Huasheng Huang
email: huasheng.huang@unifi.it.

Robert J Morley

Raymond Van Der Ham

Limi Mao

Alexis Licht

Guillaume Dupont-Nivet

Zaw Win

Wa Day

Carina Aung

Hoorn

Aung

Carina Hoorn

Grimmipollis burmanica gen. et sp. nov.: New genus of the soapberry family (Sapindaceae) from the late Eocene of central Myanmar

Keywords: Cupanieae, Morphology, Paleobotany, Paleoecology, Paleogene, Palynology

As a crossroads for plant dispersal between Laurasia and Gondwana since at least the late Eocene, the Central Myanmar Basins have yielded rich and diverse fossil pollen . Here we report Grimmipollis burmanica, a new genus and species in the soapberry family (Sapindaceae) from the upper Eocene Yaw Formation in the Central Myanmar Basins. We also investigate its similarity of pollen sculpture with other selected genera using hierarchical cluster analysis, and its source paleoecology with pollen diagrams. Grimmipollis pollen grains are characterized by parasyncolporate apertures with an apocolpial field at each side, distinct margines along the colpi, arcuate, tuberculate and rope-shaped mesocolpial ridges, and prominent, verrucate or fossulate apocolpia.

These features distinguish this genus from other syncolporate and parasyncolporate types in the extant Sapindaceae genera, and fossil pollen genera. As most syncolporate and parasyncolporate types occur in the tribe Cupanieae, G. burmanica is likely derived from an extinct Cupanieae and thus a rainforest tree species.

The hierarchical cluster analysis confirms our establishment of Grimmipollis, and reveals it is morphologically close to Meyeripollis. The sedimentary environment from which these pollen grains were retrieved suggests that G. burmanica could be a member of seasonal swamp forests.

Introduction

Sapindaceae, the soapberry family, is the largest family in the order Sapindales, including 138 genera and c.

1,900 species (http://www.theplantlist.org/1.1/browse/A/Sapindaceae/), which mostly are trees and shrubs, and less often treelets and lianas, with a distribution predominantly in tropical or subtropical zones [START_REF] Acevedo-Rodríguez | The families and genera of vascular plants. Eudicots: Sapindales, Cucurbitales, Myrtaceae[END_REF]. These genera and species belong to four subfamilies (Xanthoceroideae, Hippocastanoideae, Dodonaeoideae, and Sapindoideae) and more than 10 tribes [START_REF] Acevedo-Rodríguez | The families and genera of vascular plants. Eudicots: Sapindales, Cucurbitales, Myrtaceae[END_REF], following Harrington et al., 2005). Cupanieae, the largest tribe in Sapindaceae, contains c. 46 genera, and has a pantropical distribution in Australasia, Asia, South America and Madagascar [START_REF] Harrington | Phylogenetic inference in Sapindaceae sensu lato using plastid matK and rbcL DNA sequences[END_REF][START_REF] Buerki | Two new genera of Sapindaceae (Cupanieae) from the southern Pacific: Lepidocupania and Neoarytera[END_REF]. Cupanieae species in Southeast Asia are rain forests trees [START_REF] Yap | Sapindaceae[END_REF].

The morphology of extant Sapindaceae pollen has extensively been studied (for most of the genera in the family, Muller andLeenhouts, 1976, Acevedo-Rodríguez et al., 2011;Mischocarpus, van der Ham, 1977;Pometia, Cubilia, Otonephelium and Litchi, van den Berg, 1978;Guioa, van der Ham and van Heuven, 1989; tribe Nephelieae, van der [START_REF] Van Der Ham | Nephelieae pollen (Sapindaceae): form, function, and evolution[END_REF]Lophostigma, Acevedo-Rodríguez, 1993;Cnesmocarpon, Jagera and Trigonachras, Adema and van der Ham, 1993;tribe Paullinieae, Ferrucci and Anzótegui, 1993;Serjania, van der Ham and Tomlik, 1994;Arytera, van Bergen et al., 1995;Gongrodiscus, Turner and van der Ham, 1996; Sapindaceae pollen types in Brazilian forest fragments, [START_REF] Bellonzi | Pollen types of Sapindaceae from Brazilian forest fragments: Variations on apertures of the pollen grains[END_REF]. [START_REF] Muller | A general survey of pollen types in Sapindaceae in relation to taxonomy[END_REF] classified the apertural types of extant Sapindaceae pollen into 12 groups and regarded the spherical tricolporate type A with long colpi and small endoapertures as a basic type. [START_REF] Gonçalves-Esteves | Pollen morphology and evolutionary history of Sapindales[END_REF] concluded that tri(tetra)-colporate, which is most common, tri(tetra)-colporate, parasyncolporate, syncolporate or hemisyncolporate apertural types were found in Sapindaceae. Syncolporate and parasyncolporate types only occur in the subfamily Sapindoideae, and most commonly in the tribe Cupanieae [START_REF] Acevedo-Rodríguez | The families and genera of vascular plants. Eudicots: Sapindales, Cucurbitales, Myrtaceae[END_REF]. Besides variation in the apertural types, variation also occurs in polarity, shape, size and ornamentation of the exine [START_REF] Muller | A general survey of pollen types in Sapindaceae in relation to taxonomy[END_REF][START_REF] Acevedo-Rodríguez | The families and genera of vascular plants. Eudicots: Sapindales, Cucurbitales, Myrtaceae[END_REF].

Fossil pollen types of Sapindaceae have also been extensively reported, including the widely distributed form-genus Cupanieidites, with affinity to the tribe Cupanieae. Cupanieidites is derived from the proposed J o u r n a l P r e -p r o o f Journal Pre-proof ancestors of the tribe Paullinieae, which originated in the Late Cretaceous of Africa and dispersed to India, the Americas and Australia (see the review in [START_REF] Huang | Plant turnover in response to climate change in the Cenozoic: Palynological insights from Myanmar, Southeast Asia and beyond[END_REF][START_REF] Huang | At a crossroads: The late Eocene flora of central Myanmar owes its composition to plate collision and tropical climate[END_REF] and Europe (e.g., Krutzsch, 1959aKrutzsch, , b, 1961[START_REF] Krutzsch | Taxonomie syncolp(or)ater and morphologisch benach-barter Pollengattungen und -arten (Sporae dispersae) aus der Oberkreide und dem Tertiaer. Teil I. Syn-colp(or)ate und syncolp(or)atoide Pollenformen[END_REF]. For instance, Cupanieidites flaccidiformis (with affinity to Mischocarpus) was found in the Paleocene of Pakistan [START_REF] Frederiksen | Middle and Late Paleocene angiosperm pollen from Pakistan[END_REF], the middle Eocene of south Sulawesi, Indonesia [START_REF] Morley | Palynological evidence for Tertiary plant dispersal in the SE Asian region in relation to plate tectonics and climate[END_REF] and late Eocene of central Myanmar [START_REF] Huang | At a crossroads: The late Eocene flora of central Myanmar owes its composition to plate collision and tropical climate[END_REF]. Other fossil pollen records include Alectryon-type from the middle Eocene of Germany [START_REF] Hofmann | Scanning electron microscope investigations of pollen from an atypical mid-Eocene coal facies in Stolzenbach mine (PreußenElektra) near Borken (Kassel, Lower Hesse, Germany)[END_REF], Cardiospermum, Paullinia and Serjania from the late Eocene of Panama [START_REF] Graham | Studies in Neotropical Paleobotany. IV. The Eocene communities of Panama[END_REF][START_REF] Graham | Tropical American Tertiary floras and paleoenvironments: Mexico, Costa Rica, and Panama[END_REF], Paullinia and Serjania from the early to middle Miocene of Mexico [START_REF] Chavez | Estudio palinológico de las floras fósiles del Mioceno inferior y principios del Mioceno medio de la región de Pichucalco, Chiapas, México[END_REF], and Pometia from the Miocene of Borneo [START_REF] Muller | Palynological contributions to the history of Tertiary vegetation in N.W. Borneo[END_REF].

The Central Myanmar Basins (CMB), situated between the Indo-Burman Ranges and Sino-Burman Ranges (Fig. 1A), comprises most of Myanmar and has been a crossroads for plant dispersal between Laurasia and Gondwana since at least late Eocene, principally driven by the India-Asia collision and its tropical and protomonsoonal climate [START_REF] Huang | Plant turnover in response to climate change in the Cenozoic: Palynological insights from Myanmar, Southeast Asia and beyond[END_REF][START_REF] Huang | At a crossroads: The late Eocene flora of central Myanmar owes its composition to plate collision and tropical climate[END_REF]. This crossroads may persist until the modern day, and thus renders Myanmar as a global biodiversity hotspot [START_REF] Myers | Biodiversity hotspots for conservation priorities[END_REF] with c. 12,340 spermatophyte species (according to [START_REF] Kress | & U Chit Ko Ko on the "List of trees, shrubs, herbs and principal climbers etc. recorded from Burma")[END_REF][START_REF] Yang | Contributions to the flora of Myanmar from 2000 to 2019[END_REF]. The CMB yields highly diverse palynofloras from the Eocene onwards [START_REF] Potonié | Sporologie der eozänen Kohle von Kalewa in Burma[END_REF][START_REF] Reimann | Results of palynostratigraphical investigations of the Tertiary sequence in the Chindwin Basin/Northwestern Burma[END_REF][START_REF] Moe Lwin | Paleo-environment of the Miocene sediments, central belt, Myanmar[END_REF]Huang et al., 2020[START_REF] Huang | At a crossroads: The late Eocene flora of central Myanmar owes its composition to plate collision and tropical climate[END_REF][START_REF] Huang | Plant turnover in response to climate change in the Cenozoic: Palynological insights from Myanmar, Southeast Asia and beyond[END_REF], including among many others a diversity of Arecaceae (Huang et al., 2020) and Sapotaceae [START_REF] Huang | At a crossroads: The late Eocene flora of central Myanmar owes its composition to plate collision and tropical climate[END_REF] pollen types.

In this study, we investigate the parasyncolporate sapindaceous fossil pollen from the upper Eocene Yaw Formation in the CMB with light microscopy (LM) and scanning electron microscopy (SEM). We compare these fossil pollen with the pollen in the extant genera of Sapindaceae and (para)syncolporate fossil pollen formgenera to discuss its taxonomic position. We further seek to unravel its source paleoecology by comparing its distribution in relation to paleoecological and systems tract interpretation as presented in [START_REF] Huang | A proto-monsoonal climate in the late Eocene of Southeast Asia: Evidence from a sedimentary record in central Myanmar[END_REF]. [START_REF] Bender | Geology of Burma[END_REF][START_REF] Pivnik | Polyphase defor mation in a fore-arc/back-arc basin, Salin sub-basin, Myanmar (Burma)[END_REF][START_REF] Licht | Paleogene evolution of the Burmese forearc basin and implications for the history of India-Asia convergence[END_REF]. In the late Eocene, the CMB (1) had a near-equatorial position [START_REF] Westerweel | Burma Terrane part of the Trans-Tethyan arc during collision with India according to palaeomagnetic data[END_REF][START_REF] Westerweel | Burma Terrane collision and northward indentation in the Eastern Himalayas recorded in the Eocene-Miocene Chindwin Basin (Myanmar)[END_REF]; ( 2) was open to the Bay of Bengal and located at the margin of Eurasia [START_REF] Licht | A palaeo Tibet-Myanmar connection? Reconstructing the Late Eocene drainage system of central Myanmar using a multiproxy approach[END_REF]; (3) was predominantly sourced from the WPA and northern Myanmar [START_REF] Westerweel | Burma Terrane collision and northward indentation in the Eastern Himalayas recorded in the Eocene-Miocene Chindwin Basin (Myanmar)[END_REF]; (4) formed a crossroads for plant dispersal between Laurasia and Gondwana [START_REF] Huang | At a crossroads: The late Eocene flora of central Myanmar owes its composition to plate collision and tropical climate[END_REF].

Material and Methods

Geological Setting

The Kalewa section is situated in the Kalewa Township, Sagaing Region in northwestern Myanmar, and located at the southern margin of the Chindwin Basin (Fig. 1A; 23°14′ N, 94°15′ E). This c. 650-m thick section comprises the Yaw Formation (Fig. 1B). This Formation consists of four prominent sedimentary facies associations (FA): organic-rich lacustrine mudstones (FA2), deltaic tidal siltstones and sandstones (FA3), thick continental sandstones (FA4), and lignite layers from past swamps (FA1), reflecting a quasi-closed estuarine depositional system (Fig. 1B; [START_REF] Licht | Paleogene evolution of the Burmese forearc basin and implications for the history of India-Asia convergence[END_REF][START_REF] Westerweel | Burma Terrane collision and northward indentation in the Eastern Himalayas recorded in the Eocene-Miocene Chindwin Basin (Myanmar)[END_REF]. The underlying upper middle Eocene Pondaung Formation was deposited in a fluvial-deltaic setting and opened towards the proto-Bay of Bengal [START_REF] Naing Soe | Sedimentary facies of the Late middle Eocene Pondaung Formation (central Myanmar) and the palaeoenvironments of its Anthropoid Primates[END_REF]Licht et al., 2014). The overlying Tonhe Formation possibly with an age of late Oligocene-early Miocene (indicated from the sporomorphs in Soe Moe [START_REF] Moe Lwin | Paleo-environment of the Miocene sediments, central belt, Myanmar[END_REF], is formed of thick packages of coarse-grained sandstones and conglomerates [START_REF] Westerweel | Burma Terrane collision and northward indentation in the Eastern Himalayas recorded in the Eocene-Miocene Chindwin Basin (Myanmar)[END_REF]. The age of the Yaw Formation was estimated to be c. 37-36 Ma based on dates by a tuff layer [START_REF] Licht | Paleogene evolution of the Burmese forearc basin and implications for the history of India-Asia convergence[END_REF], magnetostratigraphy, U-Pb apatite dating, and apatite fission track dating [START_REF] Westerweel | Burma Terrane collision and northward indentation in the Eastern Himalayas recorded in the Eocene-Miocene Chindwin Basin (Myanmar)[END_REF], and palynostratigraphy (Huang et al., 2020).

Preparation of Fossil Pollen

We collected 81 samples from mudstones and sandstones in the upper Eocene Yaw Formation along the Kalewa section. These samples have been documented and published in terms of fossil palm pollen (Huang et J o u r n a l P r e -p r o o f Journal Pre-proof al., 2020), floristic composition [START_REF] Huang | At a crossroads: The late Eocene flora of central Myanmar owes its composition to plate collision and tropical climate[END_REF], climate and sequence biostratigraphy [START_REF] Huang | A proto-monsoonal climate in the late Eocene of Southeast Asia: Evidence from a sedimentary record in central Myanmar[END_REF].

Two processing methods were applied to extract as many and well-preserved pollen grains as possible: (1) 1.3 grams of rock sample was boiled in 10% sodium pyrophosphate, followed by the 10% hydrochloric acid (HCl), sieving, acetolysis mixture and bromoform-treatment; (2) 30 grams of rock sample was treated with 10% HCl and 40% hydrofluoric acid (HF), followed by a heavy liquid separation and sieving. Final residues were mounted on a slide in glycerin and sealed with paraffin for LM observation and microphotography. Residues were also used for LM and SEM analysis at the Department of Palaeontology (DoP), University of Vienna, Austria. For details on processing refer to Huang et al. (2020). All pollen slides and residues are stored at the Institute for Biodiversity and Ecosystem Dynamics (IBED), University of Amsterdam, and SEM stubs are deposited at the DoP.

Microphotography

All LM micrographs in Plate I were taken with Fujifilm X-E2 and a Zeiss Universal microscope with 63× Plan Neofluar NA1, 25 oil at the IBED, applying Nomarski Differential Interference Contrast (DIC) following [START_REF] Bercovici | Improving depth of field resolution for palynological photomicrography[END_REF]. Stacking of micrographs with different layers through manual Z-making in Helicon Focus and Photoshop CC provide a fully focused image. LM micrographs in Plates II-IV were taken by the camera ProgRes Speed XT core 5, linked with the LM Nikon Eclipse 80i; SEM micrographs in Plates II-IV were taken with the SEM JEOL JSM•6400. Plates II-IV were made with InDesign CC (Adobe, California, US).

Pollen Analysis

We compared the pollen type with the pollen micrograph database of Sapindaceae held by the third author (RvdH). We applied single-grain analysis [START_REF] Zetter | Methodik und Bedeutung einer routinemäßig kombinierten lichtmikroskopischen und rasterelektronischen Untersuchung fossiler Mikrofloren[END_REF][START_REF] Halbritter | Illustrated pollen terminology[END_REF] to investigate the details of pollen sculpture at the DoP. LM and SEM pollen terminology follows [START_REF] Punt | Glossary of pollen and spore terminology[END_REF] and [START_REF] Halbritter | Illustrated pollen terminology[END_REF], respectively. We measured the pollen with the software ImageJ (National Institute of Health, USA)

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Journal Pre-proof (pollen information and measurement results are provided in Supplementary Material 1, Tables S1 andS2).

With presence-absence data matrix (Supplementary Material 1, Tables S3 andS4), we applied hierarchical cluster analysis (HCA, with Ward's method and Euclidean distance) on PAST 4.10 software [START_REF] Hammer | PAST: paleontological statistics software package for education and data analysis[END_REF] to determine similarity of pollen sculpture among genera. We constructed the pollen diagram with the Tilia v.3.0.1 software [START_REF] Grimm | Tilia and tiliagraph[END_REF]. Software-produced images were processed with CorelDRAW 2019 (Corel Corporation, Ottawa, Canada). (格林氏粉属, gé lín shì fěn shǔ) Etymology: The generic name "Grimmipollis" is named after and in honor of the influential American paleoecologist and palynologist Dr. Eric Grimm (1951Grimm ( -2020)), for his immense contributions to palynology, particularly on developing the software Tilia [START_REF] Grimm | Tilia and tiliagraph[END_REF]; including the program CONISS, [START_REF] Grimm | CONISS: a FORTRAN 77 program for stratigraphically constrained cluster analyses by the method of incremental sum of squares[END_REF], which was also applied in this study, and the Neotoma database [START_REF] Williams | The Neotoma paleoecology database, a multiproxy, international, community-curated data resource[END_REF].

Results

Systematic and descriptive palynology

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Journal Pre-proof Generic diagnosis: Triparasyncolporate, mesocolpial ridges arcuate, granulate, rugulate, tuberculate (prominently verrucate) and rope-shaped, apocolpial fields verrucate or fossulate, mesocolpia granulate and rugulate, margines well developed along the colpi, apocolpia prominent, colpi wide.

Type species: Grimmipollis burmanica Huang, Morley et Hoorn sp. nov. Stratigraphic horizon: Upper Eocene Yaw Formation, Central Myanmar Basins (Fig. 1B).

Grimmipollis burmanica

Type locality. Western side of the Kalewa Township, Sagaing Region, Myanmar (23°14′ N, 94°15′ E).

Age: Late Eocene, c. 37-36 Ma [START_REF] Licht | Paleogene evolution of the Burmese forearc basin and implications for the history of India-Asia convergence[END_REF][START_REF] Westerweel | Burma Terrane collision and northward indentation in the Eastern Himalayas recorded in the Eocene-Miocene Chindwin Basin (Myanmar)[END_REF][START_REF] Huang | At a crossroads: The late Eocene flora of central Myanmar owes its composition to plate collision and tropical climate[END_REF]. S1.

Etymology: The specific epithet "burmanica" refers to the origin (Burma, former name of Myanmar) of the fossil pollen.

Specific diagnosis:

Outline convex-triangular to near-circular in polar view, pollen size category small, triparasyncolporate, mesocolpial ridges arcuate, granulate, rugulate, tuberculate and rope-shaped, apocolpial fields verrucate or fossulate, mesocolpia granulate and rugulate, margines well developed along the colpi, apocolpia prominent, colpi wide.

Description: Pollen grains isopolar monads, outline convex-triangular (e.g., Plates IA, IIIA) to near-circular (e.g., Plates IB, IVD) in polar view; pollen size category small, equatorial diameter 12.6-19.0 μm; triparasyncolporate, angulaperturate, ectocolpus long, wide and connected around the poles, forming apocolpia,

J o u r n a l P r e -p r o o f

Journal Pre-proof pori circular and situated equatorially, 0.4-1.7 μm in diameter; apocolpia triangular (e.g., Plate IVE) or severely corroded (Plate IVF), sculpture verrucate or fossulate (SEM), size 3.0-8.0 μm, verrucae up to 0.8 μm in diameter (SEM); mesocolpia granulate, granules up to 0.8 μm in diameter (SEM), penetrated through by ropelike sculpture (e.g., Plates IIH, IIIG), forming coarse rugulae, tuberculate rope-like structure 0.5-1.4 μm in diameter (SEM); margines psilate, rugulate and well developed along colpi, 0.4-1.6 μm in diameter, thicker around the pores (SEM); exine with indistinct columellate, 0.3-0.6 μm thick, sexine and nexine indiscernible; pollen wall tectate; pollen sculpture psilate, scabrate to rugulate in LM, verrucate (apocolpia), psilate and rugulate (margines), granulate and rugulate (mesocolpia) in SEM. 18 specimens measured in polar view (Supplementary Material 1, Table S2), as the pollen grains did not occur in equatorial view. Due to their oblate shape, syncolporate/parasyncolporate pollen grains whether extant or fossil, appear in preparations much more frequently in polar view than in equatorial view, similarly in family Myrtaceae, and genus Nymphoides (family Menyanthaceae).

Botanical affinity: Extinct and with affinity to Cupanieae, largest tribe of Sapindaceae; and see discussion below.

Discussion

Placement of Grimmipollis burmanica into Sapindaceae and comparisons with the extant parasyncolporate pollen genera of the family

For the assignment of the studied fossil pollen at the family level, qualitative characters are more important than quantitative characters (e.g., pollen size), because the latter may change during the process of deposition, particularly sediment compaction. Additionally, even within a genus of Sapindaceae, pollen grain size may vary considerably. At the very first glance, our pollen type seems to belong to Myrtaceae with its parasyncolporate aperture with arcute colpi. However, its rounded shape and ornamentation of the exine (verrucate or fossulate J o u r n a l P r e -p r o o f Journal Pre-proof apocolpial field and granulate and tuberculate mesocolpia) does not match any myrtaceous morphotype [START_REF] Thornhill | Fossil myrtaceous pollen as evidence for the evolutionary history of Myrtaceae: A review of fossil Myrtaceidites species[END_REF].

The tuberculate condition is also seen in some Leguminosae, such as Humboldtia, but the syncolporate aperture configuration is missing in this family. Pollen grains of Pachydiscus gaultherioides (= Periomphale gaultherioides in van Steenis, 1978; = Periomphale pancheri in [START_REF] Gardner | Systematic notes on the Alseuosmiaceae[END_REF] in the family Alseuosmiaceae (with micrographs from Carlos Jaramillo, Smithsonian Tropical Research Institute) resemble our fossil pollen; however, although it has margines along the colpi, it is syncolpate and has no tuberculate ridges. Pollen grains of the Madagascan endemic family Sarcolaenaceae generally are united in tetrahedral tetrads, and seem parasycolporate [START_REF] Nilsson | On the origin of the Sarcolaenaceae with reference to pollen morphological evidence[END_REF]. However, the tetrads range from c. 40 to 120 μm in diameter, with psilate ornamentation (including margines). The most comparable Sarcolaenaceae types occur in Leptolaena, Sarcolaena, Xyloolaena and Rhodolaena; nevertheless, the constituent grains are tricolporate without apocolpial fields [START_REF] Carlquist | Pollen morphology and evolution of Sarcolaenaceae (Chlaenaceae)[END_REF][START_REF] Nilsson | On the origin of the Sarcolaenaceae with reference to pollen morphological evidence[END_REF].

The combination of a syncolporate apertural pattern, coupled with the tuberculate exine (as in Magonia from tribe Harpullieae) supports the placement of the fossil pollen into Sapindaceae. Margines form just part of ornamentation and are a common feature of Sapindaceae, but do not form an independent character. Therefore, the presence of the prominent margines of the fossil pollen along the colpi is not generally an argument for its affinity. There are some connections between the mesocolpia and the margines; namely, margines are not distinctly separated from the mesocolpium. In Sapindaceae, the ornamentation is denser around the colpi.

However, the distinct margines along the colpate apertures are rare in Sapindaceae.

We compared our fossil pollen with the extant pollen archive in Sapindaceae, including 134 genera, almost covering the whole family (138 genera). We listed parasyncolporate pollen types (33 genera) and compared these types with our fossil pollen in terms of sculpture. Parasyncolporate pollen types only occur in the subfamily Sapindoideae, and mostly in the tribe Cupanieae including 28 genera; most genera of this tribe have a Southeast Asian distribution (Table 1). Other parasyncolprate pollen types occur in the tribes Nephelieae (Alectryon), Schleichereae (Schleichera) and several genera incertae sedis (Castanospora, Tristira and

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Journal Pre-proof Tristiropsis) (Table 1). None of the pollen of these 33 genera is comparable to our fossil pollen in terms of sculpture, with most of them having rugulate ornamentation of exine (Table 1).

The four genera which are not in the archive are: (1) Chonopetalum (not placed in a tribe), with a single species in central and eastern Africa; (2) Euchorium (subfamily Dodonaeoideae), with a single species in Cuba (pollen not parasyncolporate); (3) Haplocoelopsis (tribe Cupanieae), with a single species in equatorial Africa (e.g., Guinea); ( 4) Namataea (tribe Lepisantheae, subfamily Sapindoideae), with a single species in Cameroon.

They are possibly not the nearest living relatives (NLR) of our fossil pollen, as they have African and Central American distributions. However, we cannot exclude the possibility that our fossil pollen may have an affinity with African taxa facilitated by the Africa-India floristic interchange (e.g., [START_REF] Bansal | Southeast Asian Dipterocarp origin and diversification driven by Africa-India floristic interchange[END_REF][START_REF] Shi | Fire-prone Rhamnaceae with South African affinities in Cretaceous Myanmar amber[END_REF].

HCA on our fossil pollen and the 33 parasyncolporate genera in Sapindaceae (Fig. 2A) shows that Grimmipollis does not share similarity with any of these genera in terms of ornamentation of exine. This further supports the establishment of the genus Grimmipollis. Besides, HCA also suggests that the ornamentation pattern of the exine cannot be used to differentiate tribes in the subfamily Sapindoideae.

Comparisons of Grimmipollis burmanica with (para)syncolporate fossil pollen genera in Jansonius and

Hills (1976)

We compared our fossil pollen with all 16 syncolporate and parasyncolporate fossil pollen genera from Jansonius and Hills (1976) (Table 2). We included syncolporate fossil pollen genera because these genera may share similar sculptural patterns with our fossil pollen and may have developed from syncolporate to parasyncolporate apertures over geological time.

Considering the ornamentation of the exine along the colpi, it shows that the nearest form-genus is Phaseoliidites, as it has sexinous thickening which pass over the poles from aperture to aperture, while our fossil pollen grains have distinct psilate and thickened margines. However, the ornamentation of the exine of Phaseolidites pollen is psilate, scabrate to granulate, weakly verrucate or vermiculate. Moreover, Phaseolidites pollen is bigger than our fossil pollen.

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The exine of Syncolporites pollen is scabrate to micro-verrucate, psilate, or rugulate, with annulus psilate (Table 2). In the Americas, it mostly resembles Syncolporites rugucostatus Sarmiento and Syncolporites planiverrucosus Jan du Chêne (pers. comm., Carlos Jaramillo, Smithsonian Tropical Research Institute).

Nonetheless, the ornamentations of their exine (psilate, [START_REF] Sarmiento | Palinología de la Formación Guaduas-estratigrafía y sistemática[END_REF]and psilate to verrucate, Jan du Chêne, 1977) are different from that of our fossil pollen.

Meyeripollis, a genus described from the Eocene-Miocene sediments of Assam (India) [START_REF] Baksi | Meyeripollis, a new genus from the Tertiary sediments or Assam[END_REF], which has syncolporate pollen with thickened margines and has received little attention, but closely matches our fossil pollen from a broad perspective. It is thought to be derived from trees of swamp forests, and probably from Myrtaceae [START_REF] Huang | At a crossroads: The late Eocene flora of central Myanmar owes its composition to plate collision and tropical climate[END_REF]. However, the exine of its pollen is gemmate and scabrate without apocolpia and tuberculate ridges, which is different from our fossil pollen. The aforementioned genus Periomphale with an East Gondwanan distribution shows some close similarities to Meyeripollis and may be a contender for the parent plant of Meyeripollis, although the specimen of Eucalyptus spathulata (Myrtaceae) illustrated by Thornhill and MacPhail (2012, Fig. 2c), may be a closer match, suggesting a Myrtaceous affinity.

The exine of Cupanieidites is distinctly to faintly reticulate [START_REF] Cookson | Some dicotyledonous pollen types from Cainozoic deposits in the Australian region[END_REF], smooth, punctate, granulate, striate or reticulate [START_REF] Krutzsch | Taxonomie syncolp(or)ater and morphologisch benach-barter Pollengattungen und -arten (Sporae dispersae) aus der Oberkreide und dem Tertiaer. Teil I. Syn-colp(or)ate und syncolp(or)atoide Pollenformen[END_REF], scabrate to reticulate [START_REF] Chmura | Upper Cretaceous (Campanian-Maastrichtian) angiosperm pollen from the western San Joaquin Valley, California, USA[END_REF]. Therefore, the exine of our fossil pollen (with tuberculate ridges) is outside the variation of any previous emendations of Cupanieidites, namely, the circumscription for Cupanieidites does not include the exine morphology seen in our fossil pollen.

Cupanieidites has been reported from the Late Cretaceous onwards [START_REF] Muller | Fossil pollen of extant angiosperms[END_REF]. It was later found in many areas and from different time periods, including the Senonian of India, Maastrichtian of Brazil and USA, Paleocene of Pakistan, Eocene of Australia, middle Eocene of Indonesia, and the late Eocene of Myanmar, and is thought to be derived from a plant group that dispersed from Africa, into India, and then Southeast Asia [START_REF] Huang | At a crossroads: The late Eocene flora of central Myanmar owes its composition to plate collision and tropical climate[END_REF].

The other 12 genera have different sculptures compared with our fossil pollen without tuberculate exine ( HCA on our fossil pollen and the 15 (Rhamneaepites not included as without description on its sculpture) syncolporate/parasyncolporate genera (Fig. 2B) suggests that Meyeripollis and Grimmipollis are most similar in terms of ornamentation of exine, although the latter has syncolporate pollen. This similarity can be explained by their common appearancetubercles on the exine.

Botanical affinity of Grimmipollis burmanica and its source paleoecology

As parasyncolporate pollen mostly exists in Cupanieae, which are trees of rain forests [START_REF] Yap | Sapindaceae[END_REF], the parent plant of Grimmipollis burmanica may be derived from this tribe. However, considering the unique ornamentation of the exine of G. burmanica, it may have been derived from an extinct genus, and thus does not exist today.

Palynological analysis of compositional changes of G. burmanica and other selected taxa which have known environmental preferences, was performed to understand the source paleoecology of G. burmanica.

From the bottom to the top of the Kalewa section, the paleoenvironments change from a brackish, tidally influenced setting to a fully freshwater setting (Huang et al., 2020), more specifically, from a barrier-bound estuary to a fluvial setting [START_REF] Licht | Paleogene evolution of the Burmese forearc basin and implications for the history of India-Asia convergence[END_REF]. This is suggested by palynological evidence (the up-section increase in percentages of freshwater taxa, i.e., Dicolpopollis, Longapertites and Palmaepollenites kutchensis, and the decrease of coastal taxa, i.e., Spinizonocolpites, Proxapertites opercculatus, Acrosctichum, florschuetzia and Avicennia; Fig. 3; Huang et al., 2020) and sedimentological data (change of lithofacies associations; Licht et al., 2019). The reconstructed pollen diagram (Fig. 3, with pollen count data in Supplementary Material 1, Table S5) indicates that Grimmipollis burmanica is common in the basal part of the succession and missing from the uppermost samples. Otherwise, towards to the top of the section the compositional percentages of Grimmipollis J o u r n a l P r e -p r o o f Journal Pre-proof burmanica keep steady generally. The predominance of occurrences in the basal part, coinciding with the PaBart-1 lowstand, and the upper part, coinciding with the PaPr-1 lowstand and early transgressive systems tracts, suggest that the source taxon may have occurred during periods of more seasonal climate, and most likely was an element of seasonal swamp forests.

Conclusions

We report a new fossil genus and species, Grimmipollis burmanica, from the upper Eocene Yaw Formation in Central Myanmar Basins, examined using light microscopy (LM) and scanning electron microscopy (SEM).

Pollen of G. burmanica is triparasyncolporate with verrucate or fossulate apocolpial fields, and granulate, rugulate and tuberculate mesocolpia, and is characterized by distinct psilate and rugulate margines bordering the colpi. Based on its unique pollen morphology and its close similarity to the tribe Cupanieae, G. burmanica is likely to have been derived from an extinct member of the tribe Cupanieae within the soapberry family (Sapindaceae). HCA indicates that Grimmipollis is morphologically closest to Meyeripollis, whereas palynological analysis suggests that G. burmanica was most likely a member of seasonal swamp forests. from Natural Earth (https://www.naturalearthdata.com/), with ranges referring to Licht et al. (2014[START_REF] Licht | Paleogene evolution of the Burmese forearc basin and implications for the history of India-Asia convergence[END_REF] and [START_REF] Morley | Structural and tectonic development of the Indo-Burma ranges[END_REF] and red square indicating the locality of the Kalewa section, while (B) was modified after [START_REF] Licht | Paleogene evolution of the Burmese forearc basin and implications for the history of India-Asia convergence[END_REF], with geomagnetic polarity timescale (GTS16) from [START_REF] Ogg | A concise geologic time scale[END_REF], and magnetostratigraphy from [START_REF] Westerweel | Burma Terrane collision and northward indentation in the Eastern Himalayas recorded in the Eocene-Miocene Chindwin Basin (Myanmar)[END_REF] S3 andS4) based on hierarchical cluster analysis (HCA) with Ward's method of pollen sculptural characters from selected extant and fossil genera. Extant genera are those with parasyncolporate pollen in Sapindaceae, and fossil general are those with syncolporate/parasyncolporate pollen in [START_REF] Jansonius | Genera file of fossil spores and pollen[END_REF]. The dendrograms support our establishment of the new genus Grimmipollis, and its similarity with Meyeripollis. Abbreviations of tribes: C = Cupanieae, N = Nephelieae, S = Schleichereae, I = Incertae sedis.

  J o u r n a l P r e -p r o o fJournal Pre-proofThe CMB include four wide Late Cretaceous-Cenozoic Basins: the Chindwin and Shewbo Basins to the north, and the Minbu and Pegu Basins to the south. The Chindwin and Minbu Basins are in a forearc position, while the Shewbo and Pegu Basins are in a backarc setting, separated by the Wuntho-Popa Arc (WPA; Fig.1A;

  Kingdom: PLANTAE Haeckel, 1866 Division: SPERMATOPHYTA Willkomm, 1854 Subdivision: MAGNOLIOPHYTA Cronquist et al., 1966 Order: SAPINDALES Juss. Ex Bercht. and J.Presl, 1820 Family: SAPINDACEAE Juss., 1789 Subfamily: SAPINDOIDEAE Burnett Burnett, 1835 Tribe: CUPANIEAE Blume, 1847 Genus: Grimmipollis Huang, Morley et Hoorn gen. nov.

  Huang, Morley et Hoorn sp. nov. (缅甸格林氏粉, miǎn diàn gé lín shì fěn) Plates I-IV Holotype: IBED-T 40.976 (Plate IA). Paratypes: IBED-T 40.974 (Plate IB), IBED-T 41.222/1 (Plate IIA), IBED-T 41.227 (Plate IIC), IBED-T 40.968 (Plate IIIA), IBED-T 41.222/2 (Plate IIIC), IBED-T 41.189 (Plate IVA), IBED-T 41.222/3 (Plate IVC).

Repository:

  Institute for Biodiversity and Ecosystem Dynamics (IBED), University of Amsterdam, the Netherlands, and Department of Palaeontology, University of Vienna, Austria. Information about the numbers of sample/residue and England Finder coordinates of the pollen micrographs in the Figures are provided in Supplementary Material 1, Table

Fig. 1 .

 1 Fig. 1. Topographic map (A) and sedimentary log (B), showing the locality, lithology, and pollen-productive sampling layers of the upper Eocene Yaw Formation in Central Myanmar Basins. (A) was made with base map

  Abbreviations: in (A), HB = Hukawng Basin, CB = Chindwin Basin, SB = Shwebo Basin, MB = Minbu Basin, PB = Pegu Basin, WPA = Wuntho-Popa Arc; in (B), vfs = very fine sand, fs = fine sand, ms = medium sand, cs = coarse sand, B = Boulder.

Fig. 2 .

 2 Fig.2. Dendrograms (constructed with data from Supplementary Material 1, TablesS3 and S4) based on

Fig. 3 .

 3 Fig. 3. A, Details of Sedimentary log referring to Fig. 1B. B and D-F, Sequence biostratigraphic contents from Huang et al. (2023): B shows depositional sequences and systems tracts; D is pollen zones; E indicates Geomagnetic polarity timescale (GTS16) (Ogg et al., 2016) and corresponding epoch and age; F is succession of late Eocene depositional sequences according to Hardenbol et al. (1998) and the coastal onlap curve of International Commission on Stratigraphy (ICS) (2020); Abbreviations: TST = transgressive systems tract, LST = lowstand systems tract, HST = highstand systems tract, PF = planktonic foraminifer, N = calcareous nannofossil. C, Pollen diagram showing the compositional change of Grimmipollis burmanica and freshwater palms (Dicolpopollis, Longapertites and Palmaepollenites kutchensis), coastal palm(-like) taxa

Table 2 )

 2 : Cranwellia has a striate appearance; Duplopollis is minutely reticulate; Myrtaceidites is not tuberculate; granules of Nymphoideipites are often joined in striae running parallel to the side, forming finely reticulate contour; Retisyncolporites and Retitrisyncolpites are reticulate; although Rhamneaepites has no , it is not tuberculate from the micrograph; Rhoipteapollis is only faintly granulate; Sapindaceaepites is echinate or otherwise ornamented and not tuberculate; Striatellipollis is striate with reticulate apocolpia; Symplocacites is indistinctly reticulate with sinuous muri; Xinjiangpollis can also be punctate, scabrate, reticulate or striate.

	J o u r n a l P r e -p r o o f

description

Table 1

 1 Sapindaceae genera with parasyncolporate pollen types (occurring only in the subfamily Sapindoideae, and mostly in the tribe Cupanieae) and their features of aperture and sculpture, based on Acevedo-Rodríguez et al. (2011). Abbreviations: C = Cupanieae, N = Nephelieae, S = Schleichereae, I = Incertae sedis.

	Genus and tribe Aperture J o u r n a l P r e -p r o o f Sculpture
	Arytera (C)	Colporate to parasyncolporate, rarely	Rugulate to striate-rugulate
		syncolporate
	Cnesmocarpon	Syncolporate or parasyncolporate	Psilate or indistinctly rugulate
	(C)
	Cupania (C)	Syncolporate or parasyncolporate	Rugulate
	Cupaniopsis (C)	Syncolporate or parasyncolporate,	Rugulate, striate-reticulate, reticulate or

Table 2

 2 Fossil genera with syncolporate and parasyncolporate pollen types in[START_REF] Jansonius | Genera file of fossil spores and pollen[END_REF].

	Genus	Aperture	Sculpture
	Cranwellia	Occasionally syncolporate	Granulate with a striate appearance
	Cupanieidites	Syncolporate or parasyncolporate	Distinctly or faintly reticulate, smooth, punctate,
			granulate, striate or reticulate, or scabrate to
			reticulate
	Duplopollis Meyeripollis Myrtaceidites Nymphoideipites Parasyncolporate Syncolporate or parasyncolporate Syncolporate Syncolporate or parasyncolporate Phaseoliidites Syncolporate Retisyncolporites Syncolporate Retitrisyncolpites Mostly syncolporate, but also recognized as trichotomoculcate Rhamneaepites Syncolporate Rhoipteapollis Syncolporate Sapindaceaepites Syncolporate Striatellipollis Syncolp(or)ate Symplocacites Syncolporate Syncolporites Syncolporate J o u r n a l P r e -p r o o f Minutely reticulate with underlying collumellae layer Gemmate, tuberculate and verrucate Smooth, rugulate, delicately granulate or finely infratextured, never distinctly reticulate Finely granulate, granules often joined in striae running parallel to the side, contour finely reticulate Psilate, scabrate to granulate, weakly verrucate or vermiculate Reticulate, muri simplibaculate Reticulate No description Faint, granulate Reticulate, or otherwise ornamented, but never echinate or verrucate Punctate or radially striate, apocolpia reticulate Indistinctly reticulate with sinuous muri Scabrate to micro-verrucate, psilate, or rugulate,
			annulus psilate
	Xinjiangpollis	Occasionally syncolporate	Smooth, punctate, scabrate, granulate, reticulate or
			striate

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Acknowledgements

We thank the Myanmar Paleoclimate and Geodynamics Research Group members for the help on sample collection; Annemarie Philip and Malcolm Jones for processing pollen samples; Friðgeir Grímsson and Reinhard Zetter for the assistance on microphotography; Jan van Arkel for microphotography and preparing pollen plates; Andrew Thornhill, Mike Macphail, Carlos Jaramillo, Bandana Samant, Mahi Bansal and Vandana Prasad for their suggestions on pollen identification; and Eric Grimm (deceased) for organizing an instructive Tilia workshop at the University of Amsterdam in 2017 (attendees including the authors HH and CH). HH was supported by the China Scholarship Council (CSC grant 201604910677) and the University of Amsterdam; AL J o u r n a l P r e -p r o o f Journal Pre-proof and GD-N acknowledge the European Research Council Consolidator Grant (MAGIC 649081); AL was additionally supported by the ANR grant (ANR-19-ERC7-0007).

Appendix A. Supplementary data

Supplementary material 1

Declaration of interests

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

☐ The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:

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Highlights

•

Grimmipollis burmanica, a new genus and species was reported from the late Eocene central Myanmar.

• Grimmipollis pollen grains are parasyncolporate with specific sculpture.

• Hierarchical cluster analysis suggests Grimmipollis has closest similarity with Meyeripollis.

• G. burmanica is likely derived from an extinct Cupanieae and thus a rainforest tree species.

• G. burmanica could be a member of seasonal swamp forests.

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Graphics Abstract


Franck Jourdan
email: franck.jourdan@umontpellier.fr

Jonaz Vasquez-Villegas

Rania Abdel Rahman

E L Anwar

Simon Le Floc'h A

Christiane Wagner-Kocher

Semi-analytical model for stretch ratio determination in inflation test for isotropic membranes

Keywords:

This article wishes to contribute to the understanding of principal strains distribution in inflation test for circular isotropic soft membranes, combining analytical, numerical, and experimental investigations. The method developed gives a relatively simple and original solution of the deformation and thus of the stresses during an inflating test, provided that the measurement is sufficiently close to the pole. For this purpose, a semi-analytical model (SAM) is proposed and to evaluate that theory, two complementary approaches are used. The first approach is experimental. Inflation and uniaxial tensile tests are set up on medical silicone sheets to identify their hyperelastic constitutive parameters, using Digital Image Correlation. Then these mechanical properties are injected into a 3D finite element model (FEM) of membrane, simulating the inflation test, to compare the strains that had been obtained with those of the SAM. Numerical results show that the strain calculated by the SAM follows the pattern of the circumferential strain obtained by the FEM. Up to a certain distance from the center of the disk (half of the disk for our example), strain calculated by the SAM give a correct approximation of the deformation with respect to radial and circumferential FEM strains.

Introduction

The literature is very well covered in articles dealing with mechanical tests of the inflation type. As explained in [1], there has been a lot of theoretical research on the distribution of deformations. At first, the work of [2] developed a numerical solution for the principal stresses and strains, using Taylor series. Then, [3,4] determined strains, and curvatures by solving, numerically, a system of first-order differential equations.

[5] resumed these approaches to outline a method for determining functional forms of constitutive relations and to demonstrate the validity of this method through numerical experiments for a membrane having a known strain energy function. Before that, in 1981, [6] established deformation expressions by assuming that the radial deformation was constant for small deformations.

The objective of our work is to propose a simple method for the determination of stretch ratios for use in the experimental identification of material parameters. This study results from work on the characterization of the mechanical properties of biological membranes, as work of [5,[7][8][9][10]. Often these membranes are anisotropic, but some are isotropic or weakly anisotropic. This is generally the case when they are in anatomical areas where no direction of stress is preferred. Here we will focus on the case of isotropic membranes as work of [11,12]. In the literature identification of the constitutive parameters is conducted using either the external shape of the inflated membrane [13], or the displacement at the special points [14], or increasingly now, the displacement fields obtained by DIC [1,7,8,15,16]. Our identification approach is in this trend. It considers both DIC and the shape of the membrane avoiding uncertainties about the exact position of the top of the membrane, when calculate deflections.

This article wishes to contribute to the understanding of principal strains distribution combining analytical, numerical, and experimental investigations. The first section is devoted to the semi-analytical method (SAM). Then, we develop experimental approaches allowing the identification of hyperelastic material parameters, using stretch ratios developed in the SAM. Finally, we present the numerical finite element study simulating an inflation test and compare the numerical strains


Pierre Bras
email: pierre.bras@sorbonne-universite.fr

Gilles Pagès
email: gilles.pages@sorbonne-universite.fr.

Fabien Panloup
email: fabien.panloup@univ-angers.fr

Msc Classication

Total variation distance between two diffusions in small time with unbounded drift: application

Keywords: Stochastic Dierential Equation, Euler scheme, Total Variation, Richardson-Romberg extrapolation, Aronson's bounds

Introduction

The convergence properties of Euler-Maruyama schemes to approximate the solution of a Stochastic Dierential Equation (SDE) have been extensively studied, in particular for L p distances. However, the literature seems to lack some results about the convergence in total variation in small time. More specically, in this paper we consider the two following SDEs in R d starting at the same point:

X x 0 = x ∈ R d , dX x t = b 1 (t, X x t )dt + σ 1 (t, X x t )dW t , Y x 0 = x, dY x t = b 2 (t, Y x t )dt + σ 2 (t, Y x t )dW t ,
where W is a Brownian motion. We generally assume that for i = 1, 2, b i is Lipschitz continuous and that σ i is elliptic, bounded and Lipschitz continuous, but we do not assume that b i is bounded. Our objective is to give bounds of the total variation distance between the law of X x t and the law of Y x t , denoted d TV (X x t , Y x t ), as t → 0. In particular, we apply our results to the case where Y x = Xx is the one-step Euler-Maruyama scheme associated to the SDE X, given by dY x t = b 1 (0, x)dt + σ 1 (0, x)dW t .

Such bounds are well known for L p distances and their associated Wasserstein distances and are known to be of order t as t → 0. Yet the literature seems to lack results as it comes to d TV . If σ 1 = σ 2 is constant, then it is classical background that d TV (X x t , Y x t ) is of order t, using a Girsanov change of measure (see for example [PP20, Proposition 4.1]) but this strategy cannot be applied to non-constant σ. The diculty of the total variation distance in small time is the following: considering its representation formula and comparing it with the L 1 -Wasserstein distance, if x and y ∈ R d are close to each other and if f : R d → R is Lipschitz continuous, then we can bound |f (x)-f (y)| by [f ] Lip |x-y|;

whereas if f is simply measurable and bounded, then we cannot directly bound |f (x) -f (y)| in terms of |x -y|. Moreover the regularizing properties of the semi-group cannot be used in small time for the total variation distance.

Results in the literature focus on the Euler-Maruyama scheme. In [START_REF] Bally | The law of the Euler scheme for stochastic dierential equations. I. Convergence rate of the distribution function[END_REF] is proved the convergence for a xed time horizon T > 0 and as N → ∞, where N is the number of steps in the Euler-Maruyama scheme on a nite horizon. More precisely, if σ 1 is elliptic and if b 1 and σ 1 are C ∞ with bounded derivatives (but b 1 and σ 1 are not supposed bounded themselves), then ([BT96, Theorem 3.1])

∀x ∈ R d , d TV (X x T , Xx,N T ) ≤ K(T )(1 + |x| Q ) N T q ,
where Xx,N T stands for the Euler scheme with N steps, where Q and q are positive exponents and where K is a non-decreasing function depending on b 1 and σ 1 . The common strategy of proof for such bounds is to use Malliavin calculus in order to perform an integration by parts and to use bounds on the derivatives of the density. However, we cannot infer a bound as T → 0 since we do not know whether K(T )/T q → 0 as T → 0 in general. In [START_REF] Gobet | Sharp estimates for the convergence of the density of the Euler scheme in small time[END_REF] are given bounds in small time and as N → ∞. Assuming that σ 1 is uniformly elliptic and that b 1 and σ 1 are bounded with bounded derivatives up to order 3, then ([GL08, Theorem 2.3]) ∀t ∈ (0, T ], ∀x, y ∈ R d , |p(t, x, y) -pN (t, x, y)| ≤ K(T )T N t (d+1)/2 e -C|x-y| 2 /t , where p and pN denote the transition densities of X x and Xx,N respectively and where C is a positive constant depending on d and on the bounds on b 1 and σ 1 and on their derivatives. However, we cannot directly use this result for the total variation distance: taking N = 1 yields

d TV (X x t , Xx t ) = R d |p(t, x, y) -pN (t, x, y)|dy ≤ K(T )T t -1/2 R d 1 t d/2 e -C|x-y| 2 /t ,
giving a bound in t -1/2 which does not converge to 0 as t → 0. Moreover, [START_REF] Gobet | Sharp estimates for the convergence of the density of the Euler scheme in small time[END_REF] assumes that b 1 is bounded. [START_REF] Bencheikh | Convergence in total variation of the Euler-Maruyama scheme applied to diusion processes with measurable drift coecient and additive noise[END_REF] focuses on the case where b 1 is bounded and measurable but not necessarily regular and where σ 1 is constant; it proves that the convergence in total variation of the Euler scheme on a nite horizon which is regularized with respect to the irregular drift b 1 and with step h, is of order √ h. In the present paper, we rst prove a convergence rate of order t 1/3 for d TV (X x t , Y x t ), provided that for i = 1, 2, σ i is elliptic, σ i and b i are Lipschitz-continuous with respect to their time variable and that σ i is C 2 b and b i is C 1 and Lipschitz-continuous with respect to their space variable. More generally, if we furthermore assume that σ is C 2r b , then we obtain a convergence rate of order t r/(2r+1) . Letting r → ∞, we also prove that if σ ∈ C ∞ b with some technical condition on the derivatives of the densities of the random variables X x t and Y x t , then the convergence rate is of order t 1/2 exp(C log(1/t)) which is "almost" t 1/2 . Moreover, we provide an example using the geometric Brownian motion where the convergence rate is exactly t 1/2 , thus showing that we cannot achieve better bounds in general. To prove the bound in t r/(2r+1) , we use a multi-step Richardson-Romberg extrapolation [START_REF] Fry | Ix. the approximate arithmetical solution by nite dierences of physical problems involving dierential equations, with an application to the stresses in a masonry dam[END_REF] [LP17], which is a method imported from numerical analysis that we use in our case for theoretical purposes. It relies on a Taylor expansion with null coecients up to some high order. Such method can be used in more general settings with regularization arguments in order to improve the convergence rates (in our case, we improve t 1/3 into t r/(2r+1) ).

Interestingly, the dierence between the drift coecients b 1 -b 2 does not need to be small for our bounds to be valid. This is because the dominant term in d TV (X x t , Y x t ) comes from the the diusion part.

Recent results (see [START_REF] Clement | Hellinger and total variation distance in approximating Levy driven SDEs[END_REF]) establish a convergence in small time at rate t 1/2 for the Euler scheme of certain classes of diusions driven by stable Lévy processes, not directly including the Brownian case. This approach relies on Malliavin calculus techniques. In this work the "standard" drift is replaced in the Euler scheme by the ow of the associated (noiseless) ODE. This seems to be specic to Lévy driven SDEs. Adapting this approach to our general continuous framework is not as straightforward as could be expected and would deserve further investigations for future work.

The total variation distance is closely related to the estimation of the density of the solution to an SDE and this density satises a Fokker-Planck Partial Dierential Equation PDE (3.2). If the drift is bounded, then the density and its partial derivatives admit sub-gaussian Aronson's bounds (see [START_REF] Friedman | Partial dierential equations of parabolic type[END_REF] and Section 3.1). However, giving estimates and bounds for the solution of the PDE in the case of unbounded drift appears to be more dicult, see [START_REF] Lunardi | Schauder estimates for a class of degenerate elliptic and parabolic operators with unbounded coecients in R n[END_REF], [START_REF] Cerrai | Analytic semigroups and degenerate elliptic operators with unbounded coecients: a probabilistic approach[END_REF], [START_REF] Bertoldi | Estimates of the derivatives for parabolic operators with unbounded coecients[END_REF]. Recent improvements have been made in [START_REF] Menozzi | Density and gradient estimates for non degenerate Brownian SDEs with unbounded measurable drift[END_REF] using the parametrix method. Studying this case is useful to study the convergence in total variation of SDE's with unbounded drift, in particular for the Langevin equation, very popular in stochastic optimization, and which reads

dX t = -∇V (X t )dt + σ(X t )dW t ,
where in many cases, V : R d → R has quadratic growth and ∇V has linear growth (see for example [START_REF] Bras | Convergence of Langevin-Simulated Annealing algorithms with multiplicative noise[END_REF]).

In order to deal with unbounded b i , we propose two dierent methods. First, we use a localization argument and "cut" the drift b i into bi outside a compact set, so that we can use bounds from [Fri64] for the bounded drift case. We use the Girsanov formula to explicit the dependence of these bounds in ∥ bi ∥ ∞ . A second method consists in using the density estimates from [MPZ21, Section 4] to improve the dependency with respect to the bounds in x. However this second approach relies on advanced parametrix methods which require further regularity assumptions on the coecients of the SDE and which are not fully detailed for higher order derivatives. Our rst method is clearly much more elementary, starting from a quite general bound established for any pair of integrable random vectors (see Theorem 2.7) and calling upon a standard regularization strategy which combined with a multistep procedure, seems to be at least quasi-optimal in a very general framework.

Notations

We endow the space R d with the canonical Euclidean norm denoted by | • |. For x ∈ R d and for R > 0, we denote B(x, R) = {y ∈ R d : |y -x| ≤ R}.

We denote M d (R) the set of d × d matrices with real coecients. For M ∈ (R d ) ⊗k , we denote by ∥M ∥ its operator norm, i.e.

∥M ∥ = sup u∈R d×k , |u|=1 M • u. If M : R d → (R d ) ⊗k , we denote ∥M ∥ ∞ = sup x∈R d ∥M (x)∥. For k ∈ N and if f : R d → R is C k , we denote by ∇ k f : R d → (R d ) ⊗k its dierential of order k. If f is Lipschitz continuous, we denote by [f ] Lip its Lipschitz constant. If f : (t, x) ∈ R × R d → f (t, x) is
C k with respect to x, we still denote by ∇ k f its dierential with respect to x. We denote the total variation distance between two distributions π 1 and π 2 on R d :

d TV (π 1 , π 2 ) = 2 sup A∈B(R d ) |π 1 (A) -π 2 (A)|.
Without ambiguity, if Z 1 and Z 2 are two R d -valued random vectors, we also write d TV (Z 1 , Z 2 ) to denote the total variation distance between the law of Z 1 and the law of Z 2 . We have as well

d TV (π 1 , π 2 ) = sup R d f dπ 1 - R d f dπ 1 , f : R d → [-1, 1] measurable .
Moreover, we recall that if π 1 and π 2 admit densities with respect to some measure λ, then

d TV (π 1 , π 2 ) = R d dπ 1 dλ - dπ 2 dλ dλ.
We denote by W 1 the L 1 -Wasserstein distance. For x ∈ R d , we denote by δ x the Dirac mass at x. If Z is a Markov process with values in R d , we denote, when it exists, its transition probability from x to y ∈ R d between times s < t, p Z (s, t, x, y).

In this paper, we use the notation C and c to denote positive constants, which may change from line to line.

Main results

We consider the two following SDEs in R d :

X x 0 = x ∈ R d , dX x t = b 1 (t, X x t )dt + σ 1 (t, X x t )dW t , t ∈ [0, T ],
(2.1)

Y x 0 = x, dY x t = b 2 (t, Y x t )dt + σ 2 (t, Y x t )dW t , t ∈ [0, T ], (2.2)
where T is a nite time horizon, b i :

R d → R d , σ i : R d → M d (R), i = 1, 2
, are Borel functions and W is a standard R d -valued Brownian motion dened on a probability space (Ω, A, P). The one-step Euler-Maruyama scheme of X, denoted X, is dened by Xx = Y x when Y x reads

dY x t = b 1 (0, x)dt + σ 1 (0, x)dW t , t ∈ [0, T ]. (2.3)
To allievate notations, we also dene

∆b(x) := |b 1 (0, x) -b 2 (0, x)|, ∆σ(x) := |σ 1 (0, x) -σ 2 (0, x)|. (2.4)
Let us remark that if Y = X, then ∆b(x) = 0 and ∆σ(x) = 0. For g : (t, x) ∈ [0, T ]×R d → g(t, x) ∈ R q and r ∈ N, let us dene the following assumptions:

• Lip t (g): g is Lipschitz continuous with respect to t, uniformly in x.

• g ∈ C r : g is dierentiable with respect to x with continuous partial derivatives up to the order r.

• g ∈ C r b : g ∈ C r and is bounded with bounded partial derivatives up to the order r. • g ∈ C r b : g ∈ C r and has partial bounded derivatives up to the order r, but we do not assume that g is bounded itself.

• For σ :

[0, T ] × R d → M d (R), we say that σ is (uniformly) elliptic if ∃ σ0 > 0, ∀x ∈ R d , ∀t ∈ [0, T ], σ(t, x)σ(t, x) ⊤ ≥ σ2 0 I d .
(2.5)

Theorem 2.1. Let X and Y be the solutions of the SDEs (2.1) and (2.2). For i = 1, 2, assume

Lip t (b i ), Lip t (σ i ), σ i ∈ C 2 b , b i ∈ C 1 b and σ i is elliptic. Then ∀t ∈ [0, T ], ∀x ∈ R d , d TV (X x t , Y x t ) ≤ C(t 1/2 + ∆σ(x)) 2/3 + Ce c|x| 2 t 1/2 , (2.6)
where the positive constants C and c only depend

on d, T , σ0 , ∥σ i ∥ ∞ , [b i ] Lip , [σ i ] Lip and ∥∇ 2 σ i ∥ ∞ . In particular, if Y = X we have d TV (X x t , Y x t ) ≤ Ct 1/3 + Ce c|x| 2 t 1/2 .
Theorem 2.2. Let X and Y be the solutions of the SDEs (2.1) and (2.2). For

i = 1, 2, assume Lip t (b i ), Lip t (σ i ), that σ i ∈ C 2r b , b i ∈ C 1 b and that σ i is elliptic. Then ∀t ∈ [0, T ], ∀x ∈ R d , d TV (X x t , Y x t ) ≤ C(t 1/2 + ∆σ(x)) 2r/(2r+1) + Ce c|x| 2 t 1/2 , (2.7)
where the positive constants C and c only depend on d, T , σ0 , on ∥σ i ∥ ∞ , on the bounds on the derivatives of b i and σ i and on their Lipschitz constants. In particular, if Y = X we have

d TV (X x t , Y x t ) ≤ Ct r/(2r+1) + Ce c|x| 2 t 1/2 .
Remark 2.3. In Theorems 2.1 and 2.2, we can actually improve the dependency of the constants in x in small time since we have more precisely:

∀t ∈ [0, Ce -2(2r+1)c|x| 2 ], d TV (X x t , Y x t ) ≤ C(t 1/2 + ∆σ(x)) 2r/(2r+1) .
Remark 2.4. We can adapt the framework of Theorem 2.2 to the study of SDEs with homogeneous vanishing noise, i.e. where for a > 0,

dX x t = b 1 (t, X x t )dt + aσ 1 (X x t ), dY x t = b 2 (t, Y x t ) + aσ 2 (Y x t )
and we identify the dependency in a as a → 0. Namely, with the same assumptions, for a > 0 small enough we have

d TV (X x t , Y x t ) ≤ Ce ca -1 |x| 2 t 1/2 + Ca -(d+r) (t 1/2 + ∆σ(x)) 2r/(2r+1) , (2.8)
where the constant C does not depend on a. This bound is obtained adapting the proof of Theorem 2.2 in Section 3.4 and using that if Z x is the martingale

dZ x t = aσ(Z x t )dW t then (Z x t ) ∼ ( Zx a 2 t
) where Zx t = σ( Zx t )dW t which does not depend on a. We can also improve the dependency in the initial condition x ∈ R d using [START_REF] Menozzi | Density and gradient estimates for non degenerate Brownian SDEs with unbounded measurable drift[END_REF], however at the expense of further regularity assumptions on b i and σ i , i = 1, 2.

Theorem 2.5. Let X and Y be the solutions of the SDEs (2.1) and (2.2). For i = 1, 2, assume

Lip t (b i ), Lip t (σ i ), σ i ∈ C 2r+1 b , b i ∈ C 2r b and σ i is elliptic. Then ∀t ∈ [0, T ], ∀x ∈ R d , d TV (X x t , Y x t ) ≤ C t 1/2 (1 + ∆b(x)) + ∆σ(x) + t(|b 1 | + |b 2 |)(0, x) 2r/(2r+1) , (2.9)
where the positive constants C and c only depend on d, T , σ0 , on ∥σ i ∥ ∞ , on the bounds on the derivatives of b i and σ i and on their Lipschitz constants. In particular, if Y = X, we have

d TV (X x t , Y x t ) ≤ Ct r/(2r+1) 1 + t 1/2 (|b 1 | + |b 2 |)(0, x)| 2r/(2r+1)
.

Remark 2.6. Choosing Y not to be the Euler-Maruyama scheme of X but a general SDE and expressing the bounds in the Theorems in terms of ∆b(x) and ∆σ(x) allows to extend our results to more general couples of diusions with "close" coecients, for example to SDE solvers other than the genuine Euler-Maruyama scheme. Also, it is helpful to study perturbed SDEs, for example if we consider

dX x t = b 1 (t, X x t )dt + a 1 (t)σ(X x t )dW t , dY x t = b 2 (t, Y x t )dt + a 2 (t)σ(Y x t )dW t (2.10)
where

|a 1 (t) -a 2 (t)| → 0 as t → ∞. Then we have d TV (X x t+s , Y x t+s ) ≤ Ce c|x| 2 s 1/2 + C(s 1/2 + |a 1 (t) -a 2 (t)|) 2r/(2r+1)
and we obtain dierent convergence rates as t → ∞ and s → 0, depending on (t, s). A noticeable example of this is the Langevin-simulated annealing SDE [START_REF] Bras | Convergence of Langevin-Simulated Annealing algorithms with multiplicative noise[END_REF]. Furthermore we remark that in Theorems 2.1 and 2.2, the bounds do not depend on ∆b, enhancing that the dominant term in the total variation comes from the diusion part.

To improve the rate of convergence from t 1/3 in Theorem 2.1 to t r/(2r+1) in Theorem 2.2, we rely on a Richardson-Romberg extrapolation [START_REF] Fry | Ix. the approximate arithmetical solution by nite dierences of physical problems involving dierential equations, with an application to the stresses in a masonry dam[END_REF] [START_REF] Lemaire | Multilevel Richardson-Romberg extrapolation[END_REF]; this argument can also be applied in a more general framework. The following proposition gives bounds on the total variation between two random vectors, knowing bounds on the L 1 -Wasserstein distance and bounds on the partial derivatives of the densities up to some order. Theorem 2.7. Let Z 1 and Z 2 be two random vectors in L 1 (R d ) and admitting densities p 1 and p 2 respectively with respect to the Lebesgue measure. Assume furthermore that p 1 and p 2 are C 2r with r ∈ N and that ∇ k p i ∈ L 1 (R d ) for i = 1, 2 and k = 1, . . . , 2r. Then we have

d TV (Z 1 , Z 2 ) ≤ C d,r W 1 (Z 1 , Z 2 ) 2r/(2r+1) R d ∥∇ 2r p 1 (ξ)∥ + ∥∇ 2r p 2 (ξ)∥ dξ 1/(2r+1) (2.11)
where the constant C d,r depends only on d and on r.

If σ ∈ C ∞
b , then we also prove that we can "almost" get a convergence rate of order t 1/2 .

Theorem 2.8. Let X and Y be the solutions of the SDEs (2.1) and (2.2). For

i = 1, 2, assume Lip t (b i ), Lip t (σ i ), that σ i ∈ C 2r b for every r ∈ N, b i ∈ C 1 b , that σ i is elliptic and that ∆σ(x) = 0. Assume furthermore that if Z and V are the martingales dZ t = σ 1 (t, Z t )dW t and dV t = σ 2 (t, Z t )dW t , then ∀r ∈ N, ∀t ∈ (0, T ], ∀x, y ∈ R d , ∥∇ 2r y p Z (0, t, x, y)∥ + ∥∇ 2r y p V (0, t, x, y)∥ ≤ C 2r t (d+2r)/2 e -c 2r |y-x| 2 /t with lim sup r→∞ C 2r c -d/2 2r 1/(2r) < ∞.
(2.12) (see Theorem 3.1). Then

∀t ∈ (0, T ], ∀x ∈ R d , d TV (X x t , Y x t ) ≤ Ce c|x| 2 t 1/2 + Ct 1/2 e c √ log(1/t) , (2.13)
where the positive constants C and c only depend on d, T , σ0 , on ∥σ i ∥ ∞ , on the bounds on the derivatives of b i and σ i and on their Lipschitz constants.

Remark 2.9. Assumption (2.12) is satised in the case of a Brownian motion, which suggests that this assumption is satised in general provided that σ is "regular enough". Indeed, if dZ t = σdW t with σ ∈ M d (R) being non degenerate, then with Σ := σσ ⊤ we have for t > 0 and x, y ∈ R d :

p Z (0, t, x, y) = 1 det(Σ)t d/2 Φ Σ -1/2 y -x √ t , Φ(u) := 1 (2π) d/2 e -|u| 2 /2 .
Moreover for every r ∈ N and u ∈ R d we have

d r du r Φ(u) ≤ 1 (2π) d/2 | He r (|u|)|e -|u| 2 /2
where He r is the r th probabilist Hermite polynomial. Following [START_REF] Krasikov | New bounds on the Hermite polynomials[END_REF] we have

∀u ≥ 0, | He 2r (u)|e -u 2 /2 ≤ C2 -r √ r (2r)! 2 r! 2 ≤ C2 r √ r,
using the Stirling formula for the last inequality. On the other hand, using [AS64, 22.14.15] we have

∀u ≥ 0, | He 2r (u)|e -u 2 /4 ≤ 2 r+1 r!.
Then, for for every ε ∈ (0, 1],

| He 2r (u)|e -u 2 /2 = He 2r (u)e -u 2 /4 ε He 2r (u)e -u 2 /2 1-ε e -εu 2 /4 ≤ C (2 r r!) ε 2 r r 1/2 1-ε e -εu 2 /2 .
Then if we choose ε r = log -1 (r) for r ≥ 3, we have (r!) εr ≤ e εrr log(r) = e r so that d r du r Φ(u) ≤ C2 rεr e r 2 r r 1/2 e -εr|u| 2 /2 =: A r e -εr|u| 2 /2 and then for r ≥ 3 we have

∥∇ 2r y p Z (0, t, x, y)∥ ≤ ∥Σ -1/2 ∥ 2r det(Σ)t (d+2r)/2 d 2r du 2r Φ Σ -1/2 y -x √ t ≤ ∥Σ -1/2 ∥ 2r det(Σ)t (d+2r)/2 A r e -εr∥Σ -1 ∥|y-x| 2 /(2t)
where

∥Σ -1/2 ∥ 2r A r ε -d/2 r 1/(2r)
is bounded. Thus Assumption (2.12) is satised.

Remark 2.10. For the Euler-Maruyama scheme (2.3), with a slight abuse of notation, x is used both for the starting point and in the denition of the drift and diusion coecients. The transition density should be considered for constant drift and diusion coecients in this case. However the results remain valid as the Euler scheme is simply a Brownian process.

3 Proof of the Theorems

Recalls on density estimates for SDEs with bounded drift

We recall results on the bounds for the density of the solution of the SDE using the theory of partial dierential equations. Let us consider a generic SDE:

Z x 0 = x ∈ R d , dZ x t = b Z (t, Z x t )dt + σ Z (t, Z x t )dW t , t ∈ [0, T ]. (3.1)
Then under regularity assumptions on b Z and on σ Z , the transition probability p Z exists and is solution of the backward Kolmogorov PDE:

p Z (t, t, x, •) = δ x , t ∈ [0, T ], ∂ s p Z (s, t, x, y) = ⟨b Z (s, x), ∇ x p Z (s, t, x, y)⟩ + 1 2 Tr σ ⊤ Z (s, x)∇ 2 x p Z (s, t, x, y)σ Z (s, x) , s < t ∈ [0, T ]. (3.2)
Moreover, p Z and its derivatives satisfy sub-gaussian Aronson's bounds:

Theorem 3.1 ([Fri64], Chapter 9, Theorem 7). Let Z be the solution of (3.1) and let T > 0. Assume Lip t (b Z ) and Lip t (σ Z ), that b Z , σ Z ∈ C r b and that σ Z is elliptic. Then for every m 0 = 0, 1 and for every

0 ≤ m 1 + m 2 ≤ r, ∇ m 0 +m 1 x ∇ m 2
y p Z exists and

∀s < t ∈ [0, T ], ∀x, y ∈ R d , ∥∇ m 0 +m 1 x ∇ m 2 y p Z (s, t, x, y)∥ ≤ C (t -s) (d+m 0 +m 1 +m 2 )/2 e -c|y-x| 2 /(t-s) , (3.3)
where the constants C and c only depend on the bounds on b Z and σ Z and on their derivatives and their Lipschitz constants, on the modulus of ellipticity of σ Z , on d and on T .

Let us also recall the recent result from [START_REF] Menozzi | Density and gradient estimates for non degenerate Brownian SDEs with unbounded measurable drift[END_REF] giving Aronson's bounds of the partial derivatives with respect to y in the case where b Z is unbounded. Considering [MPZ21, Section 4] with [MPZ21, (3.1)], we have the following result. Theorem 3.2. Let Z be the solution of (3.1) and let

T > 0. Assume Lip t (b Z ) and Lip t (σ Z ), that b Z ∈ C r b , σ Z ∈ C r+1 b
and that σ Z is elliptic. Then for every 0 ≤ m ≤ r, ∇ m y p Z exists and

∀s < t ∈ [0, T ], ∀x, y ∈ R d , ∥∇ m y p Z (s, t, x, y)∥ ≤ C (t -s) (d+m)/2 e -c|y-x| 2 /(t-s) , (3.4)
where the constants C and c only depend on the bounds on b Z and σ Z and on their derivatives and on their Lipschitz constants, on the modulus of ellipticity of σ Z , on d and on T .

Preliminary results

In order to apply the bounds on the densities from Theorem 3.1 to Theorem 2.7, we rst "cut" the drifts b 1 and b 2 on a compact set. That is, we instead consider the processes X and Y dened by

d X x t = bx 1 (t, X x t )dt + σ 1 (t, X x t )dW t , t ∈ [0, T ], (3.5) d Y x t = bx 2 (t, Y x t )dt + σ 2 (t, Y x t )dW t , t ∈ [0, T ], (3.6)
where bi , i = 1, 2 is dened as follows. We choose R > 0 and we consider a C ∞ decreasing function ψ : R + → R + such that ψ = 1 on [0, R 2 ] and ψ = 0 on [(R + 1) 2 , ∞) and we dene bx i (t, y) := b i (t, y)ψ(|y -x| 2 ), so that bx i is bounded:

∀y ∈ R d , ∀t ∈ [0, T ], | bx i (t, y)| ≤ sup z∈B(x,R+1) |b i (t, z)| ≤ C(1 + |x|), (3.7) because b i is Lipschitz continuous. Lemma 3.3. Assume Lip t (b 1 ), Lip t (σ 1 ), b 1 ∈ C 1 b , σ 1 ∈ C 1 b . Then for every x ∈ R d and t ∈ [0, T ], d TV (X x t , X x t ) ≤ C(1 + |b 1 (0, x)| 2 )t. (3.8)
Proof. Let f : R d → R be measurable and bounded. We remark that on the event

{sup s∈[0,t] |X x s -x| 2 ≤ R 2 }, we have X x t = X x t , so that |Ef ( X x t ) -Ef (X x t )| ≤ 2∥f ∥ ∞ P sup s∈[0,t] |X x s -x| 2 > R 2 .
But using the inequality |u + v| 2 ≤ 2|u| 2 + 2|v| 2 we have

|X x t -x| 2 ≤ 2 t 0 b 1 (s, X x s )ds 2 +2 t 0 σ 1 (s, X x s )dW s 2 ≤ 2t t 0 |b 1 (s, X x s )| 2 ds+2 t 0 σ 1 (s, X x s )dW s 2 ≤ 4t[b 1 ] 2 Lip t 0 |X x s -x| 2 ds + 1 3 t 3 + 4t 2 |b 1 (0, x)| 2 + 2 t 0 σ 1 (s, X x s )dW s 2 so that E sup s∈[0,t] |X x s -x| 2 ≤ 4t[b 1 ] 2 Lip t 0 E sup u∈[0,s] |X x u -x| 2 ds + 4 3 [b 1 ] 2 Lip t 4 + 4t 2 |b 1 (0, x)| 2 + 2E sup s∈[0,t] s 0 σ 1 (u, X x u )dW u 2 .
Moreover using Doob's martingale inequality we have

E sup s∈[0,t] s 0 σ 1 (u, X x u )dW u 2 ≤ 4E t 0 σ 1 (u, X x u )dW u 2 = 4E t 0 σ 2 1 (u, X x u )du ≤ 4∥σ 1 ∥ 2 ∞ t.
Then we dene the non-decreasing deterministic process S t := E sup s∈[0,t] |X x s -x| 2 and we get the dierential inequality (using t ≤ T )

S t ≤ 4t T |b 1 (0, x)| 2 + 1 3 [b 1 ] 2 Lip T 3 + 2∥σ 1 ∥ 2 ∞ + 4t[b 1 ] 2 Lip t 0
S s ds, so the Gronwall lemma yields

S t ≤ 4t T |b 1 (0, x)| 2 + 1 3 [b 1 ] 2 Lip T 3 + 2∥σ 1 ∥ 2 ∞ e 2t 2 [b 1 ] 2 Lip ≤ C(1 + |b 1 (0, x)| 2 )t.
Using Markov's inequality, we have then

|Ef ( X x t ) -Ef (X x t )| ≤ 2∥f ∥ ∞ P sup s∈[0,t] |X x s -x| 2 > R 2 ≤ 2∥f ∥ ∞ C(1 + |b 1 (0, x)| 2 )t R 2 .
We can now apply Theorem 3.1 to X and to Y however the constants arising depend on the bound on ∥ bx i ∥ ∞ and thus on x. In order to deal with the dependency in ∥ bx i ∥ ∞ , we apply the Girsanov formula and reduce to the null drift case. Proposition 3.4. Let Z x be the solution of

Z x 0 = x, dZ x t = σ 1 (t, Z x t )dW t , t ∈ [0, T ]. (3.9) Assume Lip t (b 1 ), Lip t (σ 1 ), b 1 ∈ C 1 b , σ 1 ∈ C 1 b and σ 1 is elliptic. Then we have for every t ∈ [0, T ], x, y ∈ R d , p X (0, t, x, y) = p Z (0, t, x, y) + t 0 E U x s ⟨ bx 1 (s, Z x s ), ∇ x p Z (s, t, Z x s , y)⟩ ds, (3.10)
where X is dened in (3.5) and U x is dened as

U x s = exp s 0 ⟨g(u, Z x u ) bx 1 (u, Z x u ), dZ x u ⟩ - 1 2 s 0 ⟨g(u, Z x u ) bx 1 (u, Z x u ), bx 1 (u, Z x u )⟩du , (3.11) g = (σ 1 σ ⊤ 1 ) -1 . (3.12)
Proof. First, note that since σ 1 is elliptic and since bx 1 , σ 1 ∈ C 1 b , then p X and p Z exist as well as ∇ x p Z (Theorem 3.1). We then use [QZ04, Theorem 2.4] extended to non-homogeneous diusion processes. Following [QZ04, Remark 2.5], since σ 1 is elliptic and bounded, the assumptions of [QZ04, Theorem 2.4] hold.

We also have the following bounds on the process U x . Lemma 3.5. With the same assumptions as in Proposition 3.4, for every p ≥ 2, x ∈ R d and t ∈ [0, T ] we have

E sup s∈[0,t] |U x s | p ≤ e Cp 2 ∥ bx 1 ∥ 2 ∞ t . (3.13)
Proof. We recall that for every q ≥ 1, the process (U x ) q is a martingale with

d(U x s ) q = q(U x s ) q g(s, Z x s ) bx 1 (s, Z x s ), σ 1 (s, Z x s )dW s .
Thus, Doob's martingale inequality yields

E sup s∈[0,t] |U x s | 2q ≤ Cq 2 ∥ bx 1 ∥ 2 ∞ E t 0 |U x s | 2q ds ≤ Cq 2 ∥ bx 1 ∥ 2 ∞ t 0 E sup u∈[0,s] |U x s | 2q ds.
So with U x 0 = 1 we obtain

E sup s∈[0,t] |U x s | 2q ≤ e Cq 2 ∥ bx 1 ∥ 2 ∞ t .
Lemma 3.6. With the same assumptions as in Proposition 3.4, we have for every x, y ∈ R d and

t ∈ [0, T ], t 0 E U x s ⟨ bx 1 (s, Z x s ), ∇ x p Z (s, t, Z x s , y)⟩ ds ≤ Ce C∥ bx 1 ∥ 2 ∞ e -c|y-x| 2 /t t (d-1)/2 . (3.14)
Proof. We use Theorem 3.1 on the process Z x , which yields bounds with constants depending on σ 1 but not on bx 1 . We obtain for every q ≥ 1 and for every s ∈ [0, t]:

E |∇ x p Z (s, t, Z x s , y)| q = R d |∇ x p Z (s, t, ξ, y)| q p Z (0, s, x, ξ)dξ ≤ C q (t -s) (q+(q-1)d)/2 R d 1 (s(t -s)) d/2 exp -c q |y -ξ| 2 t -s + |ξ -x| 2 s dξ ≤ C q t d/2 1 (t -s) (q+(q-1)d)/2 e -cq|y-x| 2 /t ,
where we used Lemma A.1 in the appendix. Then for p -1 + q -1 = 1 and p ≥ 2, using the Hölder inequality we have

t 0 E U x s ⟨ bx 1 (s, Z x s ), ∇ x p Z (s, t, Z x s , y)⟩ ds ≤ ∥ bx 1 ∥ ∞ sup s∈[0,t] E|U x s | p 1/p t 0 (E |∇ x p Z (s, t, Z x s , y)| q ) 1/q ds ≤ ∥ bx 1 ∥ ∞ e Cp∥ bx 1 ∥ 2 ∞ t C q e -cq|y-x| 2 /t t d/(2q) t 0 ds (t -s) (1+(1-q -1 )d)/2 .
The integral in ds converges under the condition q < d/(d -1) if d > 1, and for any value of q > 1 if d = 1. Then performing the change of variable s = tu we obtain

t 0 E U x s ⟨ bx 1 (s, Z x s ), ∇ x p Z (s, t, Z x s , y)⟩ ds ≤ ∥ bx 1 ∥ ∞ e Cp∥ bx 1 ∥ 2 ∞ T C q e -cq|y-x| 2 /t t (d-1)/2 ≤ Ce C∥ bx 1 ∥ 2 ∞ e -c|y-x| 2 /t t (d-1)/2 .

Proof of Theorem 2.1

Lemma 3.7. We have for every x ∈ R d and t ∈ [0, T ]:

d TV (X x t , Y x t ) ≤ d TV (Z x t , V x t ) + Ce C|x| 2 t 1/2 ,
where

dZ x t = σ 1 (t, Z x t )dW t and dV x t = σ 2 (t, V x t )dW t .
Proof. Let us write

d TV (X x t , Y x t ) ≤ d TV (X x t , X x t ) + d TV ( X x t , Z x t ) + d TV (Z x t , V x t ) + d TV (V x t , Y x t ) + d TV ( Y x t , Y x t ),
where X and Y are dened in (3.5) and (3.6). Using Lemma 3.3 with (3.7), we have

d TV (X x t , X x t ) + d TV ( Y x t , Y x t ) ≤ C(1 + |x| 2 )t.
Using the formula (3.10) and the inequality (3.14), we have

d TV ( X x t , Z x t )= R d |p X (0, t, x, y)-p Z (0, t, x, y)|dy = R d t 0 E U x s ⟨ bx 1 (s, Z x s ), ∇ x p Z (s, t, Z x s , y)⟩ ds dy ≤ Ce C∥ bx 1 ∥ 2 ∞ t 1/2 R d e -c|x-y| 2 /t t d/2 dy ≤ Ce C|x| 2 t 1/2 ,
where we used (3.7). The term d TV ( Y x t , V x t ) is treated likewise.

We now prove Theorem 2.1.

Proof. Let us introduce an articial regularization. For ε > 0 and using Lemma 3.7 we have

d TV (X x t , Y x t ) ≤ Ce C|x| 2 t 1/2 + d TV (Z x t , Z x t + √ εζ) + d TV (Z x t + √ εζ, V x t + √ εζ) + d TV (V x t + √ εζ, V x t ) (3.15)
where ζ ∼ N (0, I d ) and is independent of the Brownian motion W .

• Let f : R d → R be measurable and bounded and let us dene

φ : y ∈ R d → Ef (Z x t + y) = R d f (ξ + y)p Z (0, t, x, ξ)dξ = R d f (ξ)p Z (0, t, x, ξ -y)dξ. (3.16) Then φ is C 2 with ∇ 2 φ(y) = R d f (ξ)∇ 2 y p Z (0, t, x, ξ -y)dξ.
Moreover, using Theorem 3.1, we have

∥∇ 2 y p Z (0, t, x, ξ -y)∥ ≤ C t (d+2)/2 e -c|x-ξ+y| 2 /t ,
where the constants C and c do not depend on bx 1 . This implies that for every y ∈ R d ,

∥∇ 2 φ(y)∥ ∞ ≤ C∥f ∥ ∞ t -1 R d 1 t d/2 e -c|x-ξ+y| 2 /t dξ ≤ C∥f ∥ ∞ t -1 .
Then using the Taylor formula, for every y ∈ R d there exists ỹ ∈ (0, y) such that

φ(y) = φ(0) + ∇φ(0) • y + 1 2 ∇ 2 φ(ỹ) • y ⊗2
and then for some random ζ ∈ (0, ζ) we have

|Ef (Z x t + √ εζ) -Ef (Z x t )| = |Eφ( √ εζ) -φ(0)| = √ εE[∇φ(0) • ζ] + ε 2 E[∇ 2 φ( √ ε ζ) • ζ ⊗2 ] ≤ Cε∥f ∥ ∞ t -1 ,
where we used that E[∇φ(0

) • ζ] = ∇φ(0) • E[ζ] = 0.
This way we obtain

d TV (Z x t , Z x t + √ ϵζ) ≤ Cεt -1 . The term d TV (V x t , V x t + √ ϵζ) is treated likewise.
• Let f : R d → R be measurable and bounded and let us dene

f ε : y → Ef (y + √ εζ) = 1 (2π) d/2 R d f (y + √ εξ)e -|ξ| 2 /2 dξ = 1 (2πε) d/2 R d f (ξ)e -|ξ-y| 2 /(2ε) dξ. (3.17) Then f ε is C 1 with ∇f ε (y) = 1 (2πε) d/2 R d f (ξ) ξ -y ε e -|ξ-y| 2 /(2ε) dξ = ε -1/2 (2π) d/2 R d f (y + √ εξ)ξe -|ξ| 2 /2 dξ = ε -1/2 E[f (y + √ εζ)ζ]
and then

[f ε ] Lip ≤ ∥f ∥ ∞ ε -1/2 E|ζ| ≤ C∥f ∥ ∞ ε -1/2 . (3.18) So that |Ef (Z x t + √ εζ) -Ef (V x t + √ εζ)| = |Ef ε (Z x t ) -Ef ε (V x t )| ≤ C∥f ∥ ∞ √ ε ∥Z x t -V x t ∥ 1 ≤ C∥f ∥ ∞ √ ε (t + t 1/2 ∆σ(x)), (3.19)
where we used Lemma A.3 in the Appendix. This implies that

d TV (Z x t + √ εζ, V x t + √ εζ) ≤ Cε -1/2 (t + t 1/2 ∆σ(x)).
• Conclusion : Considering (3.15), we have

d TV (X x t , Y x t ) ≤ Ce C|x| 2 t 1/2 + Cεt -1 + Cε -1/2 (t + t 1/2 ∆σ(x)).
We now choose ε = t(t + t 1/2 ∆σ(x)) 2/3 , so that

d TV (X x t , Y x t ) ≤ Ce C|x| 2 t 1/2 + C(t 1/2 + ∆σ(x)) 2/3 .

3.4

Proof of Theorem 2.2 using Theorem 2.7

We rst prove Theorem 2.7.

Proof. Let f : R d → R be measurable and bounded, let ε > 0 and let ζ ∼ N (0, I d ) be independent of

(Z 1 , Z 2 ). We have |Ef (Z 1 ) -Ef (Z 2 )| ≤ Ef (Z 1 ) - r i=1 w i Ef ε/n i (Z 1 ) + r i=1 w i Ef ε/n i (Z 1 ) - r i=1 w i Ef ε/n i (Z 2 ) + r i=1 w i Ef ε/n i (Z 2 ) -Ef (Z 2 ) , (3.20)
where f ε is dened as in (3.17) and where the n i 's and the w i 's will be dened later.

Let φ be as dened in (3.16) replacing Z x t by Z 1 . Then, φ is dierentiable up to the order 2r and for all k = 0, 1, . . . , 2r:

∇ k φ(y) = (-1) k R d f (ξ)∇ k p 1 (ξ -y)dξ.
Using the Taylor formula up to order 2r, for every y ∈ R d there exits ỹ ∈ (0, y) such that

φ(y) = φ(0) + 2r-1 k=1 ∇ k φ(0) k! • y ⊗k + ∇ 2r φ(ỹ) (2r)! • y ⊗2r .
Moreover, we have

∇ 2r φ(ỹ) • y ⊗2r ≤ C∥f ∥ ∞ |y| 2r R d ∥∇ 2r p 1 (ξ)∥dξ. (3.21)
Then there exists a random ζ ∈ (0, ζ) such that

Ef (Z 1 + √ εζ) -Ef (Z 1 ) = Eφ( √ εζ) -φ(0) = 2r-1 k=1 ∇ k φ(0) k! ε k/2 • E[ζ ⊗k ] + E[∇ 2r φ( √ ε ζ) • ζ ⊗2r ] (2r)! ε r = r-1 k=1 ∇ 2k φ(0) (2k)! ε k • E[ζ ⊗2k ] + E[∇ 2r φ( √ ε ζ) • ζ ⊗2r ] (2r)! ε r =: r-1 k=1 β k (t)ε k + βr (t, ε)ε r , (3.22) because if k is odd, then E[ζ ⊗k ] = 0.
We now rely on a multi-step Richardson-Romberg extrapolation [LP17, Appendix A]. Let us denote the reners n i = 2 i-1 and the auxiliary sequences and weights

u k := k-1 ℓ=1 (1 -2 -ℓ ) -1 , v k := (-1) k 2 -k(k+1)/2 u k+1 , w k := u k v r-k , k = 1, . . . , r. (3.23)
These weights are the unique solution to the r × r Vandermonde system r i=1

w i n -k i = 1 if k = 0, 0 else. , k = 0, 1, . . . , r -1. (3.24)
Then we have

r i=1 w i Ef (Z 1 + ε/n i ζ) -Ef (Z 1 ) = r i=1 w i r-1 k=1 β k (t)ε k n -k i + r i=1 w i βr (t, ε/n i )ε r n -r i = r-1 k=1 ε k β k (t) r i=1 w i n -k i + ε r r i=1 βr (t, ε/n i )w i n -r i = ε r r i=1 βr (t, ε/n i )w i n -r i , (3.25)
where we used (3.24) in the last equation. Now, using (3.21) we have

r i=1 βr (t, ε/n i )w i n -r i ≤ C∥f ∥ ∞ R d ∥∇ 2r p 1 (ξ)∥dξ r i=1 |w i |n -r i . Since u k → u ∞ = ℓ≥1 (1 -2 -ℓ ) -1 < ∞, the weights satisfy |w i | ≤ u 2 ∞ 2 -(r-i)(r-i+1)/2 , i = 1, . . . , r, so that r i=1 |w i | n r i ≤ u 2 ∞ r i=1 2 -(r-i)(r-i+1)/2 ≤ u 2 ∞ r i=1 2 (r-i)/2 = u 2 ∞ r-1 i=0 2 -i/2 ≤ C. (3.26)
As a consequence and since r i=1 w i = 1, we may write from (3.25)

Ef (Z 1 ) - r i=1 w i Ef ε/n i (Z 1 ) ≤ C∥f ∥ ∞ ε r R d ∥∇ 2r p 1 (ξ)∥dξ. (3.27)
The same way, we obtain

Ef (Z 2 ) - r i=1 w i Ef ε/n i (Z 2 ) ≤ C∥f ∥ ∞ ε r R d ∥∇ 2r p 2 (ξ)∥dξ.
On the other side, using (3.18) we have

r i=1 w i Ef ε/n i (Z 1 ) - r i=1 w i Ef ε/n i (Z 2 ) ≤ C∥f ∥ ∞ √ ε W 1 (Z 1 , Z 2 ) r i=1 |w i |2 (i-1)/2 . (3.28)
Moreover, for every i = 1, . . . , r,

|w i |2 (i-1)/2 ≤ u 2 ∞ 2 -(r-i)(r-i+1)/2+(i-1)/2
and then r i=1

|w i |2 (i-1)/2 ≤ u 2 ∞ r i=1 2 (i-1)/2 ≤ u 2 ∞ 2 r . (3.29)
Thus considering (3.20), we obtain for every ε > 0,

d TV (Z 1 , Z 2 ) ≤ Cε r R d ∥∇ 2r p 1 (ξ)∥ + ∥∇ 2r p 2 (ξ)∥ dξ + Cε -1/2 W 1 (Z 1 , Z 2 ).
Optimizing in ε gives

ε ⋆ = W 1 (Z 1 , Z 2 )/(2r R d ∥∇ 2r p 1 (ξ)∥ + ∥∇ 2r p 2 (ξ)∥ dξ) 2/(2r+1)
and then

d TV (Z 1 , Z 2 ) ≤ C d,r W 1 (Z 1 , Z 2 ) 2r/(2r+1) R d ∥∇ 2r p 1 (ξ)∥ + ∥∇ 2r p 2 (ξ)∥ dξ 1/(2r+1)
.

We now prove Theorem 2.2.

Proof. Using Lemma 3.7, we have

d TV (X x t , Y x t ) ≤ Ce C|x| 2 t 1/2 + d TV (Z x t , V x t ) (3.30)
We now apply Theorem 2.7 with the random vectors Z 1 = Z x t and Z 2 = V x t . Assuming that σ 1 is C 2r b and using Theorem 3.1, ∇ k y p Z exists for k = 0, 1, . . . , 2r and ∀k = 0, 1, . . . , 2r, ∀t ∈ (0, T ], ∀x, y ∈ R d , ∥∇ k y p Z (0, t, x, y)∥ ≤ C t (d+k)/2 e -c|y-x| 2 /t .

Then we have

R d ∇ 2r y p Z (0, t, x, ξ)dξ ≤ Ct -r R d 1 t d/2 e -c|x-ξ+y| 2 /t dξ ≤ Ct -r .
The same way we have

R d ∇ 2r y p V (0, t, x, ξ)dξ ≤ Ct -r .
Applying Theorem 2.7 with Lemma A.3 yields

d TV (Z x t , V x t ) ≤ C( √ t + ∆σ(x)) 2r/(2r+1) .

3.5

Proof of Theorem 2.5

For the proof of Theorem 2.5, we do not use Lemma 3.7; instead we directly apply Theorem 2.7. Using Theorem 3.2, ∇ k y p X and ∇ k y p Y exist for k = 0, 1, . . . , 2r and satisfy the same bounds as previously. Then using Theorem 2.7 with Lemma A.3 we obtain

d TV (X x t , Y x t ) ≤ C( √ t(1 + ∆b(x)) + ∆σ(x) + t|b(x)|) 2r/(2r+1) .

3.6

Proof of Theorem 2.8

Proof. We use Lemma 3.7 again and rework the bound on d TV (Z x t , V x t ) by paying attention to the dependency of the constants in r in the proof of Theorem 2.7 with Z 1 := Z x t and Z 2 := V x t . Since σ 1 ∈ C 2r b for every r ∈ N, we write (3.22) for any r ∈ N and we have

| βr (t, ε)| ≤ C 2r ∥f ∥ ∞ t -r E[|ζ| 2r ] (2r)! , C 2r := C 2r c -d/2 2r ,
where C 2r and c 2r are dened in (2.12) and where

E[|ζ| 2r ] = 2 r Γ(d/2 + r) Γ(d/2) = r-1 i=0 (d + 2i).
Using (3.26) we get

r i=1 βr (t, ε/n i )w i n -r i ≤ C C 2r ∥f ∥ ∞ t -r r-1 i=0 (d + 2i) (2r)!
and we obtain as in (3.27):

Ef (Z x t ) - r i=1 w i Ef ε/n i (Z x t ) ≤ 1 2 κ 1 ∥f ∥ ∞ ε r t -r , κ 1 := C C 2r r-1 i=0 (d + 2i) (2r)! Ef (V x t ) - r i=1 w i Ef ε/n i (V x t ) ≤ 1 2 κ 1 ∥f ∥ ∞ ε r t -r .
On the other hand, considering (3.28) and (3.29) with Lemma A.3 with ∆σ(x) = 0 we have

r i=1 w i Ef ε/n i (V x t ) - r i=1 w i Ef ε/n i (Z x t ) ≤ κ 2 ∥f ∥ ∞ √ ε t, κ 2 := C2 r . (3.31) We now minimize κ 1 ε r t -r + κ 2 ε -1/2 t in ε, giving ε ⋆ = t (2r+2)/(2r+1) (2rκ 1 ) 2/(2r+1) κ 2/(2r+1) 2
and then

κ 1 ε r ⋆ t -r + κ 2 ε -1/2 ⋆ t ≤ Cκ 2r/(2r+1) 2 κ 1/(2r+1) 1 t r/(2r+1)
with as r → ∞:

κ 2r/(2r+1) 2 κ 1/(2r+1) 1 ∼ C 1/(2r+1) 2r r-1 i=0 (d + 2i) 1/(2r+1) 1 (2r)! 1/(2r+1) 2 2r 2 /(2r+1) with r-1 i=0 (d + 2i) 1 (2r+1) = exp r 2r+1 1 r r-1 i=0 log(d + 2i) ≤ exp r 2r+1 log(d+(r-1)) ≤ √ d+r-1, 1 (2r)! 1/(2r+1) ∼ e 2r , lim sup r→∞ C 1/(2r+1) 2r < ∞
where we used Assumption (2.12), so that

κ 2r/(2r+1) 2 κ 1/(2r+1) 1 ≤ C √ d + r -1 e 2r 2 r .
Then we have d TV (Z x t , V x t ) ≤ C2 r r -1/2 t r/(2r+1) and we choose r(t) = ⌊log 1/2 (1/t)⌋ so that as t → 0,

d TV (Z x t , V x t ) ≤ Ct 1/2 exp C log(1/t) .

Counterexample

In this section we give a counter-example showing that we cannot achieve a bound better than t 1/2 in general. More specically, we show that we cannot achieve a bound better than t 1/2 for the total variation between an SDE and its Euler-Maruyama-scheme in general. For x > 0 and σ > 0, let us consider the one-dimensional process

Y x t = xe σWt , (4.1)
where W is a standard Brownian motion. The process Y is solution of the SDE dY x t = (σ 2 /2)Y x t dt + σY x t dW t and its associated Euler-Maruyama schemes reads

Ȳ x t = x + (σ 2 /2)xt + σxW t ∼ N x(1 + tσ 2 /2), σ 2 x 2 t . (4.2)
Proposition 4.1. Let Y be the process dened in (4.1). Then for small enough t we have

d TV (Y x t , Ȳ x t ) ≥ C x t 1/2 . (4.3) Proof. We have p Y (t, x, y) = 1 √ 2πσ 2 t exp -1 2σ 2 t log 2 (y/x) y 1 y≥0 (4.4) so that d TV (Y x t , Ȳ x t ) = 1 √ 2πσ 2 t R exp - log 2 (y/x) 2σ 2 t y -1 1 y≥0 -exp - (y -x -xtσ 2 /2) 2 2σ 2 x 2 t x -1 dy ≥ 1 √ 2πσ 2 ∞ -x/ √ t 1 x + √ ty exp - log 2 (1 + √ ty/x) 2σ 2 t - 1 x exp - (y -x √ tσ 2 /2) 2 2σ 2 x 2
dy.

But we have as (t, y) → 0:

1 1 + √ ty/x exp - log 2 (1 + √ ty/x) 2σ 2 t -exp - (y -x √ tσ 2 /2) 2 2σ 2 x 2 = (1 - √ ty/x + O(ty 2 )) exp - 1 2σ 2 t ( ty 2 x 2 - t 3/2 y 3 x 3 + O(t 2 y 4 )) -exp - y 2 2σ 2 x 2 - tσ 2 8 + √ ty 2σ 2 x = e -y 2 2σ 2 x 2 (1 - √ ty/x + O(ty 2 )) 1 + √ ty 3 2σ 2 x 3 + O(ty 4 ) -1 + √ ty 2σ 2 x - tσ 2 8 + O(t 2 ) + O(ty 2 ) = e -y 2 2σ 2 x 2 - √ ty x - √ ty 2σ 2 x + √ ty 3 2σ 2 x 3 + tσ 2 8 + O(ty 2 ) + O(t 2 ) .
Thus there exists ϵ > 0 and t 0 such that for every t ≤ t 0 :

d TV (Y x t , Ȳ x t ) ≥ 1 √ 2πσ 2 x 2 e -ε 2 2σ 2 x 2 √ t 2 ε -ε - y x - y 2σ 2 x + y 3 2σ 2 x 3 dy, so that d TV (Y x t , Ȳ x t ) is of order t 1/2 as t → 0.
However, the process Y does not satisfy the assumptions of Theorem 2.2 as its noise coecient is not elliptic neither bounded on (0, ∞). We then prove the following result. Proposition 4.2. There exists a diusion process X on R with C 1 and Lipschitz continuous drift, with C ∞ b and elliptic diusion coecient σ and there exists T > 0 and ε ∈ (0, 1) such that

∀t ∈ [0, T ], ∀x ∈ (ε, ε -1 ), d TV (X x t , Xx t ) ≥ C x t 1/2
where X is the Euler-Maruyama scheme of X and where the positive constant C x depends on x.

Proof. We construct from the geometric Brownian motion Y dened in (4.1), a process X with elliptic and bounded drift and such that d TV (X x t , Xx t ) ≥ C x t 1/2 . For ε ∈ (0, 1/2), let us consider ψ : R → R + a C ∞ b approximation of

ψ : x ∈ R -→    x if x ∈ [ε, ε -1 ], ε if x ≤ ε ε -1 if x ∈ [ε -1 , ∞)
such that ψ = ψ on [2ε, ε -1 /2] ∪ (-∞, ε/2] ∪ [2ε -1 , ∞). Then we dene the process with elliptic and bounded noise coecient

dX x t = - σ 2 2 X x t dt + σψ(X x t )dW t .
Then for x ∈ (2ε, ε -1 /2) we have Xx t = Ȳ x t and P(Y x t ̸ = X x t ) ≤ P sup s∈[0,t] Y x s ≥ ε -1 /2 + P inf s∈[0,t] Y x s ≤ 2ε .

With a proof similar to the proof of Lemma 3.3, we show that P sup s∈[0,t] Y x s ≥ ε -1 /2 ≤ C x,ε t.

Moreover, we remark that (Y x ) -1 ∼ x -2 Y x in law so P inf s∈[0,t] Y x s ≤ 2ε = P sup s∈[0,t] (Y x s ) -1 ≥ ε -1 /2 = P sup s∈[0,t] Y x s ≥ x 2 ε -1 /2 ≤ C x,ε t.

Then we obtain

d TV (X x t , Xx t ) ≥ d TV (Y x t , Ȳ x t ) -d TV (X x t , Y x t ) ≥ C x √ t.
Remark 4.3. We could also consider the process X with "cut" bounded drift b and get the same bounds, proving then that we cannot achieve better bounds in general than the ones established in Theorem 2.8 even if we assume that b is bounded.

A Appendix Lemma A.1 ([Fri64], Chapter 9, Lemma 7). For a > 0, 0 < u < t ≤ T , x ∈ R d , ξ ∈ R d , let

I a := R d 1 (u(t -u)) d/2 exp -a |x -y| 2 t -u + |y -ξ| 2 u dy.
Then there exists a constant C > 0 depending only on d and T such that for every 0 < ε < 1,

I a ≤ C (εat) d/2 exp -a(1 -ε) |x -ξ| 2 t .
Let us recall [PP20, Lemma 3.4(a)], with an immediate adaptation to the non-homogeneous case.

Lemma A.2. Let Z be solution to the generic SDE:

Z x 0 = x ∈ R d ,

  dZ x t = b(t, Z x t )dt + σ(t, Z x t )dW t , t ∈ [0, T ],where b and σ are Lipschitz continuous in (t, x) and where σ is bounded. Then for p ≥ 1,∀t ∈ [0, T ], ∀x ∈ R d , ∥Z x t -x∥ p ≤ C(p, T, [b] Lip , [σ] Lip , ∥σ∥ ∞ ) t|b(0, x)| + t 1/2 .Lemma A.3. Let X and Y be the solution to the two general SDEs (2.1) and (2.2). Assume that b and σ are Lipschitz continuous in (t, x) and that σ is bounded. Then for every p ≥ 1,∀t ∈ [0, T ], ∀x ∈ R d , ∥X x t -Y x t ∥ p ≤ C t(1 + ∆b(x)) + t 3/2 (|b 1 | + |b 2 |)(0, x) + ∆σ(x)t 1/2 (A.1)Proof. We rst deal with the case p ≥ 2. We have But using the Burkholder-Davis-Gundy and the generalized Minkowski inequalities, we have constant which only depends on p and where we used Lemma A.2. So that∥X x t -Y x t ∥ p ≤ [b 1 ] Lip Y x s ) -(0, x)∥ p + t∆b(x) + C(t + t 3/2 (|b 1 | + |b 2 |)(0, x)) + ∆σ(x) √ t∥W 1 ∥ p ≤ C t(∆b(x) + 1) + t 3/2 (|b 1 | + |b 2 |)(0, x) + ∆σ(x) √ twhich completes the proof for p ≥ 2. For p ∈ [1, 2), the inequality is still true remarking that ∥ • ∥ p ≤ ∥ • ∥ 2 .

	t	(σ 1 (s, X x s ) -σ 1 (0, x))dW s	≤ C BDG	t	|(s, X x s ) -(0, x)| 2 ds	1/2
	0						p	0	p/2
	≤ C BDG p	[σ 1 ] Lip	0	t	∥(s, X x s ) -(0, x)∥ 2 p ds	1/2	≤ C(t + t 3/2 |b 1 (0, x)|),
	where C BDG
				t				t
				0	∥(s, X x s ) -(0, x)∥ p ds + [b 2 ] Lip	0	∥(s,
		t						t
	∥X x t -Y x t ∥ p ≤	(b 1 (s, X x s ) -b 2 (s, Y x s ))ds	+	(σ 1 (s, X x s ) -σ 2 (s, Y x s ))dW s
		0						p	0

p ≤ t 0 (b 1 (s, X x s ) -b 1 (0, x))ds p + t∆b(x) + t 0 (b 2 (s, Y x s ) -b 2 (0, x))ds p + t 0 (σ 1 (s, X x s ) -σ 1 (0, x))dW s p + t 0 ∆σ(x)dW s p + t 0 (σ 2 (s, Y x s ) -σ 2 (0, x))dW s p p [σ 1 ] Lip p is a