕ β (y)dy ≥ x 1 -y 1-β - 1 -β 2 y -2β
k := 2 -k k-1 j=0 ((1 -β) -j(1 + β)), for k ≥ 1 integer.
By the choice of n, we can see that d n > 0. If we iterate n times the integration by parts, we get:

ϕ β (x) = - n k=0 d k x (1-β)-k(1+β) e -2 β+1 x β+1 + d n +∞ x z (1-β)-(n+1)(β+1) (-2z β e -2 β+1 z β+1 )dz.
Since (1 -β) -(n + 1)(β + 1) ≤ 0 we can write

ϕ β (x) ≥ - n k=0 d k x (1-β)-k(1+β) + d n x (1-β)-(n+1)(β+1) e -2 β+1 x β+1 .
By integrating, we have

x 1 e 2 β+1 y β+1 ϕ β (y)dy ≥ x 1 n k=0 d k y (1-β)-k(1+β) + d n y (1-β)-(n+1)(β+1) dy,
and we easily deduce (2.3). The proof of (2.3) is complete for all β ∈ (-1, ∞).

Proof of Lemma 3.5. Recall that h p,γ (x) = (1 + |x| pγ ) 1 /p and assume firstly that |x| < 1. Since h p,γ is continuously differentiable and equivalent to |x| γ at infinity, there exists k > 0 such that The desired inequality is then clear. Secondly, assume that |x| ≥ 1. It is a simple computation to see that for all z ≥ 0 and r > 0, there exists c r > 0, such that

(1 + z) r -1 ≤ c r z1 {z≤1} + z r 1 {z>1} .
We deduce that, for all (u, v) ∈ [0, ∞) × [0, ∞), there exist k r > 0 such that Since |x| ≥ 1, i(x) /|x| is bounded, and since p > 1, ( 1 /|x| pγ + 1) (1-p) /p ≤ 1. Using that pγ > 2 and the fact that |y| /|x| is bounded, we have the existence of a k ′ > 0 such that

1 + |y| |x| pγ -1 ≤ k ′ |y| |x| .
Taking k = max(k1 /p k pγ , k1 /p k ′ ), we get the second inequality in the first part of Lemma 3.5.

We proceed with the second part and we note that, since pγ > 2, h p,γ is twice differentiable with h ′′ p,γ (x) = γ |x| pγ-2 (γ -1)|x| pγ + pγ -1 (1 + |x| pγ ) 1 /p-2 .

Moreover, since γ < α < 2, h ′′ p,γ ∈ L ∞ . We split (L α,β h p,γ )(x) into three terms The first term on the right hand side is equivalent to -γ|x| γ+β-1 at infinity, while for the second term, since |y| ≤ 1, we have

Remark 2 . 4 .

 24 Let us note that if β = 1 (Ornstein-Uhlenbeck case), g β (x) = -x, κ 2,β = 1 and the result of Theorem 1.1 coincides with the result of Proposition 2.1, p. 268, in[START_REF] Hintze | Small noise asymptotics and first passage times integrated Ornstein-Uhlenbeck process driven by α-stable Lévy process[END_REF].

. 14 )

 14 Introduce the Lyapunov functionh p,γ (x) := (1 + |x| pγ ) 1/p .(3.15)

0 T

 0 Vu || Vs |. Using Markov's property and that Vu and V0 follow the invariant law, we get, for u < s, E | Vs || Vu | = E | Vs-u || V0 | . Therefore E u |id|( V0 )du .

1 / 2 α 2 [

 122 ,β B, where B is a standard Brownian motion andκ α,β := R ĝ(x + y) -ĝ(x)] 2 ν(dy)m α,β (dx) > 0.

Proposition 3 . 6 . 2 R 2 = 2 -0

 36222 The constant κ α,β in Theorem 1.1 given in (3.28) satisfiesκ α,β = -xĝ(x)m α,β (dx) > 0.(3.29)Proof. Since, by (3.26) and (3.28), κ α,β = 1 t E M 2 t , for all t > 0, by taking t = ε αθ and using Itô's formula, we getκ α,β = ε -αθ E ĝ Vε αθ -ĝ( V0 ) -ε -αθ E ĝ Vε αθ -ĝ( V0 ) 2E ĝ Vε αθ -ĝ( V0 ) ε αθVs ds .(3.30)

  .33) Putting together (3.30)-(3.32), we obtain that κ α,β = -2 xĝ(x)m α,β (dx) + o(1), as ε → 0.

  |h p,γ (x + y) -h p,γ (x)| ≤ |y| sup z∈[-2,2] |h ′ p,γ (z)|1 {|y|≤1} + k|y| γ 1 {|y|>1} .

(r- 1 ≤- 1 p 1

 111 u + v) r -u r = u r 1 + v u k r vu r-1 1 {v≤u} + v r 1 {u<v} . (3.34) Since x = 0, |h p,γ (x + y) -h p,γ (x)| = |x| γ 1 |x| pγ + 1 + 34) with u = 1 |x| pγ + 1, v = 1 + y x pγ and r = 1 p , we obtain |h p,γ (x + y) -h p,γ (x)| ≤ k1 /p |x| γ {|y|<i(x)} .Since i(x) > |x|, we can use again (3.34) to estimate the first term in the bracket on the right hand of the latter inequality. We let u = 1, v = y x and r = pγ and we get|h p,γ (x + y) -h p,γ (x)| ≤ k1 /p k pγ |y| γ 1 {i(x)≤|y|} + k1 /p |x| γ 1 + y x pγ -1 1 |x| pγ + 1 1-p p 1 {|y|<i(x)} .

L

  α,β h p,γ (x) = -γ |x| pγ+β-1 (1 + |x| pγ ) 1-1 /p + |y|≤1 h p,γ (x + y) -h p,γ (x) -yh ′ p,γ (x) ν(dy) + |y|>1h p,γ (x + y) -h p,γ (x) ν(dy).

  h p,γ (x + y) -h p,γ (x) -yh ′ p,γ (x) ≤ y 2 sup |z|≤1 |h ′′ p,γ (x + z)| ≤ y 2 h ′′ p,γ ∞ . Hence |y|≤1 h p,γ (x + y) -h p,γ (x) -yh ′ p,γ (x) ν(dy) ≤ c α h ′′ p,γ ∞ ,where c α := |y|≤1 y 2 ν(dy). We use the first part of the lemma to estimate the third term on the right hand side. There are two situations : if |x| ≥ 1, we geth p,γ (x + y) -h p,γ (x) ≤ k(|y||x| γ-1 1 {|y|≤i(x)} + |y| γ 1 {i(x)<|y|} ). Hence |y|>1 h p,γ (x + y) -h p,γ (x) ν(dy) ≤ k|x| γ-1 {i(x)≥|y|>1} |y|ν(dy) + k {max(1,i(x))≤|y|} |y| γ ν(dy) ≤ k|x| γ-1 {i(x)≥|y|>1} |y|ν(dy) + kc ′ α,γ , where c ′ α,γ := {|y|>1} |y| γ ν(dy). Since i(x) = O(|x|), as |x| → ∞, k|x| γ-1 {i(x)≥|y|>1} |y|ν(dy) = O(|x| γ-1 ) + O(|x| γ-α ), as |x| → ∞. If |x| < 1, since |y| > 1, |h p,γ (x + y) -h p,γ (x)| ≤ k|y| γ , so |y|>1 h p,γ (x + y) -h p,γ(x) ν(dy) ≤ |y|>1 |y| γ ν(dy) < +∞. Denote by u the continuous function -L α,β h p,γ . Putting together the previous estimates, since β > 1 and 2 p < γ < α, we obtain that u(x) ∼ |x| γ+β-1 , as |x| → ∞, and since γ > 2 -β, 1 + |x| = o(u(x)), as |x| → ∞.

  Girsanov's theorem gives the existence of a weak solution to equation (2.2 2 ). For both situations, the solution is defined until an explosion time τ e , but it is no difficult to prove that τ e = ∞ a.s. by using Theorem 10.2.1, p. 254, in[START_REF] Stroock | Multidimensional Diffusion Processes[END_REF] and a convenient Lyapunov function (for instance h(x) = 1 + x 2 for all |x| ≥ 1, h(x) = 1 for all |x| ≤ 1/2 and h ≥ 1 ). Introduce the scale function and the speed measure associated to the diffusion

	whereas if -1 < β < 1, s β (x) :=
	2.1.1 Existence and uniqueness
	If β ≥ 1, the drift coefficient in (2.2 2 ) is a locally Lipschitz function hence by well known results (see, for instance, Theorem 12.1, p. 132 in [12]), we get a pathwise unique strong solution V to equation (2.2 2 ),

  dy, and (2.3) follows. More generally, assume β ∈ [-n n+2 , 1-n n+1 ), for an integer n ≥ 0. Set d 0 = 1 and d

Acknowledgements

The authors are grateful to Ilya Pavlyukevich for many instructive and stimulating discussions on the subject and also for pointed out to us the reference [13].

Then relations (3.17)-(3.18) hold true and the proof is complete.