Lise Ropars
email: lise.ropars@imbe.fr

Laurence Affre

Matthieu Aubert

Catherine Fernandez

Floriane Flacher

David Genoud

Frédéric Guiter

Coline Jaworski

Xavier Lair

Clémentine Mutillod

Gabriel Nève

Lucie Schurr

Benoît Geslin

Pollinator Ecology and Management Pollinator Specific Richness and Their Interactions With Local Plant Species: 10 Years of Sampling in Mediterranean Habitats

Keywords: Apoidea, Syrphidae, Bombyliidae, biodiversity, conservation

In the context of global pollinator decline, little is known about the protection status and ecology of many species. This lack of knowledge is particularly important for Mediterranean protected areas that harbor diverse pollinator communities and are subject to considerable anthropogenic pressures. Calanques National Park (85 km 2 ), which is located near Marseille (France), is dominated by Mediterranean low-vegetation habitats, such as phrygana and scrublands. These habitats offer favorable conditions for pollinator species due to the important amount of floral resources. Within a 10-yr period, we recorded bee (Hymenoptera: Apoidea: Anthophila), hover fly (Diptera: Syrphidae), and bee fly (Diptera: Bombyliidae) species and their interactions with the local flora through 10 field campaigns. We caught 250 pollinator species, including 192 bees, 38 hover flies, and 20 bee flies, for a total of 2,770 specimens. We recorded seven threatened bees (six near threatened and one endangered). Among the bee species, 47.9% were below-ground nesting species, and 54.7% were generalist species. Analysis of the pollination network showed that generalist and specialist pollinators do not share the same floral resources. The Cistaceae plant family (Malvales: Cistaceae) acted as a central node in the plant-pollinator network, interacting with 52 different pollinator species, which shows the importance of large open flowers that could be easily visited by both short and long-tongued pollinators in Mediterranean habitats. The occurrence of pollinator species and their ecological traits should strongly contribute to reinforcing the available information to provide or ameliorate the conservation statuses determined by IUCN Red List.

Insect pollinators are essential for the reproduction of wild plants and crops [START_REF] Ollerton | How many flowering plants are pollinated by animals?[END_REF]. In Europe, the species richness of insect pollinators increases along a north-south gradient, with a maximum reached around the Mediterranean basin [START_REF] Nieto | European Red List of Bees[END_REF]. The Mediterranean basin is considered a biodiversity hotspot [START_REF] Médail | Hot-spots analysis for conservation of plant biodiversity in the Mediterranean basin[END_REF] and hosts a large diversity of plant species, which can benefit a wide richness of pollinator species [START_REF] Dafni | Pollination syndromes in the Mediterranean: generalizatoins and peculiarities[END_REF]. This hotspot, however, suffers from many types of anthropogenic pressure, such as land use change (fragmentation, increasing urbanization), pollution (nearby cities, past industrial activity), agricultural practices (grazing), climate change [START_REF] Giorgi | Climate change projections for the Mediterranean region[END_REF] and intentional or accidental wildfires [START_REF] Petanidou | Interdependence of native bee faunas and floras in changing Mediterranean communities[END_REF]. To address the conservation of species within Mediterranean ecosystems, large areas have been included in protected national parks. However, there are still knowledge gaps related to the pollinating fauna in Mediterranean protected ecosystems. As an example, the IUCN European Red List of bees indicates that, for at least 38% of the wild bee species of French Mediterranean ecosystems, there is not enough data to state their conservation status [START_REF] Nieto | European Red List of Bees[END_REF][START_REF] Rasmont | Addition to the checklist of IUCN European wild bees (Hymenoptera: Apoidea)[END_REF]. Considering the current biodiversity loss and the crucial importance of pollinators in the functioning of ecosystems, it is urgent to develop a better understanding of the community composition of pollinating fauna in addition to their distributions, ecological traits, and interactions with plant species.

In European and Mediterranean ecosystems, bees and flies are the most dominant pollinators responsible for the reproduction of many flowering plants [START_REF] Ollerton | Pollinator diversity: distribution, ecological function, and conservation[END_REF]. Bees are key pollinators because they rely on nectar and pollen at both adult and larval stages. Each bee family is characterized by specific functional traits related doi: 10.1093/ee/nvaa061 to the way they collect pollen and nectar, such as their tongue length and specific pollen collection structures (e.g., pollen baskets on honey bees and bumble bee species, flocculus on Andrena species, and abdominal scopa on Megachilidae). In France, the INPN national taxonomic database referenced 962 wild bee species in 2019 [START_REF] Gargominy | TAXREF v13, référentiel taxonomique pour la France : méthodologie, mise en oeuvre et diffusion[END_REF]. The IUCN European Red List estimated the occurrence of approximately 720 species in southern France, most of which are endemic to the Mediterranean basin [START_REF] Nieto | European Red List of Bees[END_REF]. Furthermore, as well as harboring a large diversity of plants, Mediterranean habitats offer many nesting resources for wild bee species due to the dry climate, which limits vegetation cover and thus results in more available bare soil [START_REF] Nielsen | Assessing bee species richness in two Mediterranean communities: importance of habitat type and sampling techniques[END_REF]. Consequently, the composition of wild bee species and their ecology and interactions with the local floral might be very specific but are still poorly known.

Regarding nonbee species such as hover flies, the 540 known species in the French territory also strongly contribute to pollination [START_REF] Rader | Non-bee insects are important contributors to global crop pollination[END_REF]. Hover flies are considered bio-indicators of natural habitats because of their various ecological niches, mainly during their larval stage, role as pollinators, and sensitivity to pesticides and landscape composition [START_REF] Sommaggio | Syrphidae: can they be used as environmental bioindicators?[END_REF]. As larvae, hover fly species have various diets, including saprophagous, insectivorous, phytophagous, or mycophagous diets, and may then occupy many ecological niches. As adults, most hover fly species seek floral resources (pollen and nectar). Among nonbee species, bee flies also contribute to pollination services. Their larvae parasitize the brood of wild hymenopteran species (Merritt et al. 2009), whereas adults collect and feed on nectar and pollen. According to the INPN national taxonomic database, 150 bee fly species have been recorded in France [START_REF] Gargominy | TAXREF v13, référentiel taxonomique pour la France : méthodologie, mise en oeuvre et diffusion[END_REF]), but only two recent works on their distribution or ecology have been published [START_REF] Zaitzev | Contributions to the Palaearctic fauna of the dipteran families Bombyliidae and Mythicomyiidae (Diptera): I[END_REF][START_REF] Zaitzev | Contributions to the Palaearctic fauna of the dipteran families bombyliidae and mythicomyiidae (Diptera): II[END_REF].

Generalist species are of particular importance as they occupy a central position in plant-pollinator networks. They are usually considered as key species because they have an important role to support the coexistence of both plants and pollinators [START_REF] Pocock | Succinctly assessing the topological importance of species in flower-pollinator networks[END_REF][START_REF] Bartomeus | Understanding linkage rules in plant-pollinator networks by using hierarchical models that incorporate pollinator detectability and plant traits[END_REF]. Indeed, considering their floral diet, some pollinator species are generalists and collect floral resources from a wide range of plant species, whereas specialist pollinator species rely on few plant species. The arrangement, diversity, and strength of links between plant and pollinator species are indicators of the stability of pollination networks and may be linked with the sustainability of ecosystems [START_REF] Memmott | Tolerance of pollination networks to species extinctions[END_REF]). In plant-pollinator networks, generalist plant and pollinator species could contribute to increasing the stability of the network, especially by multiplying the number of links with the other levels, including specialist plant species [START_REF] Martín González | Centrality measures and the importance of generalist species in pollination networks[END_REF]. In contrast, pollinator species that forage on only one plant species and plant species visited by only one pollinator species are the most vulnerable [START_REF] Weiner | Land-use impacts on plant-pollinator networks: interaction strength and specialization predict pollinator declines[END_REF]. Each change in these interactions could affect the structure of the whole network and, therefore, plant and pollinator populations. However, in Mediterranean ecosystems, despite that few plant-pollinator networks have been described, these generalist species are poorly known.

In this study, we intend to improve the knowledge on pollinator communities by focusing on bee, hover fly, and bee fly species and their relationships with plant species in Mediterranean low-vegetation habitats. Through a 10-yr sampling effort (from 2009 to 2018) in a protected area (Calanques National Park), we explored the following questions: (i) What is the species richness of bees, hover flies, and bee flies in low-vegetation Mediterranean habitats and what is their protection status? (ii) What are the dominant ecological and functional traits of these pollinating species? (iii) What floral resources are included in their diets, and what is structure of the whole pollination network?

Materials and Methods

Study Site and Pollinator Surveys

All observations and captures were conducted in Calanques National Park (near Marseille, France-43°13′N, 5°28′E). This area is characterized by typical Mediterranean low-vegetation habitats (western Mediterranean phrygana, Astragalo-Plantaginetum subulatae, and shrubland) and by Mediterranean forests and rocky habitats. The western Mediterranean phrygana corresponds to the sclerophyll cushion associations typical of Mediterranean coastal areas [START_REF] Molinier | Etudes phytosociologiques et écologiques en Provence Occidentale[END_REF]. This protected biodiversity hotspot of 85 km 2 hosts more than 900 plant species [START_REF] Pires | La Flore remarquable des Bouches du Rhône. Plantes, milieux naturels et paysages[END_REF], and the dominant entomophilous species are Rosmarinus officinalis L. (Lamiales: Lamiaceae), Cistus albidus L. (Malvales: Cistaceae), and Thymus vulgaris L. (Lamiales: Lamiaceae) in shrubland [START_REF] Flo | Yearly fluctuations of flower landscape in a Mediterranean scrubland: consequences for floral resource availability[END_REF]. The Mediterranean climate is characterized by hot and dry summers, rainy autumns, and mild winters. The main flowering season stretches from February to June.

Over a period of 10 years (2009)(2010)(2011)(2012)(2013)(2014)(2015)(2016)(2017)(2018), 10 sampling campaigns were carried out within this protected area (Table 1). These sampling campaigns were all conducted when meteorological conditions were favorable for pollinator activity (i.e., only on sunny days with wind < 20 km/h and temperature > 15°C). Pollinators were caught either through entomological nets or pan traps or both (Table 1). The net collection of insects consisted of catching insects in densely flowered patches for 5 to 10 min and repeating the action at least two times for each sampling site (Fig. 1). The pan trap collection was carried out by laying three colored pan traps for a 24 h period following the method of [START_REF] Westphal | Measuring bee diversity in different European habitats and biogeographical regions[END_REF]. Plastic bowls of 500 ml were painted with blue, white, or yellow paint that reflected UV light to mimic flowers and passively attract flower visitors. Each pan trap was filled with soapy water and placed on the ground; the pans were placed 1 m apart from each other in a triangle arrangement. Pan trap collection underestimates the occurrence of large pollinators because they are better able to escape the trap than smaller species. Net collection underestimates the presence of small pollinators due to the reduced ability of humans to detect them [START_REF] Wilson | Sampling bee communities (Hymenoptera: Apiformes) in a desert landscape: are pan traps sufficient?[END_REF][START_REF] Popic | Evaluation of common methods for sampling invertebrate pollinator assemblages: net sampling out-perform pan traps[END_REF]). These two methods are thus complementary and can be combined for a better estimation of pollinator diversity [START_REF] Westphal | Measuring bee diversity in different European habitats and biogeographical regions[END_REF]. All ecological traits, such as nesting habits and lectism of wild bee species, are reported in Supp Table 1 (online only) according to the expertise of D. Genoud and M. Aubert [START_REF] Carré | Biodiversité, paysages et conservation de la communauté d'abeilles dans les agrosystèmes[END_REF][START_REF] Lugassy | Systèmes de pollinisation et perturbations anthropiques: de l'échelle paysagère à l'échelle macroécologique[END_REF][START_REF] Genoud | Synthèse des connaissances sur les hyménoptères apoïdes --Prélèvements 2002 -2009 et étude[END_REF][START_REF] Ropars | La diversité des abeilles parisiennes[END_REF][START_REF] Ropars | La diversité des abeilles parisiennes[END_REF].

The first four sampling campaigns were conducted by X. Lair and M. Aubert between 2009 and 2011. They consisted of onetime bee surveys (Samplings 1 to 4). In May 2016, a collective 3-d sampling campaign was completed as part of a training course on wild bees (Sampling 5) [START_REF] Ropars | La diversité des abeilles parisiennes[END_REF]. Between 2017 and 2018 (Samplings 6 to 9), more intensive surveys were carried out every week for 2-5 mo [START_REF] Schurr | Pollination insights for the conservation of a rare threatened plant species, Astragalus tragacantha (Fabaceae)[END_REF][START_REF] Ropars | Land cover composition, local plant community composition and honeybee colony density affect wild bee species assemblages in a Mediterranean biodiversity hot-spot[END_REF]. Finally, D. Genoud performed a one-time bee species survey in May 2018 (Sampling 10). The pan traps were only used in Sampling 9 in 2018 in addition to the net surveys. Details of each sampling campaign are provided in Table 1.

The total species richness was estimated with the Chao and jackknife methods and included all captures [START_REF] Gotelli | Estimating species richness[END_REF]. The Chao1 index estimates the potential maximum species number in an area based on the species that are observed only once or twice. The jackknife index is a resampling technique that decreases the estimator bias by deleting some of the observations. We calculated these indexes using the function ChaoSpecies() within the Spade-R package in R version 3.6 software [START_REF] Chao | SpadeR (Speciesrichness Prediction And Diversity Estimation in R): an R package in CRAN. Program and User's Guide[END_REF][START_REF] Core | R: A language and environment for statistical computing[END_REF].

Plant-Pollinator Interactions

The plant-pollinator network was built using two successive methods: (i) direct observations and individual insect captures on plant species and (ii) reconstruction of the a posteriori network through an analysis of pollen grains carried by pollinator species. For part (i), we based our global qualitative plant-pollinator network on net samplings on plant species (Samplings 6 to 9 only). Collectors also conducted sessions in which they observed insect visits on plant species without catching them to assess the floral diet of pollinators. For part (ii), pollen grains carried by the sampled bees during Samplings 7 and 9 were collected on their pollen collecting structures. Pollen grains were then acetolysed following the protocol of [START_REF] Faegri | Textbook of pollen analysis[END_REF] to reveal their ornamentation for identification. They were mixed with glycerol and mounted on slides to be identified to the family and, if possible, to the genus and species using a reference atlas [START_REF] Reille | Pollen et spores d'Europe et d'Afrique du nord[END_REF]. The pollen analysis was conclusive for only 72 bee individuals, and few links completed the whole pollination network which was not sufficient to compare both pollen and capture networks. The overall plant-pollinator interaction network for the bee, hover fly and bee fly species, including all the interactions from (i) and (ii), was built with the bipartite package [START_REF] Dormann | Indices, graphs and null models: analyzing bipartite ecological networks[END_REF]) available on the R software (R Core Team 2019). For the remaining samplings, pollinators not linked with a plant species (e.g., caught by pan trap or while flying) were not considered.

Using the obtained network, we calculated its modularity, nestedness, number of compartments, and connectance as main descriptors using the computeModules() and networklevel() functions from the bipartite package [START_REF] Dormann | Indices, graphs and null models: analyzing bipartite ecological networks[END_REF][START_REF] Core | R: A language and environment for statistical computing[END_REF]. The modularity highlights the groups of species that interact preferentially together, and its value ranged between 0 (random links between species) and 1 (strong community structure) [START_REF] Barber | Modularity and community detection in bipartite networks[END_REF]. Nestedness is indicated when a specialist species interacts with a subset of the species used by generalists species [START_REF] Mariani | Nestedness in complex networks: observation, emergence, and implications[END_REF]. It is expressed as a percentage where the maximum value corresponds to a perfect nested structure. Finally, the connectance corresponds to the proportion of observed interactions relative to the maximum number of potential interactions in the network and ranges between 0 and 1 [START_REF] Poisot | When is an ecological network complex? Connectance drives degree distribution and emerging network properties[END_REF].

Results

Bee Species Diversity

From 2009 to 2018, 192 bee species (Hymenoptera: Apoidea: Anthophila) were caught in Calanques National Park via both net and pan trap collections. They belonged to six families and 38 genera. Megachilidae was the most species-rich family, followed by Apidae and Halictidae, with 53, 44, and 43 species, respectively. Andrena Fabricius, 1775 was the richest genus, with 33 species, whereas eight genera were represented by only one species (Stelis Panzer, 1806;Protosmia Ducke, 1900;Pseudoanthidium Friese, 1898;Coelioxys Latreille, 1809;Chelostoma Latreille, 1809;Anthidiellum Cockerell, 1904; Amegilla Friese, 1897; Panurgus Panzer, 1806; Supp Table 1 [online only]). Using the jackknife 1 and 2 indexes, the total bee species richness was estimated to be between 240.98 ± 9.89 and 268.96 ± 17.13. The observed richness thus represented 71.3 to 79.6% of the estimated bee richness (Table 2). The Chao1 method estimated the observed richness to be 76.8% of the potential maximum richness (Table 2). The most abundant species was Apis mellifera L. (276 individuals), followed by Rhodanthidium septemdentatum (Latreille, 1809), Rh. sticticum (Fabricius, 1787) (174 and 168 individuals, respectively), and Lasioglossum transitorium (Shenck, 1868) (113 individuals). Finally, 50 species (26%) were represented by only one individual (singletons).

Ecological Traits of Bees

Regarding the ecological traits of nesting preferences, we identified 59 solitary bee species nesting above ground, 92 solitary bee species nesting below ground and 23 cleptoparasitic species, corresponding to 30.7, 47.9, and 12.0% of the total bee species richness, respectively. We detected four social species (bumble bees and honey bees-2.1%). Finally, the nesting preferences of 14 species remain unknown (7.3%-Supp Table 1 [online only]). Regarding the floral diet of bee species, we found that 105 species (54.7%) foraged on a wide range of plant species (i.e., generalist species, polylectic) and 51 species (26.5%) foraged on one plant family or one/few plant species (i.e., specialist species, oligolectic). For the remaining 36 bee species (18.8%), the floral diet was not known (e.g., for cleptoparasitic species). However, most generalist species showed a preference for one or a few plant families (Supp Table 1 [online only]). For the 49 singletons, 14 bee species (28%) were specialists, and 9 bee species (18%) were cleptoparasitic (Supp Table 1 [online only]).

An IUCN status was available for 178 species (excluding the species complexes and unidentified species). Among those, 47 species were classified as data deficient (DD-26.4%), 124 species were least concerned (LC-69.6%), six species were near threatened (NT-3.4%; Andrena ovatula (Kirby, 1802), Colletes albomaculatus (Lucas, 1849), Dasypoda argentata Panzer, 1809, Halictus quadricinctus (Fabricius, 1776), Lasioglossum prasinum (Smith, 1848), and L. pygmaeum (Schenck, 1853)), and one species was considered endangered (EN-0.6%; L. soror (Saunders, 1901)-Supp Table 1 [online only]). Three oligolectic species were among the seven threatened bee species. A. ovatula and C. albomaculatus are specialists on Fabaceae, D. argentata is a specialist on Caprifoliaceae.

Hover Fly and Bee Fly Species Diversity

From 2009 to 2018, 38 species of hover flies (Diptera: Syrphidae) were caught in Calanques National Park. They belonged to 24 genera (Supp Table 2 [online only]). The most abundant genera were Paragus Latreille, 1804 (33 individuals), Meliscaeva Frey, 1946 (14 individuals), and Sphaerophoria Lepeletier de Saint-Fargeau & Audinet-Serville in Latreille, 1828 (13 individuals), whereas 20 species (53%) were represented by only one individual (Supp Table 2 [online only]). The bee flies (Diptera: Bombyliidae) were represented by 20 species, including five singletons (25%) (Supp Table 3 [online only]). The estimation of total species richness of hover flies and bee flies was between 83 and 97 species using the jackknife 1 and 2 indexes, respectively. Consequently, the observed richness represented 59.8-69.9% of the estimated hover fly and bee fly richness (Table 2).

Ecological Traits of Hover Flies and Bee Flies

Bee fly species correspond to parasitic species of Hymenoptera species including wild bees, but their hosts and their ecology are not well known. Regarding the captured hover flies, their larvae were mostly zoophagous (23 species), only two species were phytophagous, and three species were both phytophagous and zoophagous. Hover fly larvae of 24 species (63%) were nontolerant to inundations. At the adult stage, 27 species of hover flies preferentially foraged on pollen

Plant-Pollinator Interaction Network

The complete network included 133 pollinator species in interaction with 62 plant species, representing 314 different links (Fig. 2). The network had six compartments: five blocks with one plant species linked to one or two pollinator species and one block with the remaining plant and pollinator species. The modularity, nestedness, and connectance of the network were estimated at 0.50, 12.96, and 0.04 respectively. The highly generalist pollinator species were A. mellifera, Rh. septemdentatum, Rh. sticticum, L. transitorium, L. bimaculatum (Dours, 1872), and Paragus tibialis (Fallén, 1817), which foraged on 14, 12, 11, 11, 10, and 10 plant species, respectively (Fig. 2). On the other hand, the plant species visited by a high diversity of pollinators were C. albidus and Ro. officinalis, with 37 and 35 pollinator species, respectively (Fig. 2). Twenty plant species were visited by only one pollinator species, and the mean shared pollinator species per plant was 5.06 species.

Discussion

Species Richness of the Mediterranean Pollinator Community

For this study to the best of our knowledge, we compiled all available surveys of pollinators within Calanques National Park from 2009 to 2018. We found 250 pollinator species, including 192 bee, 38 hover fly, and 20 bee fly species. This corresponded to 20, 7, and 13% of the bee, hover fly, and bee fly species, respectively, that have been described in France. The highest abundance and diversity of pollinators was sampled when both weekly net and pan trapping were used to catch pollinators in the Mediterranean low-vegetation habitats [START_REF] Nielsen | Assessing bee species richness in two Mediterranean communities: importance of habitat type and sampling techniques[END_REF].

The species richness found in this study indicates that Mediterranean low-vegetation habitats can support a high richness of pollinators. In another protected area located at 70 km east of Calanques National Park (Port-Cros National Park), Coiffait-Gombault et al. (2018) found a similar species richness of 162 bee species and 26 hover fly species [START_REF] Gombault | Inventaires des abeilles sauvages sur les sites de Port-Cros, Bagaud et du cap Lardier[END_REF]. In Greece, from 4 yr of net and Malaise samplings within Mediterranean low-vegetation habitats, 262 bee species, 59 hover fly species, and 47 bee fly species were documented (Petanidou andEllis 1993, Petanidou et al. 2011). This richness of pollinator assemblages is, therefore, close to the scores given by the Chao and jackknife indexes for bees (Chao1 = 250 species; Jackknife2 = 269 species), hover flies and bee flies (Chao1 = 85 species; Jackknife2 = 96 species). Given the high spatio-temporal variability of the composition of the pollinator community, our species list might be incomplete, but the Chao and jackknife indexes indicated that we caught a high proportion of the species richness of these Mediterranean low-vegetation habitats. Nevertheless, a thorough understanding of the pollinator species on Calanques National Park might also include other sampling methods, such as the use of Malaise traps, which is particularly efficient for Diptera [START_REF] Campbell | Efficiency of Malaise traps and colored pan traps for collecting flower visiting insects from three forested ecosystems[END_REF], and field work should also be intensified during periods of the year that are rarely sampled, such as autumn. Likewise, due to the difference in sampling effort across years, we cannot explore long-term population trends.

With the Mediterranean dry climate, bee nesting sites seem to be very diverse, with good opportunities for both belowground-nesting and cavity-nesting bee species [START_REF] Potts | Role of nesting resources in organising diverse bee communities in a Mediterranean landscape[END_REF]. Indeed, the large available areas of bare soil cover in these areas offers advantageous nesting sites for belowground-nesting species. Cavity-nesting species can build their nests with available dry stems or dead branches [START_REF] Petanidou | Pollinating fauna of a phryganic ecosystem: composition and diversity[END_REF]. Hover fly species were also characteristic of dry Mediterranean low-vegetation habitats, as the larvae of most of the recorded species (63%) are intolerant of very wet soils.

Protection Status of Mediterranean Pollinator Species

According to the IUCN European Red List [START_REF] Nieto | European Red List of Bees[END_REF], seven (3%) threatened bee species were recorded in our surveys. This number is similar to that found in Port-Cros National Park [START_REF] Gombault | Inventaires des abeilles sauvages sur les sites de Port-Cros, Bagaud et du cap Lardier[END_REF]). Among those seven species, three have been found in both protected areas (A. ovatula, L. pygmaeum, and H. quadricinctus). We also sampled 47 bee species (24%) that are considered data deficient by the IUCN Red List, and no conservation statuses are currently attributed to hover fly or bee fly species. This still demonstrates the need for pollinator surveys to provide population trends and conservation statuses. Indeed, these data are essential for IUCN European Red List updates as well as National Red List establishment, especially because little is known about pollinator species presence in southern France. As few examples, in Calanques National Park, we provided the first national record of Nomada rubricoxa Scharz, 1977 (Hymenoptera: Apidae) in May 2016 [START_REF] Ropars | La diversité des abeilles parisiennes[END_REF], and specimens of a new Pelecocera Meigen, 1822 (Diptera: Syrphidae) species were collected in 2017 and 2018. The description of this new Pelecocera species is underway and a publication is in preparation (see [START_REF] Lair | Revision of Pelecocera from France, and description of a new species (Diptera : Syrphidae), p. Poster[END_REF].

Interaction Network of the Mediterranean Plant-Pollinator Community

A plant-pollinator network was built based on two types of data from captures and visitations on plant species and from analyses of the pollen carried by the bees. Pollen analysis may give an important added value to networks. However, in our study, an insufficient number of individuals were analyzed to compare pollen and capture networks.

We found that the Cistaceae species in Calanques National Park acted as a central node in the network and were especially represented by three species of Cistus (Malvales: Cistaceae): C. albidus L., C. monspeliensis L., and, to a lesser extent, C. salviifolius L. These species are known to provide both pollen and nectar resources for both bees and hover flies [START_REF] Bosch | Floral biology and pollinators of three co-occurring Cistus species (Cistaceae)[END_REF], and in the present network, they support 52 pollinator species. Cistus species produce large open flowers and can be easily visited by short and long-tongued pollinators, which makes them generalist species. Our results show that these species are of great importance for a high richness of pollinators in Mediterranean habitats. The Lamiaceae family is also known to be attractive for many pollinator species within Mediterranean habitats, especially because tubular flowers provide high amounts of nectar (Petanidou andVokou 1993, Petanidou et al. 2000). Among Fabaceae, Astragalus tragacantha L. (Fabales: Fabaceae) was the species visited by the highest richness of pollinator species because the field campaign focused on it. This endangered metal-tolerant species is the target of a LIFE Habitats Calanques conservation programme to reinforce in situ populations [START_REF] Salducci | How can a rare protected plant cope with the metal and metalloid soil pollution resulting from past industrial activities? Phytometabolites, antioxidant activities and root symbiosis involved in the metal tolerance of Astragalus tragacantha[END_REF]. Additionally, the morphology of the A. tragacantha flower is not adapted to small pollinators, which have difficulty opening its keel and collecting pollen and nectar. Therefore, larger wild bee species are more able to forage on this species. A deeper knowledge of the pollinator guild has been indicated to be essential to provide management advice to build an efficient conservation programme [START_REF] Schurr | Pollination insights for the conservation of a rare threatened plant species, Astragalus tragacantha (Fabaceae)[END_REF]. We also found 20 plant species that interacted with only one pollinator species. This can be explained by the low abundance of ruderal plant species, such as Papaver rhoeas L. (Papaverales: Papaveraceae), Rubus fructicosus L. (Rosales: Rosaceae), Medicago lupulina L. (Fabales: Fabaceae), or Crataegus monogyna Jacq. (Rosales: Rosaceae), within the sampled area of Calanques National Park or by the low sampling effort on the nondominant plant species, such as Teucrium chamaedrys L. (Lamiales: Lamiaceae), T. flavum L., Iris lutescens Lam. (Liliales: Iridaceae), and Asphodelus ramosus L. (Liliales: Liliaceae) in the Mediterranean low-vegetation habitats within our sampling sites. As a complement, we found weak nestedness in the network, which indicates that the interactions performed by specialist species were not entirely included as a subset of the interactions realized by generalist species. This shows the importance of specialist species in our network [START_REF] Almeida-Neto | A consistent metric for nestedness analysis in ecological systems: reconciling concept and measurement[END_REF][START_REF] Ulrich | A consumer's guide to nestedness analysis[END_REF]), as they represent 26.5% of the samplings, including three of the seven threatened bee species. As an example, within the network, we observed D. argentata foraging on Scabiosa (Dipsacales: Caprifoliaceae) species. The other six threatened bee species were generalists or were not linked with a particular plant species in these present surveys. The network indexes, such as modularity and connectance, were consistent with the average metrics of a mutualistic network composed of 195 species (Thebault and Fontaine 2010). Within the mutualistic networks, several studies indicated that the average value of connectance was low to moderate (approximately 0.11 or 0.18), and according to Jordano 1987, the connectance values decreased with species richness [START_REF] Jordano | Patterns of mutualistic interactions in pollination and seed dispersal: connectance, dependence asymmetries, and coevolution[END_REF][START_REF] Landi | Complexity and stability of ecological networks: a review of the theory[END_REF]. The connectance value obtained (0.04) remains difficult to interpret regarding its impact on the stability of the plant-pollinator network. For Thébault and Fontaine (2010), highly connected mutualistic networks are related to high resilience and persistence of the network (Thebault and Fontaine 2010), whereas [START_REF] Allesina | Stability criteria for complex ecosystems[END_REF] indicated that connectance destabilizes mutualistic networks (May 1973, Allesina and[START_REF] Allesina | Stability criteria for complex ecosystems[END_REF].

The presence of numerous singletons is very common in ecological surveys, including insect pollinator surveys [START_REF] Lim | Determining species boundaries in a world full of rarity: singletons, species delimitation methods[END_REF]).

Here, specialist and cleptoparasitic bee species represented half of the bee singletons. Indeed, specialized bee species must feed on their preferred plant species, which decreases the frequency of their captures, especially because some surveys focused on certain plant species in Mediterranean low-vegetation habitats. Cleptoparasitic bees are more difficult to catch because they only visit flowers to gather nectar and fly rapidly close to the ground to find their host's nests [START_REF] Oertli | Ecological and seasonal patterns in the diversity of a species-rich bee assemblage (Hymenoptera: Apoidea: Apiformes)[END_REF]. However, whereas only 0.5% of cleptoparasitic bee species were found during the 3-d sampling in 2016 [START_REF] Ropars | La diversité des abeilles parisiennes[END_REF], we increased this figure to 12% with the present compilation of surveys. Several studies recorded approximately 17% of the parasitic bee species in urban or grassland habitats [START_REF] Oertli | Ecological and seasonal patterns in the diversity of a species-rich bee assemblage (Hymenoptera: Apoidea: Apiformes)[END_REF][START_REF] Fortel | Decreasing abundance, increasing diversity and changing structure of the wild bee community (Hymenoptera: Anthophila) along an urbanization gradient[END_REF]. The high richness of cleptoparasitic species may indicate that populations of wild bees and their diversity are large enough to support the presence of cleptoparasitic species in this area [START_REF] Sheffield | The potential of cleptoparasitic bees as indicator taxa for assessing bee communities[END_REF]. Furthermore, no wild bee species were drastically dominant in our survey (Supp Table 1 [online only]), which might suggest a well-structured community [START_REF] Sheffield | The potential of cleptoparasitic bees as indicator taxa for assessing bee communities[END_REF][START_REF] Geslin | The bee fauna of large parks in downtown Paris, France[END_REF].

Conclusion

This survey was an attempt to provide a comprehensive overview of bee, hover fly, and bee fly species richness in a protected area typical of Mediterranean habitats across a 10-yr period [START_REF] Medail | Biodiversity hotspots in the Mediterranean basin: setting global conservation priorities[END_REF]. In addition to these surveys, we believe that some supplementary data coming from museums or private collections could be used to improve this list. In the global context of pollinator decline (Sánchez-Bayo and Wyckhuys 2019), surveys such as ours are especially important, as these taxa represent the main pollinator species in Mediterranean habitats such as phrygana and shrublands [START_REF] Herrera | Pollination relationships in southern Spanish Mediterranean shrublands[END_REF][START_REF] Herrera | Daily patterns of pollinator activity, differential pollinating effectiveness, and floral resource availability, in a summer-flowering mediterranean shrub[END_REF]. The occurrence and ecological traits of pollinator species should strongly contribute to reinforcing information about species to provide, for example, a conservation status for species still considered as data deficient by the IUCN Red List. We built the plant-pollinator network associated with these habitats to document pollinator diet and their dependency upon floral resources. To further refine this biodiversity inventory, future studies in this area should focus on habitats of community interest, such as Brachypodium (Poales: Poaceae) grasslands or grassland on dolomites, which could be strategic habitats for less abundant or rare pollinator species in our surveys. Preserved or hard-to-reach habitats, such as the small islands (the Maïre, Riou, Plane, and Jarre islands) or the military camps, within the park should also be inventoried. Diversifying the season or the time slot of surveys (e.g., autumnal, nocturnal surveys), as well as the capture methods (e.g., Malaise traps), should lead to more inventoried species. Finally, the networks built from the pollen carried by bees have still been insufficiently explored and could greatly improve and supplement our knowledge of bee floral diets.

Fig. 1 .

 1 Fig. 1. Terrestrial limits and habitats of Calanques National Park. Crosses and dots correspond to each sampling plot. Crosses correspond to one-time inventories (only one session performed), whereas dots correspond to regular inventories (at least two sessions per year).

Fig. 2 .

 2 Fig. 2. Interaction network between plant species (left part) and pollinator species (right part) within Calanques National Park. For color figure refer online version.Threatened pollinator species and their links are highlighted in red, while threatened plant species are highlighted in green. Cistus albidus, Rosmarinus officinalis, and Thymus vulgaris, which correspond to the dominant flowering plant species, are colored pink, blue, and purple, respectively. The most generalist and dominant bee species, Apis mellifera, is highlighted in yellow, as are its links.

Table 1 .

 1 Details of each sampling effort conducted from 2009 to 2018 within Calanques National Park

	Sampling	Taxa caught	Capture method	Number of	Date	Number of	Collector	Pollen	Networks
	campaign			capture sessions		sites		analyses	data
	1	Bees and hover flies	Net	One-time	April 2009	4	X. Lair
	2	Bees	Net	One-time	May and June 2009	3	M. Aubert
	3	Bees	Net	One-time	May and July 2010	2	M. Aubert
	4	Bees	Net	One-time	Feb. and April 2011	3	M. Aubert
	5	Bees	Net	One-time	May 2016	3	Bee course attendants
							(Geslin et al. 2018)
	6	Bees and hover flies	Net	Weekly	Mar.-April 2017	6	L. Schurr		X
							(Schurr et al. 2019)
	7	Bees, hover flies, and	Net	Weekly	Feb.-June 2017	17	L. Ropars	X	X
		bee flies
	8	Bees, hover flies, and	Net	Weekly	Mar.-June 2018	6	C. Jaworski		X
		bee flies
	9	Bees, hover flies,	Net and Pan traps	Weekly	Feb.-June 2018	17	L. Ropars	X	X
		and bee flies
	10	Bees	Net	One-time	May 2018	2	D. Genoud

Table 2 .

 2 Richness estimations of hover fly, bee fly, and bee species

	Groups	Hover flies and bee flies		Bees
	Variables	Values	95%	Values	95%
			Confidence		Confidence
			interval		interval
	Abundance	219	-	2,551	-
	Number of species	58	-	192	-
	Chao1	86	[69; 134]	250	[220; 310]
	Jackknife1	83	[72; 101]	241	[225; 264]
	Jackknife2	97	[79; 129]	269	[242; 310]

Acknowledgments

We are thankful to Calanques National Park, the Observatoire Hommes-Milieux Littoral méditerranéen (OHM-Littmed), and the AXA postdoctoral fellowship for the financial support. We also thank Calanques National Park and its territory owners for the access permission. We are grateful to E. Dufrêne for the cleptoparasitic bee identifications. We are thankful to C. Lauriaut, B. Tissot, and T. Toth and for hover fly and bee fly identifications. We thank M. Berenger, A. Bideau, P. Caillault, V. Calvert, M. Hamon, A. Maléchaux, J.-Y. Meunier, B. Oursel, M. Panchot, C. Scagliola, A. Syx, and M. Zakardjian, who participated in bee samplings in the field. We are thankful to B. Gauzens for his help in the description of the network. We thank the two anonymous reviewers who made helpful comments on the previous version of the manuscript.

Supplementary Data

Supplementary data are available at Environmental Entomology online.


Pierre Etoré
email: pierre.etore@univ-grenoble-alpes.fr

Miguel Martinez
email: miguel.martinez@u-pem.fr

Stochastic processes associated to multidimensional parabolic transmission problems in divergence form

Keywords: 2010 Mathematics Subject Classification. 60G0, 60G12, 35K67, Probabilistic interpretation of divergence form operators, Dirichlet forms, skew diffusions, transmission problems. Miguel

 for general divergence form operators. We show that X is a semimartingale and that it is solution of a stochastic differential equation involving partial reflections in the co-normal directions along the transmission boundary.

Introduction

In this note we aim at tying -with all the necessary rigor -various theoretical results that come from different approaches concerning the probabilistic study of divergence form operators. We also provide a probabilistic representation of the underlying process when the divergence operator is a transmission operator involving a transmission condition across some smooth interface: in this particular case, we show that the process is solution of a stochastic differential equation involving partial reflections in the co-normal directions along the transmission boundary. All of these results are natural but we could not find them in the existing literature and would like to record them in print with a proof as they ground the foundations for the study of probabilistic numerical methods for transmission problems (see e.g. [START_REF] Bossy | Probabilistic interpretation and random walk on spheres algorithms for the Poisson-Boltzmann equation in molecular dynamics[END_REF]; [START_REF] Étoré | A transformed stochastic Euler scheme for multidimensional transmission PDE[END_REF]).

Our starting point is a : R d → M d (R) some measurable symmetric matrix valued coefficient satisfying the following ellipticity and boundedness condition (E -B):

Assumption 1.1. (E-B): There exists λ, Λ ∈ (0, ∞) such that

∀x ∈ R d , ∀ξ ∈ R d , λ|ξ| 2 ≤ ξ * a(x)ξ ≤ Λ|ξ| 2 .
(1.1)

Let us associate to the coefficient a the following unbounded operator A :

D(A) ⊂ L 2 (R d ) → L 2 (R d ) defined by D(A) = u ∈ H 1 (R d ) with d i,j=1 D i (a ij D j u) ∈ L 2 (R d ) and ∀u ∈ D(A), Au = d i,j=1 D i (a ij D j u).
There exists a closed symmetric Dirichlet form (E, D [E]) and its corresponding semigroup (T t ) on L 2 (R d ) that are naturally in link with (A, D(A)).

We define rigorously these objects and study their relations in Section 2. Using the spectral resolution of the identity associated to (A, D(A)), we study the regularity in the 'time variable t' of E(T t f, g), f ∈ L 2 (R d ), g ∈ D[E] (Subsection 2.1). This permits to establish rigorously in Subsection 2.2 the connection with the results in [START_REF] Stroock | Diffusion semigroups corresponding to uniformly elliptic divergence form operators[END_REF] that are exposed by D.W. Stroock in the C b (R d ) setting (Feller semigroup) and to assert the validity of Aronson's estimates for the transition function of (T t ) (see [START_REF] Aronson | Bounds for the fundamental solution of a parabolic equation[END_REF], [START_REF] Stroock | Diffusion semigroups corresponding to uniformly elliptic divergence form operators[END_REF], [START_REF] Bass | Diffusions and elliptic operators[END_REF]).

We then aim at providing tractable (from a numerical perspective) stochastic representations for the Hunt process X associated to (E, D, E).

Of course, we are in the ideal setting to apply the stochastic calculus for symmetric Dirichlet forms, and we give the Fukushima decomposition of X via the Revuz correspondence for additive functionals that is presented in Subsection 3.1. In this general setting a representation of X may be also provided by the so-called Lyons-Zheng decomposition involving reversed-time martingale increments: we give a brief insight of the ideas behind this theoretical decomposition in Subsection 3.2. Please note that none of the results presented in Section 3 are new (see [START_REF] Fukushima | Dirichlet forms and symmetric Markov processes[END_REF], [START_REF] Lyons | Diffusion processes with nonsmooth diffusion coefficients and their density functions[END_REF]) and we have tried our best to present the ideas in a coherent and assimilable way for a reader that might not be familiar with the subject.

Then, gradually moving from broad issues to more specific ones, we focus in Section 4 on the particular case where (A, D(A)) is a transmission operator across some transmission boundary: we present a Skorokhod representation of the Hunt process X associated to (E, D[E]) in this case. This result is new and constitutes the main contribution of this note (see Subsection 4.1). Finally, we give a special attention to the particular case of a diagonal coefficient matrix a that remains constant on each side of the transmission boundary and compare our result to the one obtained in the pionneering paper [START_REF] Bossy | Probabilistic interpretation and random walk on spheres algorithms for the Poisson-Boltzmann equation in molecular dynamics[END_REF] (see Subsection 4.2). We show that, when reduced to this very specific context, our description essentially matches the stochastic differential equation considered in [START_REF] Bossy | Probabilistic interpretation and random walk on spheres algorithms for the Poisson-Boltzmann equation in molecular dynamics[END_REF].

Dirichlet form and Markovian semigroup associated to general elliptic divergence form operators

2.1. Definitions and first properties. To the coefficient matrix a, we may associate a closed symmetric Dirichlet form (E,

D[E]) defined on L 2 (R d ) by          D [E] = H 1 (R d ), E(u, v) = d i,j=1 R d a ij D j u D i v, u, v ∈ D [E]
(see Fukushima et al. (1994), p111). This closed symmetric Dirichlet form is the starting point of our construction.

On the underlying Hilbert space L 2 (R d ), we denote within this subsection by (A, D(A)) the (unique) self-adjoint operator associated to (E, D [E]) and characterized by (Fukushima et al. (1994), Theorem 1.3.1 and Corollary 1.3.1 p.21).

D(A) ⊂ D [E] , E(u, v) = -Au, v L 2 (R d ) , u ∈ D(A), v ∈ D [E]
We aim at identifying this operator -as expected it will turn out that (A, D(A)) is nothing else than the operator defined in the Introduction, therefore the common notation.

By the very definition of (A, D(A)), we have for any f ∈ D(A) and any

g ∈ C ∞ c (R d ) -Af, g L 2 (R d ) = E(f, g) = d i,j=1 R d a ij D j f D i g = - d i,j=1 D i (a ij D j f ), g H -1 (R d ),H 1 (R d )
where

d i,j=1 D i (a ij D j f ) is understood in the distributional sense as an element of H -1 (R d ). But as Af ∈ L 2 (R d ) by the definition of D(A) the above equality shows that d i,j=1 D i (a ij D j f ) ∈ L 2 (R d ) (for any f ∈ D(A)).
Thus, it is proved that

D(A) ⊆ {f ∈ H 1 (R d ) with d i,j=1 D i (a ij D j f ) ∈ L 2 (R d )}. In turn (by the density of C ∞ c (R d ) in L 2 (R d ))
the equality permits to identify for any f ∈ D(A),

Af = d i,j=1 D i (a ij D j f ).

Let us now prove the reverse inclusion

{f ∈ H 1 (R d ) with d i,j=1 D i (a ij D j f ) ∈ L 2 (R d )} ⊆ D(A). Let f ∈ {f ∈ H 1 (R d ) with d i,j=1 D i (a ij D j f ) ∈ L 2 (R d )}.
By the symmetry of the coefficient matrix a and integration by parts, it is not hard to prove that for any v ∈ D(A), [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF]). So that we get the reverse inclusion

Av, f L 2 (R d ) = -E(v, f ) = - d j,i=1 R d a ji D i f D j v = j,i D j (a ji D i f ), v L 2 (R d ) and in particular f ∈ D(A * ) def = {g ∈ L 2 (R d ) | ∃h g ∈ L 2 (R d ) s.t. Av, g = v, h g , ∀ v ∈ D(A)} (see
{f ∈ H 1 (R d ) with d i,j=1 D i (a ij D j f ) ∈ L 2 (R d )} ⊆ D(A * ) = D(A)
where the equality comes from the fact that (A, D(A)) is self-adjoint. Finally, we have proved

D(A) = {f ∈ H 1 (R d ) with d i,j=1 D i (a ij D j f ) ∈ L 2 (R d )} (2.1)
and (A, D(A)) is fully identified as being the same operator of the Introduction 1. Note that since a is only assumed to be measurable, C ∞ c (R d ) -which is a core for the Dirichlet form (E, D(E)) -is not even a subset of D(A).

We now turn to the study of the spectral resolution and the semigroup associated to (E, D[E]) and (A, D(A)). For the sake of conciseness we denote (•,

•) = •, • L 2 (R d ) and || • || = || • || L 2 (R d )
till the end of the section.

Since (-A, D(A)) is a self-adjoint operator on the Hilbert space L 2 (R d ) that is non-negative definite, it admits a spectral resolution of the identity

{E γ : γ ∈ [0, ∞)}. For any γ ≥ 0 the operator E γ : L 2 (R d ) → L 2 (R d ) is a self-adjoint projection operator with (E γ f, f ) ≥ 0, f ∈ L 2 (R d ),
and the E γ 's form a spectral family with in particular E µ E γ = E µ∧γ , (see [START_REF] Fukushima | Dirichlet forms and symmetric Markov processes[END_REF] p18 for a list of properties). The link with (-A, D(A)) is through Fukushima et al. (1994) paragraph 1.3.4 p.18).

(-Af, g) = [0,∞) γd(E γ f, g) ∀f ∈ D(A), g ∈ L 2 (R d ) and D(A) = f ∈ L 2 (R d ) : [0,∞) γ 2 d(E γ f, f ) < ∞ (see
Consequently, the family of operators {T t def = e tA : t > 0} is a strongly continuous semigroup of self-adjoint contractions acting on L 2 (R d ) (Fukushima et al. (1994) Lemma 1.3.2 p.19) and

(T t f, g) = [0,∞) e -γt d(E γ f, g) ∀f ∈ L 2 (R d ), g ∈ L 2 (R d ).
Note that for any γ ≥ 0, t > 0, and any functions f ∈ L 2 (R d ) and g ∈ L 2 (R d ), we have the commutation property

(T t E γ f, g) = (E γ f, T t g) = [0,∞) e -ξt d ξ (E γ f, E ξ g) = [0,γ] e -ξt d ξ (E ξ E γ f, g) + [γ,∞) e -ξt d ξ (E ξ E γ f, g) = [0,γ] e -ξt d ξ (E ξ f, g) = [0,∞) e -ξt d ξ (E ξ f, E γ g) = (T t f, E γ g) = (E γ T t f, g).
Note also that for any f ∈ L 2 (R d ) and any t > 0,

[0,∞) γ 2 d(E γ T t f, T t f ) = [0,∞) γ 2 d γ [0,∞) e -ξt d ξ (E γ E ξ f, T t f ) = [0,∞) γ 2 d γ [0,∞) e -ξt d ξ [0,∞) e -θt d θ (E γ E ξ f, E θ f ) = [0,∞) γ 2 e -2γt d γ (E γ f, f ) ≤ 4 t 2 e -2 [0,∞) e -γt d γ (E γ f, f ) = 4 t 2 e -2 (T t f, f ) ≤ 4 t 2 e -2 ||f || 2 < +∞,
where we have used the spectral family property, the associativity of the Stieltjes integral and the inequality γ 2 e -γt ≤ 4e -2 /t 2 . The above inequality ensures that T t f ∈ D(A) for any t > 0.

From the fact that | d dt e -γt | ≤ γ is integrable w.r.t. d(E γ h, g) whenever h ∈ D(A), we deduce from the commutation property that for any f, g ∈ L 2 (R d ) and for any s > 0

- d dt (T t f, T s g) = [0,∞) γe -γt d(E γ f, T s g) ---→ t 0+ [0,∞) γd(E γ T s f, g) = (-AT s f, g)
where the limit exists and is well defined (since we have shown that

T s f ∈ D(A)). If moreover g ∈ D [E] then - d ds (T s f, g) = - d dt (T s+t f, g)| t=0+ = - d dt (T t f, T s g)| t=0+ = (-AT s f, g) = E(T s f, g). (2.2)
And sinced ds (T s f, g) = -d ds (T s g, f ) by the symmetry property of T s , we deduce that

E(T s f, g) = E(f, T s g) for any f, g ∈ D [E].
Consequently, for any f ∈ D [E] and using the ellipticity of the coefficient matrix a,

λ||∇T s f || 2 ≤ E(T s f, T s f ) = E(T 2s f, f ) = (-AT 2s f, f ) = [0,∞) γe -2γs d(E γ f, f ) ≤ e -1 s [0,∞) e -γs d(E γ f, f ) = e -1 s (T s f, f ) ≤ ||f || 2 s ,
from which we deduce the fundamental estimate

||∇T s f || ≤ ||f || √ λ s , ∀s > 0.
(2.3)

In turn this estimate implies that for any

f ∈ L 2 (R d ), g ∈ D [E], the function s → E(T s f, g) is integrable on (0, t],
and from (2.2) and the right continuity of s → T s f at time s = 0+ (one may extend

T 0 f = f as long as no differentiation of s → T s f is implied at s = 0+ when f / ∈ D(A)), we deduce the integrated version of (2.2) namely ∀f ∈ L 2 (R d ), ∀g ∈ D [E], (T t f, g) -(f, g) = - t 0 d i,j=1 (a ij D j T s f, D i g) ds = - t 0 E(T s f, g)ds, t ∈ (0, ∞).
(2.4) 2.2. Link with the results of D.W. [START_REF] Stroock | Diffusion semigroups corresponding to uniformly elliptic divergence form operators[END_REF]. In his celebrated article Diffusion semigroups corresponding to uniformly elliptic divergence form operators D.W. Stroock constructs via a regularization procedure a Feller continuous semigroup {P t : t > 0} associated to a with the properties that (with our notations)

(1) the map t ∈ [0, ∞) → P t φ ∈ H 1 (R d ) is a weakly continuous map for each φ ∈ C ∞ c (R d ). (2) ∀φ, ψ ∈ C ∞ c (R d ), (P t φ, ψ) -(φ, ψ) = - t 0 (a∇P s φ, ∇ψ) ds = - t 0 E(P s φ, ψ)ds, t ∈ (0, ∞).
(2.5) (Nota: please note that there is a sign error in the original version of [START_REF] Stroock | Diffusion semigroups corresponding to uniformly elliptic divergence form operators[END_REF]). In fact, {P t : t > 0} determines a unique strongly continuous semigroup

{ Pt : t > 0} of self-adjoint contractions on L 2 (R d ).
The aim of this subsection is to prove the following equality, which clarifies the relationship between the results obtained in [START_REF] Stroock | Diffusion semigroups corresponding to uniformly elliptic divergence form operators[END_REF] and the those provided by the theory of Dirichlet forms [START_REF] Fukushima | Dirichlet forms and symmetric Markov processes[END_REF].

Proposition 2.1. { Pt : t > 0} = {T t : t > 0} on L 2 (R d ).
(2.6)

Proof : The semigroup { Pt : t > 0} is strongly continuous on H 1 (R d ). Moreover, for each t > 0, the operator Pt maps L 2 (R d ) into H 1 (R d ) and for each f ∈ H 1 (R d ) = D [E],
we have the fundamental estimate

||∇ Ps f || ≤ 1 √ λ ||f || √ s ∧ ||∇f ||, ∀s > 0.
(2.7) (See [START_REF] Stroock | Diffusion semigroups corresponding to uniformly elliptic divergence form operators[END_REF] Theorem II.3.1. p.341). This estimate implies that for for each f, g ∈ D [E] and any t, s > 0,

|E( Pt f, g) -E( Ps f, g)| ≤ Λ||∇g|| || Pt∨s-t∧s f -f || λ (t ∧ s) --→ s→t 0, which ensures the continuity of s → E( Ps φ, ψ) for any φ, ψ ∈ C ∞ c (R d ). Since ( Pt ) t>0 and (P t ) t>0 coïncide on C ∞ c (R d ), we may differentiate in (2.5) (as long as t > 0) to find that d dt ( Pt φ, ψ) = -E( Pt φ, ψ), t ∈ (0, ∞). (2.8)
This has to be compared to (2.2).

Let us now justify rigorously that for any t > 0, s ∈ (0, t) and φ, ψ

∈ C ∞ c (R d ), d ds (T s φ, Pt-s ψ) = d du (T u φ, Pt-s ψ)| u=s - d du (T s φ, Pt-u ψ)| u=s .
(2.9)

We have for sufficiently small 0 ≤ h < t -s and using the strong continuity of

(P t ) t>0 | T s+h φ -T s φ, Pt-s+h ψ -Pt-s ψ | ≤ || Pt-s+h ψ -Pt-s ψ|| ||T s+h φ -T s φ|| ≤ ε ψ (h) [0,∞) e -2γs (e -γh -1) 2 d(E γ φ, φ) 1/2 ≤ ε ψ (h) [0,∞) e -2γs (γh) 2 d(E γ φ, φ) 1/2 ≤ hε ψ (h) [0,∞) e -γs γ 2 e -γs d(E γ φ, φ) 1/2 ≤ hε ψ (h) 2e -1 s ||φ||,
where as usual ε ψ (•) denotes some positive continuous function vanishing at zero. We deduce that

1 h | T s+h φ -T s φ, Pt-s+h ψ -Pt-s ψ | ---→ h→0 0,
implying (2.9). Hence, from (2.9) and applying (2.2) and (2.8), we have that

d ds (T s φ, Pt-s ψ) = 0, s ∈ (0, t).
(2.10)

Integrating the identity (2.10) on (0, t) and using the time continuity of both semigroups (T t ) and ( Pt ) up to time s = 0+ gives

(T t φ, ψ) = (φ, Pt ψ) = ( Pt φ, ψ) (2.11) which holds for any φ, ψ ∈ C ∞ c (R d ). Since C ∞ c (R d ) is dense in L 2 (R d
), using the strong continuity of both semigroups (T t ) and ( Pt ), we finally deduce from (2.11) the identification (2.6).

Consequently, all results in [START_REF] Stroock | Diffusion semigroups corresponding to uniformly elliptic divergence form operators[END_REF] that are valid for { Pt : t > 0} are true for {T t : t > 0}. For example, identifying abusively {T t : t > 0} with its Feller restriction

{P t : t > 0} on C ∞ c (R d ), we deduce that there is a p ∈ C (0, ∞) × R d × R d such that [T t φ] (x) = R d φ(y)p(t, x, y)dy, (dx) -a.e., φ ∈ C ∞ c (R d ).
(2.12)

Moreover, the fundamental function p satisfies the well-known Aronson's estimates for the fundamental solutions of elliptic divergence form operators, namely there exists a constant

M (λ, Λ, d) ∈ [1, ∞) such that 1 M t d/2 exp -M |x -y| 2 /t ≤ p(t, x, y) ≤ M t d/2 exp -|x -y| 2 /M t .
(2.13) Finally, we have the convergence result of Stroock (1988) (Theorem II.3.1. p.341) that we state roughly without introducing the necessary notations (see [START_REF] Stroock | Diffusion semigroups corresponding to uniformly elliptic divergence form operators[END_REF] for details): if {a n } ∞ 1 ⊂ A(λ, Λ) and a n -→ a almost everywhere, then p n (t, x, y) -→ p(t, x, y) uniformly on compacts

(in (0, ∞) × R d × R d ) and for each t ∈ [0, ∞) and φ ∈ C ∞ c (R d ), T n t φ -→ T t φ in H 1 (R d ).

Associated stochastic processes

Since

(E, D [E]) is a regular Dirichlet form (with the space D [E] ∩ C c (R d ) or C ∞ c (R d
) as a special standard core, see e.g. Exercice 1.4.1 in [START_REF] Fukushima | Dirichlet forms and symmetric Markov processes[END_REF]), we are in position to apply Theorem 7.2.1 p. 380 of [START_REF] Fukushima | Dirichlet forms and symmetric Markov processes[END_REF].

We may associate to (E, D [E)] and its corresponding semigroup (T t ) a Hunt process, symmetric w.r.t the Lebesgue measure (dx) on R d . We shall denote by

M = (Ω, (F t ) t≥0 , F, (X t ) t≥0 , (P x ) x∈R d ) this Hunt process, with X = (X 1 , . . . , X d ). The correspondence with (E, D [E]) and (T t ) is through E x [f (X t )] = T t f (x), ∀f ∈ L 2 (R d ), ∀t ≥ 0, ∀x ∈ R d (3.1)
(see the discussion p160 in [START_REF] Fukushima | Dirichlet forms and symmetric Markov processes[END_REF], at the beginning of Section 4.2). The aim of this section is to present various representations of M in various contexts. We start with the general case and then present a Skorokhod decomposition in the specific case where our Dirichlet form is associated to a transmission parabolic operator in divergence form.

3.1. The Revuz correspondence for additive functionals and the Fukushima decomposition.

α > 0, f ∈ B b (R d ) and x ∈ R d , R α f (x) = R d r α (x, y)f (y)dy with r α (x, y) = ∞ 0 e -αt p(t, x, y)dt. Denote by S the set of positive Radon measures on (R d , B(R d )). For µ ∈ S define R 1 µ(x) = R d r 1 (x, y)µ(dy) (x ∈ R d )
and introduce the subset of finite energy measures

S 0 := µ ∈ S : sup v∈D[E]∩Cc(R d ) R d |v(x)| ||v|| E 1 µ(dx) < ∞ ,
(where we follow the notations of [START_REF] Fukushima | Dirichlet forms and symmetric Markov processes[END_REF]).

Finally, introduce

S 00 := {µ ∈ S 0 : µ(R d ) < ∞, ||R 1 µ(.)|| ∞ < ∞}.
Let us denote respectively by A + c and A + c,1 the families of all Positive Continuous Additive Functionals (PCAF in short) (resp. the family of all PCAF in the strict sense) associated to M (for the distinction between A + c and A + c,1 , see [START_REF] Fukushima | Dirichlet forms and symmetric Markov processes[END_REF] the introduction of Section 5.1). The Revuz correspondence asserts that there is a one-to-one correspondence (up to equivalence of processes) between A + c and S. This correspondence permits to construct for any µ ∈ S 00 a unique PCAF in the strict sense

A ∈ A + c,1 such that ∀x ∈ R d , E x ∞ 0 e -t dA t = R 1 µ(x). (3.2)
(see for e.g. Theorem 5.1.4 in [START_REF] Fukushima | Dirichlet forms and symmetric Markov processes[END_REF]).

In order to get a bijective map, introduce a new subset S 1 of S defined by µ ∈ S 1 if there exists a sequence (E n ) n≥0 of Borel finely open sets increasing to R d satisfying that I En .µ ∈ S 00 for each n. Then, there is a one-to-one correspondence between S 1 and A + c,1 (up to equivalence) which is given by relation (3.2) whenever µ ∈ S 00 . The set of measures S 1 is called the set of smooth measures (in the strict sense). For u ∈ D [E] b , we may associate a unique positive Radon measure µ u ∈ S, satisfying

R d f (x)µ u (dx) = 2E(uf, u) -E(u 2 , f ), ∀f ∈ D [E] ∩ C c (R d ). (3.3)
Observe that the positivity of the right hand side in (3.3) comes from

0 ≤ E x (u(X t ) -u(x)) 2 = E x u 2 (X t ) -2u(x)u(X t ) + u 2 (x) = (T t u 2 -u 2 )(x) -2u(x)(T t u -u)(x).
Taking the scalar product with

f ∈ D [E] ∩ C c (R d ) and dividing by t gives 0 ≤ T t u 2 -u 2 t , f -2 T t u -u t , uf
which tends to 2E(uf, u) -E(u 2 , f ) as t tends to 0+ whenever u ∈ D(A); the positivity in the case u ∈ D [E] b may be obtained by a density argument.

If u ∈ D [E] b,loc , we may construct µ u ∈ S with the help of a sequence (G n ) n≥0 of relatively compact open sets such that G n ⊂ G n+1 and n≥0 G n = R d . Let (u n ) n≥0 a sequence of functions in D [E] b satisfying u n = u on G n .
There is no ambiguïty in defining µ u = µ un on G n because the construction is consistent (since µ un = µ u n+1 on G n ). For an account on the above assertions, please refer to [START_REF] Fukushima | Dirichlet forms and symmetric Markov processes[END_REF] Section 3.2.

3.1.2. The Fukushima decomposition. Note that obviously (E, D [E]) is strong local, so we may apply Theorem 5.5.5 in [START_REF] Fukushima | Dirichlet forms and symmetric Markov processes[END_REF].

Suppose that a function u satisfies the following conditions:

(1) u ∈ D [E] b,loc , u is finely continuous on R d .

(2) I G .µ u ∈ S 00 for any relatively compact open set G.

(3) ∃ = (1) -(2) with I G .

(1) , I G .

(2) ∈ S 00 for any relatively compact open set G and

E(u, v) = ( , v), ∀v ∈ C ∞ c (R d ). (Note that even though u is not formally in D [E], the quantity E(u, v) is well-defined because v has compact support and u ∈ D [E] b,loc ).
Let A (1) , A (2) , and B be PCAF's in the strict sense with Revuz measures (1) , (2) , and µ u respectively. Then, Theorem 5.5.5 in [START_REF] Fukushima | Dirichlet forms and symmetric Markov processes[END_REF]

asserts that u(X t ) -u(X 0 ) = M [u] t + N [u] t , P x -a.s, ∀x ∈ R d .
(3.4)

Here,

N [u] = -A (1) + A (2) , P x -a.s, ∀x ∈ R d (3.5)
and M [u] is a local Additive Functional in the strict sense such that for any relatively compact set G,

E x M [u] t∧τ G = 0, ∀x ∈ G and E x (M [u] t∧τ G ) 2 = E x B t∧τ G , ∀x ∈ G, where τ G = inf(s > 0 : X s /
∈ G) stands for the first leaving time from G (with the convention inf ∅ = ∞) and B denotes the PCAF in the strict sense with Revuz measure µ u .

3.2. An insight to the Lyons-Zheng decomposition for diffusions associated to divergence form operators. For the seek of completeness, in this paragraph we present briefly an insight on the Lyons-Zheng decomposition [START_REF] Lyons | Diffusion processes with nonsmooth diffusion coefficients and their density functions[END_REF] for the stochastic process in correspondence with a divergence form operator.

3.2.1. Time reversal for diffusions associated to divergence form operators. Assume for a moment that a ∈ A (λ, Λ) is very smooth and belongs to

C ∞ R d → M d (R) .
In this smooth case, the process {X t = (X 1 t , . . . , X d t ) : 0 ≤ t ≤ T } in correspondence with (E, D [E]) and constructed in the previous section becomes a diffusion process with values in R d and well-known results from the general theory for solutions of stochastic differential equations ensure that (X t ) is solution of

X k t = x k + t 0 d j=1 σ kj (X s )dW j s + t 0 d j=1 ∂ j a kj (X s )ds t ≥ 0, P x -a.s., ∀x = (x 1 , . . . , x d ) ∈ R d , (3.6)
where W = (W 1 , . . . , W d ) denotes a d-dimensional standard Brownian motion starting from zero and σ : R d → M d (R) denotes the positive square-root of coefficient 2a i.e. the positive matrix real valued coefficient satisfying

σσ * (x) = 2a(x), ∀x ∈ R d .
We will denote by L the classical generator of (X t ),

L = d i,j=1 ∂ i (a ij ∂ j ) = d i,j=1 a ij ∂ 2 ij + d i,j=1 (∂ i a ij ) ∂ j (3.7)
acting on C 2 (R d ) real valued functions. Fix x = (x 1 , . . . , x d ) ∈ R d . For any fixed 0 ≤ t < 1, denote L

x t the second order differential operator

L x t = i,j=1 a ij ∂ 2 ij - d i,j=1 (∂ i a ij ) ∂ j + [p(1 -t, x, .)] -1 d i,j=1 ∂ j ((2a ij ) p(1 -t, x, .)) ∂ i
acting also on C 2 (R d ) real valued functions and where p denotes the fundamental solution (2.12).

(Nota: the reader should be careful with the 2 factor appearing in the last integral, that is due to the fact that we are considering σ as the square root of 2a and not a. Also, since a is assumed to be smooth, classical estimates for p ensure that the term p(1 -t, x, .

) -1 ∂ j ((2a ij ) p(1 -t, x, .)) ∂ i is well defined).
Consider {X t := X 1-t : t ∈ [0, 1)} the time reversed process of (X t ). The fact that the time reversal of a Markov process is again a (weak) Markov process with respect to the reversed filtration traces back to the seminal result of K.L. Chung and J.B. Walsh [START_REF] Chung | To reverse a Markov process[END_REF]. Moreover, it is proved in [START_REF] Haussmann | Time reversal of diffusions[END_REF] that X t 0≤t<1 is a diffusion process with generator L up to time 1 excluded. The process

{X t = (X 1 1-t , . . . , X d 1-t ) : 0 ≤ t < 1} is solution of X k t = X 1 + t 0 d j=1 σ kj (X s )dβ j s - t 0 d j=1 ∂ j a kj (X s )ds + t 0 p(1 -t, x, X s ) -1 d i,j=1 ∂ j (2a ij ) p(1 -s, x, X s ) ds, P x -a.s., ∀x = (x 1 , . . . , x d ) ∈ R d ,
where β = (β 1 , . . . , β d ) denotes a d-dimensional standard Brownian motion starting from zero and adapted to the filtration

F X t := σ (X 1-u : u ∈ [0, t]) | t ∈ [0, 1) .
3.2.2. The Lyons-Zheng decomposition. We now make the following observation. Fix 0 < ε < 1,

then for any ϕ ∈ C ∞ c (R d ) and arbitrary t ∈ [ε, 1] ϕ(X t ) -ϕ(x) = 1 2 (ϕ(X t ) -ϕ(x)) + 1 2 ϕ(X 1-(1-t) ) -ϕ(X 1-(1-ε) ) + (ϕ(X ε ) -ϕ(x)) = 1 2 (ϕ(X t ) -ϕ(x)) - 1 2 ϕ(X 1-ε ) -ϕ(X 1-t ) + 1 2 (ϕ(X ε ) -ϕ(x))
and applying Itô's formula, we find that for all t ∈ [ε, 1]

ϕ(X t ) -ϕ(x) = 1 2 t 0 (∇ϕ(X u ), σ(X u )dW u ) + t 0 Lϕ(X u )du - 1 2 1-ε 1-t ∇ϕ(X u ), σ(X u )dβ u - 1-ε 1-t L x u ϕ(X u )du + 1 2 (ϕ(X ε ) -ϕ(x)) = 1 2 M ϕ t + 1 2 Ñ ϕ,ε t + 1 2 t ε (L -L x 1-u )ϕ(X u )du + 1 2 ϕ(X ε ) -ϕ(x) + ε 0 Lϕ(X u )du (3.8)
where we have set

M ϕ t := t 0 (∇ϕ(X u ), σ(X u )dW u ) and Ñ ϕ,ε t := - 1-ε 1-t ∇ϕ(X u ), σ(X u )dβ u .
We may write

Ñ ϕ,ε t = - 1-ε 1-t ∇ϕ(X u ), σ(X u )dβ u = 1-t 0 ∇ϕ(X u ), σ(X u )dβ u - 1-ε 0 ∇ϕ(X u ), σ(X u )dβ u = N ϕ 1-t -N ϕ 1-ε ,
where N ϕ θ stands for the

F X martingale θ 0 ∇ϕ(X u ), σ(X u )dβ u θ∈[0,1) . Now for any z ∈ R d 1 2 L -L x 1-u ϕ(z) = d i,j=1 (∂ i a ij ) ∂ j ϕ(z) -[p(u, x, z)] -1 d i,j=1 ∂ j (a ij p(u, x, z)) ∂ i ϕ(z) =   d i,j=1 (∂ i a ij ) ∂ j ϕ(z) - d i,j=1 (∂ j a ij ) ∂ i ϕ(z)   -[p(u, x, z)] -1 d i,j=1 a ij ∂ j p(u, x, z)∂ i ϕ(z) = -[p(u, x, z)] -1 d i,j=1 a ij ∂ j p(u, x, z)∂ i ϕ(z)
where terms cancel due to the symmetry of the coefficient matrix a.

Of course lim

ε 0+ ϕ(X ε ) -ϕ(x) + ε 0
Lϕ(X s )ds = 0. Applying the martingale convergence theorem ensures that the martingale term N ϕ 1-ε tends P x -a.s. to

N ϕ 1 = 1 0 ∇ϕ(X u ), σ(X u )dβ u
as ε tends to zero.

In turn, this ensures from (3.8) that the limit

lim ε 0+ t ε [p(u, x, X u )] -1 d i,j=1
a ij ∂ j p(u, x, X u )∂ i ϕ(X u )du (3.9) exists P x -a.s.

Coming back to (3.8) and taking limits as ε 0+ yields the Lyons-Zheng decomposition of (X t ) when the coefficient matrix a is smooth:

for all ϕ ∈ C ∞ c , ∀t ∈ [0, 1], P x -a.s. ϕ(X t ) -ϕ(x) = 1 2 M ϕ t + 1 2 Ñ ϕ t - t 0 [p(u, x, X u )] -1 d i,j=1 a ij ∂ j p(u, x, X u )∂ i ϕ(X u )du (3.10) with Ñ ϕ t = N ϕ 1-t -N ϕ 1
an increment of a time-reversed martingale. The quadratic variations of the martingales involved in the decomposition are given by

M ϕ t = t 0 d i,j=1 2a ij (X s )∂ i ϕ(X s )∂ j ϕ(X s )ds, N ϕ t = t 0 d i,j=1 2a ij (X s )∂ i ϕ(X s )∂ j ϕ(X s )ds.
We make the following observations:

• If we have proved (3.9), it is important to notice however that there is no limit to the deterministic quantity

t ε [p(u, x, z)] -1 d i,j=1
a ij ∂ j p(u, x, z)∂ i ϕ(z)du as ε tends to 0+. This can be easily seen by performing the computation from the explicit Laplacian case where p is just the Gaussian transition density of some Browian motion started from x.

• For any ε > 0 the random variable Ñ ϕ,ε t = -

1-ε 1-t ∇ϕ(X u ), σ(X u )dβ u is measurable w.r.t the sigma field σ X u : u ∈ [1 -t, 1 -ε] = σ (X u : u ∈ [ε, t])
so that Ñ ϕ t and all terms in (3.10) are adapted to F X t t∈[0,1] the natural filtration of X. • Only ϕ and its first order partial derivatives appear in equality (3.10). Using a density argument and a little work we may prove that the equality holds for ϕ ∈ H 1 (R d ).

• None of the quantities in the right hand side of (3.10) involve the derivatives of the coefficient matrix a: the dependence on the derivatives of the coefficient a is totally encompassed in the logarithmic derivative of the fundamental solution p(t, x, y). The idea is now to pick a measurable in A (λ, Λ) and to take a sequence of smooth {a n } ∞ 1 ⊂ A(λ, Λ) such that a n -→ a almost everywhere and to prove that there is convergence in law for the decomposition (3.10) for any ϕ belonging to a the widdest possible class of functions. This programm has been successively performed in Rozkosz (1996a) and Rozkosz (1996b) in a more general setting of an inhomogeneous divergence operator.

Though theoretically powerful, the Lyons-Zheng decomposition is unlikely to be directly exploitable from a numerical perspective, as we do not have access to the logarithm derivative of the transition density, even in mild cases. We denote Assume the a ij 's satisfy (a ± ) ij ∈ C( D± ; R). We may define then the co-normal vector fields γ + (x) := a + (x)ν(x) and γ -(x) := a -(x)ν(x), for x ∈ Γ.

Stochastic dynamics associated to transmission operators in divergence form

D = D + ∪ D -= R d \ Γ ⊂ R d .
We shall consider restricted operators and bilinear forms in the following sense. We define A + :

H 1 (D + ) → H -1 (D + ) by ∀v ∈ H 1 (D + ), A + v = d i,j=1 D i (a + ) ij D j v .
We define A -: H 1 (D -) → H -1 (D -) in the same manner (note that we do not specify here any domain D(A ± )). Further, we define

E ± (u, v) = d i,j=1 D ± (a ± ) ij D j u D i v, ∀u, v ∈ H 1 (D ± ).
We have, for

u ± ∈ H 1 (D ± ) with A ± u ± ∈ L 2 (D ± ), E ± (u ± , v) = D ± (-A ± u ± )v, ∀v ∈ H 1 0 (D ± ). (4.1)
Imagine now that in (4.1) we wish to take the test function in H 1 (D ± ) instead of H 1 0 (D ± ). There will still be a link between A ± and E ± , but through Green type identities, involving conormal derivatives and boundary integrals.

We introduce a specific notation for the one-sided conormal derivatives on

Γ of u ∈ L 2 (R d ) with u ± ∈ H 2 (D ± ). Provided the (a ± ) ij are in C 1 b ( D± ; R)
and Γ is bounded and Lipschitz we set

B ± ν u = ν * Tr ± (a ± ∇u ± ) = d i=1 d j=1 ν i Tr ± (a ± ) ij D j u ± on Γ (4.2)
where Tr ± : H 1 (D ± ) → H 1/2 (Γ) stand for the usual trace operators on Γ.

For g ∈ H -1 2 (Γ) and f ∈ H 1 2 (Γ) we denote by g, f Γ the action of g on f . If both f, g are in H 1 2 (Γ) the quantity g, f Γ coincides with the surface integral Γ gf dς. Let us recall the version of the Green identity that is used in the sequel.

2 . Let u ∈ L 2 (R d ) with u + ∈ H 2 (D + ) and u -∈ H 2 (D -). Assume that the coefficients (a ± ) ij are in C 1 b (D ± ; R). Then E ± (u ± , v) = D ± (-A ± u ± )v ∓ B ± ν u, Tr ± (v) Γ , ∀v ∈ H 1 (D ± ).
We have the following result.

Theorem 4.2. Assume Γ is bounded and C 2 . Assume that Assumption (E -B) is fulfilled and that for all

1 ≤ i, j ≤ d, a ij ∈ C 1 b (D + ; R) ∩ C 1 b (D -; R)
with a ij possessing a possible discontinuity on Γ. Then, the Hunt process M associated to (E, D [E]) is a diffusion which possesses the following Skorokhod decomposition: for any k ∈ {1, . . . , d},

X k t = x k + t 0 d j=1 σ kj (X s )dW j s + t 0 d j=1 ∂ j a kj (X s )I Xs∈D ds + 1 2 t 0 γ +,k (X s )dK s - 1 2 t 0 γ -,k (X s )dK s , t ≥ 0, P x -a.s., ∀x = (x 1 , . . . , x d ) ∈ R d . (4.3)
In the above equality σ : R d → R d×d denotes the positive square-root of coefficient 2a i.e. the positive matrix real valued coefficient satisfying

σσ * (x) = 2a(x), ∀x ∈ D.
(Note that this coefficient exists because a(x) is non-negative definite for all x ∈ D). The process W = (W 1 , . . . , W d ) is a d-dimensional standard Brownian motion starting from zero and (K t ) t≥0 denotes the unique PCAF associated to the surface measure ς(dξ) ∈ S on Γ through the Revuz correspondence. The process (K t ) increases only at times where X lies on Γ,

t 0 I Xs∈Γ dK s = K t , t ≥ 0.
Proof : We apply the results of Theorem 5.5.5 in [START_REF] Fukushima | Dirichlet forms and symmetric Markov processes[END_REF] in this context for the coordinate functions

p k (x 1 , . . . , x d ) := x k (k ∈ {1, . . . , d}).
f k ∈ D [E] b such that p k = f k on G. Let M [f k ] the square bracket of M [f k ] .
Then, an easy computation from (3.3) shows that the energy measure of

M [f k ] (the Revuz measure of M [f k ] ) is µ f k (dy) = µ M [f k ] (dy) = 2a(y)∇f k (y), ∇f k (y) (dy) (4.4)
and we know that µ f k = µ p k on G. It is easy to show that I G .µ p k is a finite Radon measure belonging to S 00 and that µ p k is a smooth measure. Then, an easy computation from (3.2) shows that

M [f k ] t = t 0 2a(X s )∇f k (X s ), ∇f k (X s ) ds, k ∈ {1, . . . , d}
and by the well-known results on stochastic representation of martingales, there exists a d dimensional Brownian motion W = (W 1 , . . . , W d ) such that

M [f k ] t = t 0 σ(X s )∇f k (X s ) * dW s , P x -a.s. ∀x ∈ R d , k ∈ {1, . . . , d}
(see for e.g. [START_REF] Revuz | Continuous martingales and Brownian motion, volume 293 of Grundlehren der mathematischen Wissenschaften[END_REF] Chapter V Theorem 3.9 and the remark following its proof).

Moreover, for any v ∈ C ∞ c (R d ), using the Green Identities of Proposition 4.1 and taking into account that v is of compact support, we have:

E(f k , v) = E + (f k,+ , v) + E -(f k,-, v) = D + (-A + f k,+ )v -B + ν f k,+ , Tr + (v) Γ + D - (-A -f k,-)v + B - ν f k,-, Tr -(v) Γ = - D d i,j=1 D i (a ij (y)D j f k (y)) v(y)I y∈D (dy) - Γ ν * Tr + (a + ∇f k,+ ) -Tr -(a -∇f k,-) γ(v)dς = - D d j=1 ∂ j a kj (y)v(y)I y∈D (dy) - Γ Tr + ((a + ν) k ) -Tr -((a -ν) k ) vdς = - D d j=1 ∂ j a kj (y)v(y)I y∈D (dy) - Γ [(ã + ν) k -(ã -ν) k ] vdς = ( + k , v) -( - k , v) with ± k (dy) := - d j=1 [∂ j a kj (y)] ± I y∈D (dy) + [(γ -) k -(γ + ) k ] ± (y)I y∈Γ ς(dy).
(here, the notation [a] + (resp. [a] -) stands for the positive (resp. negative) part of some real number a).

Let us now proceed to show that the measures I G . ± k belong to S 00 . Note that ||∂ j a kj I D∩G || ∞ < ∞ and from the definition of S 00 and the Revuz correpondence (3.2), it is not difficult to prove that the measures -[∂ j a kj ] ± (y)I y∈D (dy) are smooth with their corresponding additive functional writing as -

Thus, for all

h ∈ D [E] ∩ C c (R d ), we have Γ |h(y)|ς(dy) ≤ C 0 D + (|∇h(x)| + |h(x)|) (dx) ≤ C 0 (2 (D + )) 1/2 R d (|∇h(x)| 2 + |h(x)| 2 ) (dx) 1/2 ≤ C 0 (2 (D + )) λ (E(h, h) + (h, h)) 1/2
so that the surface measure ς(dy) belongs to S 0 . Since

∀y ∈ Γ, [| (γ -) k -(γ + ) k |(y)] ± ≤ 2|ã ± (y)ν(y)| ≤ 2Λ, the surface measures ζ ± k (dy) := [(γ -) k -(γ + ) k ] ± (
y)I y∈Γ ς(dy) belong also to S 0 .

Note that from Aronson's estimates (2.13) we retrieve the following estimations

r 1 (x, y) ≤ C|x -y| -(d-2) if d > 2 ; r 1 (x, y) ≤ C (ln(1/|x -y|) ∨ 1) if d = 2.
Then, using the same arguments as in [START_REF] Fukushima | Dirichlet forms and symmetric Markov processes[END_REF] (Example 5.2.2 p.255), we can assert that the measures ζ ± k (dy) belong to S 00 . Moreover, let (K t ) t≥0 denote the PCAF associated to ς(dy); in regard of the results stated in the original article of D. Revuz (cf. Revuz (1970) p.507) we may assert that

t 0 [(γ -) k -(γ + ) k ] ± (X s )I Xs∈Γ dK s t≥0
is the PCAF associated to ζ ± k (dy) via the Revuz correspondence.

By application of Theorem 5.5.5 in [START_REF] Fukushima | Dirichlet forms and symmetric Markov processes[END_REF] and since all the necessary hypothesis are fulfilled, we get the decomposition (4.3) on the set {t ≥ 0 : t ≤ τ Gq } where G q := {x ∈ R d : |x| < q}. The identification of the process for all times follows by letting q tend to infinity.

Let u 0 ∈ D(A). From the Hille-Yosida theorem [START_REF] Brezis | Analyse fonctionnelle. Collection Mathématiques Appliquées pour la Maîtrise[END_REF] Theorems VII.4 and VII.5) we can prove that there exists a unique function

u ∈ C 1 [0, T ]; L 2 (R d ) ∩ C [0, T ]; D(A) satisfying du dt = Au, u(0) = u 0 . (4.5)
where the first equality in (4.5) has to be understood in the weak sense. Under the hypothesis of Theorem 4.2, we deduce the following Corollary.

Corollary 4.3. Let 0 < T < ∞. Under the conditions of Theorem 4.2, for any u 0 ∈ D(A), we have

E x [u 0 (X t )] = u(t, x), ∀t ∈ [0, T ], ∀x ∈ R d , (4.6)
where X is the diffusion considered in Theorem 4.2 and u is the solution of (4.5).

In particular, the following transmission condition Proof : In view of (3.1) and since d dt T t u 0 = AT t u 0 [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF] Thm 2.4-c)) the function (t, x) → E x [u 0 (X t )] is solution of (4.5). We refer to the proofs of Proposition 3.14 and Theorem 3.1 in [START_REF] Étoré | A transformed stochastic Euler scheme for multidimensional transmission PDE[END_REF] for the verification of the other assertions.

a + ∇ x u + (t, y) -a -∇ x u -(t,
In the light of (4.6) and in order to compute an approximate value of u(t, x), one could think of producing a Monte Carlo method. Our article [START_REF] Étoré | A transformed stochastic Euler scheme for multidimensional transmission PDE[END_REF] is an attempt to tackle this issue. 4.2. The diagonal case: link with the results of [START_REF] Bossy | Probabilistic interpretation and random walk on spheres algorithms for the Poisson-Boltzmann equation in molecular dynamics[END_REF]. We wish to compare the result of our Theorem 4.2 with the ones in [START_REF] Bossy | Probabilistic interpretation and random walk on spheres algorithms for the Poisson-Boltzmann equation in molecular dynamics[END_REF]. For this purpose we restrict once more the assumption on the diffusion coefficient a(x) given in the setting of the preceding subsection. Namely we assume a

(x) = I d ε(x) with ε(x) := ε + 1 D+ + ε -1 D -(x), I d the identity matrix and ε + = ε -∈ R * + .
We are therefore considering the case of a diagonal diffusion matrix. We will also assume in the forthcoming theorems that Γ is bounded and closed, and will consider that D + is the interior domain delimited by Γ.

We start by summing up the notions and results in [START_REF] Bossy | Probabilistic interpretation and random walk on spheres algorithms for the Poisson-Boltzmann equation in molecular dynamics[END_REF] that we need in order to do the comparison.

In the sequel we denote (C, C) the usual canonical space, i.e. C = C([0, ∞); R d ) and C = B(C) (see Pb 2.4.2 in [START_REF] Karatzas | Brownian motion and stochastic calculus[END_REF] for details). We also denote (C t ) t≥0 the usual canonical filtration (see Eq. (5.3.19) in [START_REF] Karatzas | Brownian motion and stochastic calculus[END_REF]).

We shall denote by ω the canonical process defined on (C, C). Note that ω = (ω t ) t≥0 is (C t )adapted.

Let us define the transmission operator L acting on functions ϕ

∈ C(R d ) ∩ C 2 ( D+ ) ∩ C 2 ( D-) by Lϕ(x) = d i,j=1 ∂ i [I d ε] ij ∂ j ϕ (x) = d i=1 ε(x) ∂ 2 ii ϕ(x), ∀x ∈ R d \ Γ, Lϕ(x) = δ(x), ∀x ∈ Γ (4.8)
where for any x ∈ Γ, δ(x) ∈ R is an arbitrary value of no importance in our computations.

ϕ ∈ C b (R d ) ∩ C 2 b ( D+ ) ∩ C 2 b ( D-) (4.9) ε + ∇ x ϕ + (y) -ε -∇ x ϕ -(y), ν(y) = 0, ∀y ∈ Γ, ( ) (4.10)
one has that the t-indexed process defined by

ϕ(ω t ) -ϕ(ω 0 ) - t 0 Lϕ(ω s )ds, ∀t ≥ 0 is a (C t )-martingale under P x .
ii) The martingale problem is said to be well-posed if there exists a unique family of probability measures (P x ) x∈R d which solves the martingale problem for the operator L.

Theorem 4.5 (Theorems 2.4, 2.10 and 2.12 in [START_REF] Bossy | Probabilistic interpretation and random walk on spheres algorithms for the Poisson-Boltzmann equation in molecular dynamics[END_REF]). For all

x ∈ R d consider the SDE        X t = x + t 0 2ε(X u )dB u + ε + -ε - 2ε - t 0 ν(X u )dL 0 u (Y ) Y t = ρ(X t ) (4.11)
where B is a d-dimensional Brownian motion, ρ(z) is the distance from z ∈ R d to the boundary Γ and L 0 t (Y ) stands for the (right) local time at point zero of the local martingale Y . If Γ is of class C 3 and compact we have: i) There exists a weak solution to (4.11). For any x ∈ R d consider the law P x of this weak solution on (C, C). The family (P x ) is a solution to the martingale problem for L.

ii) Conversely let (P x ) be a solution to the martingale problem for L, and let x ∈ R d . There exists a (C t )-Brownian motion B under P x , such that ω is a weak solution to (4.11) driven by B, under P x . Remark 4.6. Note a change of sign in the weight in front of the local time term in (4.11), compared to [START_REF] Bossy | Probabilistic interpretation and random walk on spheres algorithms for the Poisson-Boltzmann equation in molecular dynamics[END_REF]. This is because for us the exterior normal to the interior domain D + is -ν.

Remark 4.7. Note that in [START_REF] Bossy | Probabilistic interpretation and random walk on spheres algorithms for the Poisson-Boltzmann equation in molecular dynamics[END_REF] the authors work with right local time at point zero of Y = ρ(X). Working with the symmetric local time instead, as it is often the case in the study of asymetric diffusions (e.g. [START_REF] Lejay | On the constructions of the skew Brownian motion[END_REF]) would lead to different coefficients in front of the local time term.

Theorem 4.8 (Theorem 2.14 in [START_REF] Bossy | Probabilistic interpretation and random walk on spheres algorithms for the Poisson-Boltzmann equation in molecular dynamics[END_REF]). Assume Γ is of class C 3 and compact. Then the martingale problem for L is well posed and in particular there is a unique weak solution to (4.11) in the sense of probability law.

We then have the following main result. Theorem 4.9. Assume Γ is of class C 2 and compact. Consider the Hunt process X (M) in Theorem 4.2. For any starting point x ∈ R d consider the law P x of X on (C, C). The family (P x ) is a solution to the martingale problem for L.

Proof : Let M = (Ω, (F t ) t≥0 , F, (X t ) t≥0 , (P x ) x∈R d ) the Hunt process considered in Theorem 4.2. Let x ∈ R d and let ϕ satisfying (4.9) and (4.10). We aim at applying Theorem 5.5.5. in [START_REF] Fukushima | Dirichlet forms and symmetric Markov processes[END_REF] with ϕ in order to check that ϕ(X t ) -ϕ(x) -t 0 Lϕ(X s )ds is a martingale under P x w.r.t (F t ).

Indeed, proceeding as in [START_REF] Karatzas | Brownian motion and stochastic calculus[END_REF] p314, this implies that the law P x induced on (C, C) by X (under P x ) is such that if we consider the family (P x ) this is a solution to the martingale problem for L.

The fact that ϕ is finely continuous is clear. In order to check that ϕ ∈ D [E] b,loc it suffices to notice that for any compact G ⊂ R d the functions (ϕ1 G ) + and (ϕ1 G ) + are respectively in H 1 (D+) and H 1 (D -). As ϕ1 G is in L 2 (R d ) and continuous across Γ, Exercise 4.5 in [START_REF] Mclean | Strongly elliptic systems and boundary integral equations[END_REF] implies

that ϕ1 G ∈ H 1 (R d ) = D [E].
We now check that 1 G • µ ϕ ∈ S 00 for any relatively compact set G. We first check that 1 G • µ ϕ is a positive and finite Radon measure. Indeed one has, as in Eq. (3.3),

1 G • µ ϕ (R d ) = G µ ϕ (dx) = 2E(1 G ϕ, ϕ) -E(1 G ϕ 2 , 1) = 2E(1 G ϕ, ϕ)
and 0 ≤ E(1 G ϕ, ϕ) < ∞. We now check that ||R 1 1 G • µ ϕ || ∞ < ∞. For any x ∈ R d we have, with a constants m, M depending on λ, Λ, Here we have used (2.13) at the second line. At the third line we have used (3.3) and computations similar to the ones leading to (4.4). At the last line we have used (E-B). As the bound in (4.12) does not depend on x we have proven

|R 1 1 G • µ ϕ (x)| ≤ ∞ 0 e -t
||R 1 1 G • µ ϕ || ∞ < ∞.

  3.1.1. A reminder of the Revuz correspondence for additive functionals. Denote by {R α : α > 0} the Markovian resolvent kernel of the Markovian transition function {p(t, x, dy) := p(t, x, y)dy : t > 0}. Then, for any

  Let us introduce D [E] b (resp. D [E] b,loc ) the space of essentially bounded functions belonging to D[E] (resp. locally to D[E]. A function u is in D [E] b,loc if for any compact set G, there exists a bounded function ω ∈ D [E] such that u = ω, (dx)-a.e. on G).

4. 1 .

 1 Skorokhod representation of the Hunt process associated to a transmission operator in divergence form. Consider R d = D+ ∪ D -with D + and D -two open connected subdomains separated by a transmission boundary Γ that is to say Γ = D+ ∩ D-.

For a point

  x ∈ Γ we denote by ν(x) ∈ R d the unit normal to Γ at point x, pointing to D + . In the following, "f ∈ C p ( D+ ; R) ∩ C p ( D-; R)" means that the restriction f + of the real valued function f to D + (and the restriction f -of f to D -) coincides on D + (resp. D -) with a function f+ of class C p (R d ) (resp. f-). Furthermore C p b (E) = C p b (E; R) will denote the set of real valued functions on E of class C p , bounded with bounded derivatives up to order p.

  Proposition 4.1 (First Green identity, first version; McLean (2000), Lemma 4.1). Assume Γ is bounded and C

  and follow the ideas of Trutnau (2005) Theorem 5.2. Of course p k ∈ D [E] b,loc and p k is finely continuous on R d . Let G a relatively compact open set containing Γ and a function

t 0 [

 0 ∂ j a kj ] ± (X s )I Xs∈D ds t≥0 . We now turn to the surface measures ζ ± k (dy) := [(γ -) k -(γ + ) k ] ± (y)I y∈Γ ς(dy). It is well-known (see e.g. Evans and Gariepy (2015) p.134 3. ( ), ( )) that there exists a universal constant C 0 > 0, depending only on the Lipschitz domain D + , such that for all h ∈ C 1 (D + ), Γ |h(y)|ς(dy) ≤ C 0 D + (|∇h(x)| + |h(x)|) (dx).

  y), ν(y) = 0, for a.e. (t, y) ∈ (0, T ] × Γ ( )

Definition 4. 4 (

 4 [START_REF] Bossy | Probabilistic interpretation and random walk on spheres algorithms for the Poisson-Boltzmann equation in molecular dynamics[END_REF]). i ) A family of probability measures (P x ) x∈R d on (C, C) solves the martingale problem for the operator L if, for all x ∈ R d one has P x (ω(0) = x) = 1 and, for all ϕ satisfying

  , x, y)|1 G • µ ϕ |(dy) dt 2 e -|y-x| 2 M t |1 G • µ ϕ |(dy) 2 e -|y-x| 2 M t | 2a(y)∇ϕ(y), ∇ϕ(y) |dy = M ∞ 0 dt e -t E x | 2a(mW t )∇ϕ(mW t ), ∇ϕ(mW t ) |1 Wt∈G

Acknowledgements

The authors would like to thank the anonymous referee for his valuable remarks about preliminary versions of this paper.

We have, using in particular Proposition 4.1, the smoothness of ϕ, Tr + (v) = Tr(v+) = Tr -(v) = Tr(v-) = Tr(v) and (4.10),

The function Lϕ is piecewise continuous and bounded and proceeding as above one may check that the positive and negative parts (-Lϕ) ± satisfy 1 G • (-Lϕ) ± dx ∈ S 00 for any relatively compact set G.

We denote A ± the PCAF's related to Revuz measures (-Lϕ) ± dx. We set

and notice that following Example 5.1.1 in [START_REF] Fukushima | Dirichlet forms and symmetric Markov processes[END_REF] we have N

)ds (note that we use the fact that t 0 1 Xs∈Γ ds = 0 P x -a.s., therefore the arbitrary value of Lϕ(z) for z ∈ Γ causes no issue).

We now apply Theorem 5.5.5. in [START_REF] Fukushima | Dirichlet forms and symmetric Markov processes[END_REF]. We have ϕ] which a martingale (as for example the M [f k ] 's are martingales in the proof of Theorem 4.2). The proof is completed.

In view of Theorems 4.5, 4.8 and 4.9 we immediately get the following corollary.

Corollary 4.10. Assume Γ is of class C 3 and compact. The solutions of (4.3) and (4.11), with starting point x ∈ R d , have the same distribution on (C, C).

In particular there is uniqueness in the sense of probability law of the weak solutions of (4.3).

Going a bit further in the analysis we may do an identification in the strong sense of the terms appearing in (4.3) and (4.11).

Corollary 4.11. Assume Γ is of class C 3 and compact.

Consider the Hunt process

Then one also has   

with B(X) a Brownian motion under P x (that is X is also a weak solution to (4.11)).

Moreover one has B(X) = W (with W the Brownian motion driving (4.3)) and

3) in the matrix form in the case of interest. One has

and notice that G(•) is obviously measurable and that

Noting that Φ(•) is measurable and remembering the definition of P x as the law of X on (C, C) under P x it is obvious that one has P x (Φ(X) = 0) = 1 = P x (Φ(ω) = 0).

Therefore under P x one has a.s.

where B(ω) is a (C t )-Brownian motion under P x . In the notation we have stressed that B(ω) is constructed from the paths of ω (through a measurable mapping).

Applying G(•) viewed as in (4.14) to X we get the first part of the corollary, that is that X solves (4.11) (4.11) driven by B(X) (to check that B(X) is a B.m. under P x we have to check that the increments of B(X) are independent and that B t (X) -B s (X) for any s < t and distributed as a N d (0, (t -s)I d ), by identifying the law of X under P x with the one of ω under P x ).

Using the uniqueness of the decomposition of a semimartingale and recalling that γ ± = a ± ν we have

From (4.15) and (1.1) one has

where the bracket has to be understood in the multidimensional sense (matrix of brackets). Therefore W ≡ B(X) using Proposition IV.1.12 in [START_REF] Revuz | Continuous martingales and Brownian motion, volume 293 of Grundlehren der mathematischen Wissenschaften[END_REF]. It remains to use the componentwise meaning of (4.16) in order to check (4.13). For the use of exposure we assume for a while that d = 2, with ν = (ν 1 , ν 2 ) T . Using the fact that if ν 1 (y) = 0 then ν 2 (y) = 0, y ∈ Γ, we get for any t ≥ 0,

We claim that the above reasoning can easily be extended to d > 2. The proof is completed.