J Brocchieri

L De Viguerie

C Sabbarese

M Boyer

Combination of non-invasive imaging techniques to characterize pigments in Buddhist

Keywords: MA-XRF, Hypersepctral imaging, identification and mapping of pigments, Thangkas

Scanning macro-XRF, XRF point measurements and hyperspectral reflectance imaging in the visible and near-infrared range were combined to characterize the pigments of a Tibetan painting.

The complementary use of the different investigation techniques allowed us to (i) clearly identify the pigments used by the artist, (ii) strengthen the few data present in historical and contemporary sources and (iii) provide data necessary for any accurate conservation treatment of the painting. The proposed methodology allows an in-depth characterization of the materials used in thangkas, with the identification of all the pigments used by Tibetan artists, as shown by the analyses of reference samples. In the present study traditional pigments, commonly cited in historical sources, were identified such as vermilion, minium, orpiment and azurite, but also more unusual compounds such as brochantite and antlerite (in blue/green areas). Elemental distribution images confirmed that some parts of the thangka were restored, without taking into account the original pigments or the original design. Underlying annotations and a carbon black under-drawing were highlighted by the application of near-infrared hyperspectral imaging technique, which proved to be particularly useful for the study of such paintings.

INTRODUCTION

Thangkas are sacred paintings, usually depicting a Buddhist deity [START_REF] Béguin | Les peintures du bouddhisme tibétain[END_REF] . They constitute a unique art form that belongs to Tibetan culture and is linked to the nomadic lifestyle of early Buddhist monks.

Painting a thangka remains a religious practice in itself, and therefore follows ancestral traditions.

The constituent materials have changed little since their appearance eight centuries ago [START_REF] Jackson | Tibetan Thangka painting: methods & materials[END_REF][START_REF] Elgar | [END_REF]4,[START_REF] Ricciardi | Tibetan Printing: Comparison, Continuities, and Change[END_REF] : a cotton cloth or Chinese silk brocade, a natural glue sizing and a coating traditionally composed of chalk or kaolin, then a coloured layer made of mineral pigments and organic dyes. The technique is similar to that of the painters of the western Middle Ages. The main binder for these pigments was size or hide glue, the same gelatinous solution that was used for the preparation layer for the ground.

General information on Tibetan pigments is mainly based on literary sources and oral testimonies of modern-day painters. The now indispensable work of D.P. and J.A. Jackson brings together information on traditional techniques and materials from the testimonies of some twenty painters in India and Nepal, as well as from sources written by scholars [START_REF] Jackson | Tibetan Thangka painting: methods & materials[END_REF] . A few analytical results on thangkas have also been published [START_REF] Béguin | Les peintures du bouddhisme tibétain[END_REF][START_REF] Duffy | Historical painting techniques, materials and studio practice, Prepints of a symposium[END_REF][START_REF] Duffy | Scientific Research in the Field of Asian Art[END_REF][START_REF] Tabasso | DIPINTI TIBETANI dalle spedizioni di Giuseppe TucciMateriali e tecniche alla luce delle indagini non invasive[END_REF][START_REF] Mass | Proceedings of the Forum on the Conservation of Thangkas, Special Session of the ICOM-CC 15th Triennial Conference[END_REF][START_REF] Richard | [END_REF]11,[START_REF] Colinart | ICOM Comittee for conservation, 11th triennal meeting[END_REF] . Duffy and Elgar, published in 1995 their extensive work on twenty-three thangkas from the 17th century combining in-situ X-ray fluorescence (XRF) and an indepth characterization of micro cross-sections [START_REF] Duffy | Historical painting techniques, materials and studio practice, Prepints of a symposium[END_REF] . Ten years later, a broad study was carried out by an Italian collaboration of researchers on more than thirty-five thangkas from the famous scientific missions of Giuseppe Tucci [START_REF] Tabasso | DIPINTI TIBETANI dalle spedizioni di Giuseppe TucciMateriali e tecniche alla luce delle indagini non invasive[END_REF] . A series of investigations was performed with complementary in-situ non-destructive techniques (radiography, infrared reflectography and chemical analyses by colorimetry and X-ray fluorescence) focusing on the painting technique and the materials employed.

The study of thangkas is very often rendered particularly challenging by several factors: the presence of surface alteration products, degradation issues, later restorations and a general lack of information. We present here the first application of chemical imaging techniques to the study of thangka paintings. In the past few years, spectroscopic imaging techniques have proved their value in the investigation of historical paintings [START_REF] Alfeld | [END_REF]14,15,16] . Scanning Macro-XRF (MA-XRF) and Hyperspectral Imaging (HSI) provide more representative results than in-situ punctual analyses or characterization of micro cross-sections. They can be a great help in the characterization of pigments and the possibility of mapping their presence in a painting. To set a methodology for the analysis of thankgas, the first step was to analyse reference pigments. Thereafter, one thangka from a French private collection was investigated during its conservation/restoration, using MA-XRF, XRF point measurements and HSI in the visible and near-infrared range. We provide detailed data relating to the materials and structure of the painting, the artist's palette, and the use of colour and pigment mixtures.

THE SAMPLES

Reference pigments

A set of 8 pigments, traditionally used in thangkas, was used as a reference, all purchased from Kremer Pigmente, Germany: malachite (10300), azurite (10200), vermilion (42000), orpiment (10700), minium (42500), indigo (36000), red aluminium lake of carminic acid (42100), extracted from cochineal, and calcite (58720). Kaolin was borrowed from the MNHN collection (geological sample from Combes des Eyzies, in France). They were analysed as pellets via XRF, visible and near-infrared HSI. X-ray diffraction was used to ensure the composition of the mineral pigments; the kaolin sample was found to be composed of kaolinite and quartz.

The Thangka

The thangka analysed (Figure 1), probably dated from the 18th century, and belongs to a French private collection. In transmitted light, the object was observed to consist of a canvas made of two pieces. As to its state of conservation, the gluing was weakened by multiple rollings and unrollings.

Loss of colour and blackening is particularly observed particularly in the blues and greens, while the red, yellow, and orange areas are quite well preserved. Moreover, dirt due to smoke and handling covers the whole surface, disturbing its legibility.

The main deity is a Vaisravana: the king of the North and god of wealth [START_REF] Tcheuky | Petite encyclopédie des divinités et symboles du boushisme tibetain[END_REF][START_REF] Béguin | [END_REF] . Traditionally, he is mounted on a white lion, with the traditional mongoose in the left hand, regarded in Indian mythology as the animal that spits out jewels. He is accompanied by a suite known as the eight horsemen (Ashvapatis) [START_REF] Béguin | [END_REF] . Figure 1: Photograph of the thangka analysed (French private collection).

EXPERIMENTAL

Point XRF and HSI measurements (visible and near infrared), were used on reference samples, complemented by X-Ray Diffraction (XRD) in the case of inorganic pigments.

On the thangka, 40 XRF spot analyses and 7 MA-XRF scans using various sizes and parameters (detailed below) were carried out. Visible and near-infrared HSI was also carried out over the entire painting, and in certain areas with a better resolution through "zoom" optics. Macrophotographs (Olympus OM-D E-M5 Mark II with a Olympus 60 mm macro lens) of the analysed areas were also recorded.

MA-XRF

The X-ray fluorescence apparatus was designed and developed in the laboratory, and is described elsewhere [19] . In-house built MA-XRF instrument using a Pd anode transmission tube (Moxtek MAGNUM, Orem, UT) operated at 30 kV and 0.1 mA (3 W) and a SD-detector (active area of 25 mm 2 , X-123FAST SDD, Amptek, Bedford, MA). The X-ray tube was connected to the detector via a holder produced by 3D printing (fixing the angle between both to 45° for single point analysis, or 32° for imaging). A digital microscope (Dino-lite, AnMo Electronics Corporation, New Taipei City, Taiwan) and a laser distance measurement device (OADM20, Baumer, Frauenfeld, Switzerland) allow for measuring the position of the primary beam on the surface of the object. The beam size was approximately 1.2 mm and the typical working distance of 1 cm. The system was used with manual translation stages for single-point analysis with acquisition time of 300 s.

For XRF imaging the measurement head was mounted on a motorized XY stage with 20 cm travel range in both directions (M-403.8PD, Physik Instrumente (PI) GmbH & Co. KG, Karlsruhe, Germany) with an additional manual Z translation. In all scans, the lateral resolution was defined by a step size of 0.8 mm. The central area was investigated using a dwell time per pixel 0.2 s.

For the evaluation of the raw full spectral XRF data, the dedicated PyMCA software was used [20] .

HSI

VNIR (Visible Near InfraRed) and SWIR (Short Wave InfraRed) hyperspectral camera by Specim (Oulu, Finland), mounted on translation stagewere used. Illumination was provided by two 20 W halogen lamps 1 m away. For the VNIR camera, the spectral range is 400 nm to 1000 nm with 212 wavelength channels (rebinned from 1200) with a spectral sampling of 2.8 nm. For the SWIR camera, the spectral range is 1000 to 2500 nm with 280 wavelength channels and a spectral sampling of 12 nm. The different acquisition parameters, camera and optics characteristics are reported in Table 1; for the SWIR camera, two optics with different spatial resolution were used.

During the investigation, diffuse illumination was provided by two 20 W halogen lamp, placed at 0.4 m away. The dedicated ENVI software (Harris Corporation, Melbourne, Florida, USA) was used for data treatment. The data was normalized with dark and bright field images using the Specim plug-in in ENVI. The spectral angle mapper (SAM) algorithm of ENVI was used to obtain reflectance map considering only a portion of the spectral range that contained the key spectral feature/s characteristic of the material to be mapped.

RESULTS AND DISCUSSION

Reference Pigments

According to Jackson [START_REF] Jackson | Tibetan Thangka painting: methods & materials[END_REF] , thangkas are painted with a set of pigments, sourced from a number of sites in the Tibetan area, presented here by colour:

-blue and green: copper carbonates, respectively azurite and malachite. It is said that the artist's first task was to fill in large areas of sky and landscape with blues and greens.

-red: native cinnabar or its synthetic form vermilion (HgS).

-orange: minium or its synthetic form red lead (Pb 3 O 4 ) ; realgar being mainly used for wall paintings.

-yellow: orpiment (As 2 S 3 ).

Yellow and red ochres are also cited, but said to be used mainly for wall paintings (and not in thangkas), or as an undercoat for gold.

-white: calcium minerals, consisting mainly of calcium carbonate (CaCO 3 ).

-black: charcoal (wood chars) or lamp black.

Gold was also widely used as well as different organic dyes, such as indigo, and red « lac dye » (from the resin secreted by lac insects).

We present here the results obtained on the main pigments, as reference spectra. The complementary use of XRF and HSI (VNIR and SWIR) allow the complete identification of the whole palette (Table 2). Figure 2 shows the visible reflectance spectra of orpiment, minium, vermilion, malachite, azurite, indigo and a cochineal red lake (as a substitute of lac dye, with similar spectral features [21] ). The SWIR reflectance spectra of malachite, azurite, calcite and kaolin, which display characteristics bands in this domain, are reported in Figure 3.

Reflectance Wavelength (nm) !0,005% 0,005% 0,015% 0,025% 0,035% 0% 0,2% 0,4% 0,6% 0,8% 1% 1,2% 400% 500% 600% 700% 800% 900% !0,005% 0,005% 0,015% 0,025% 0,035% 0,045% 0,055% 0% 0,5% 1%
(2)

(3)

Figure 3: SWIR reflectance spectra of: a) reference samples, as pellets: kaolin (grey), calcite (black), azurite (blue) and malachite (green) (the spectrum of kaolin was offset for clarity).

Identification and mapping of pigments

The pigments identified by the different analytical methods, according to the characteristics indicated in table 2, are reported in table 3 (with typical spectra in Supporting Information).

Elemental maps acquired by XRF (figure 4 and7) allow a straightforward visualization of the pigments distribution, showing subtle differences in peak intensities. Distribution maps of pigments on restricted area could also be obtained thanks to hyperspectral data treatment as discussed in 4.3 and shown in figure 8. * no characteristic feature observed by this technique; ** results difficult to interpret due to the degradation state of the thangka; ***possible hypothesis that needed to be confirmed thanks to XRF (Figure S1). The main yellow pigment, actually used in almost all the whitish areas, is orpiment, as indicated by its reflectance spectrum (figure S2) and the presence of As and S.

Golden areas in the flesh of the main deity are mainly composed of gold (Au) as detected by X-ray fluorescence (figure 4 andS1) and suggested by the inflection point at 510 nm of the visible reflectance spectra (figure S1). Orpiment (As and S) is also present, probably applied as an underlayer of the gold. An ochre base is traditionally used but other bases exist such as mixtures of orpiment and calcium white [START_REF] Jackson | Tibetan Thangka painting: methods & materials[END_REF] . On the contrary, the gold colour of the secondary divinities, such as dots (Figure 5) in the wrathful deity (below the main deity), correspond to orpiment only, being less precious zones or more altered ones.

The orange used to decorate clothing ornaments in the deity and horsemen is minium, identified thanks to the presence of Pb and its reflectance spectrum (figure S2).

Red and Pink in the lips of the main deity, in the tongue of snow lion, in the aura of some of the horsemen and in many decorations of the painting, are composed of Hg and S, which indicates the use of vermilion (confirmed by the reflectance spectrum, figure S2) Some red parts, rather darkened today, indicate a strong K signal as visible in figure 4: in clothes but also in the eyes of the deity. This can be linked to the use of a lake pigment, prepared by precipitating or adsorbing an organic dye onto an inert inorganic substrate such as alum, traditionally potash alum KAl(SO 4 ), 12H 2 O. The presence of aluminium in these zones would confirm this use; however as the measurements were done in-situ, the absorption in air does not permit access to information for elements below silicon. The VNIR spectra of these areas do not contain any characteristic feature (except for a rise in the slope between 500 and 800 nm) probably because of their darkening. We can thus only assume the use of a lake pigment, in accordance with the painting technique of thangkas: used for the eyes, clothes and also as a thin glaze on minium in the necklace.

The brown colour used for the flesh is composed of minium and orpiment, indicated by the presence of As and Pb detected by X-ray fluorescence, and by the presence of the corresponding two inflection points in the visible reflectance spectrum.

Blue/dark blue colours mainly correspond to azurite identified thanks to its characteristic nearinfrared spectrum and to the presence of Cu detected by XRF. The XRF maps in figure 4 reveal the co-localization of Cu with significant traces of manganese (Mn), probably linked to the azurite source (a natural ore). Here again, different shades of blue and paint aspects are visible, such as in the Lotus flowers (figure 5d): changing the pigment/binder ratio or the size of the particles can modify the aspect of the painting although based on the same pigment. In [START_REF] Jackson | Tibetan Thangka painting: methods & materials[END_REF] , the authors report how Tibetan artists and pigment-makers obtain four kinds of blue, corresponding to four ranges of particle sizes, thanks to a "progressive sedimentation".

More greenish zones, or mixtures of darkened blue and green can also be found. A strong darkening of azurite is very commonly observed on thangkas, although not as yet clearly understood. In the thangka under investigation, if some zones could be attributed easily to azurite, such as the halo of the main deity, some blue/greenish areas presented more complex spectra, probably originating from mixtures of Cu-compounds (figure 6b). In particular, it is possible to identify basic copper (II) sulfates, brochantite [Cu 4 SO 4 (OH) 6 ], and antlerite [Cu 3 SO 4 (OH) 4 ]: according to [START_REF] Catelli | Can hyperspectral imaging be used to map corrosion products on outdoor bronze sculptures?[END_REF] , the brochantite SWIR spectrum shows a peak at 2334 nm, and antlerite is characterized by a doublet at 2304 and 2352 nm. Brochantite has been identified in thangkas by Duffy and Elgar [START_REF] Duffy | Historical painting techniques, materials and studio practice, Prepints of a symposium[END_REF] , but mixed with malachite in green zones. The authors explained it by their natural combination in malachite deposits. Here it is unclear whether these minerals were present in the original pigments (found in natural azurite and malachite ores, or produced for painting purposes) or are secondary products formed by the degradation of azurite and malachite [23] .

The pigments used for white and in the preparation layer were difficult to analyse due to the degraded and darkened aspect of the painting. Even for white, most of the areas analysed were in fact yellow, made with orpiment. One area, a white mountain covered with snow (visible clearly in figure 8c) is most probably painted with a white pigment; its XRF spectrum shows the presence of Fe, Ti, K, Ca, S and Si but also Cu (probably originating in the surrounding blue sky), and traces of Hg and As that may have been added to the ground. As calcite and kaolin are indicated in the sources as used for white pigments and in the preparation layer, we compare in figure 6a the SWIR spectrum of this area to the ones of the calcite and kaolin reference samples. A calcite compound, together with the textile support, can explain the main bands observed. According to [24,25] , a cellulose fibre (such as cotton, most probable in the case of a Tibetan artwork) exhibits features at 2275 nm, 2105 nm and 1930 nm, and a very broad band centered at ca. 1490 nm. Minor supplementary bands and slight displacements observed in our spectrum can also be due to the addition of other clay compounds (i.e. calcium, magnesium and aluminium silicates).

(2)

Figure 6: SWIR reflectance spectra of: a) reference samples as pellets: kaolin (grey, offset for clarity), calcite (black) and a white zone corresponding to snow on a mountain (grey dots); b) blue/greenish zones in the analysed thangka: (1) blue halo around the head of the deity, ( 2) and (3) greenish area indicated by a red circle in figure 8.

Revealing inaccurate past restorations

Hg(L) Ag(K) Zn(K) Au(L) As(K) Cu(K) Ag (K) Zn (K) Hg (L)

Figure 7: MA-XRF elemental distribution images of the snow lion, with a brighter hue indicating a stronger signal. 1 st line, 1 st column: corresponding photograph of the zone; 1 st line, 2 nd column: overlay of Ag-K (green), Hg-L (red) and Zn-K (blue). In the Cu-K and As-K maps, arrows highlight bright areas, original position of respectively the eyebrows and the eyes.

When painting a thangka, the artist uses variations in mattness and gloss to create perspective and depth (by playing with pigments of different particle size and therefore different proportions of binder). The entire thangka now appears matt, as a result of the surface dust, except for the lion which stands out, glossy. This visual perception seems to be linked to a later restoration, as indicated by:

-the presence of silver in the fur of the snow lion, which is traditionally white and appears now of a metallic grey colour with blue reflections(figure 7and S3).

We can assume here that a restorer used metallic silver, probably crushed silver leaf, to render the luminous aspect of the snow lion. Silver is not commonly found in thangkas as it was more prone to tarnish and darker than gold, but widely used in Tibetan sculpture.

-the presence of zinc in one of the snow lion's eye, probably indicating zinc white (used only from the 19 th century). This restoration did not take into account the original pigments and/or the original drawing and thus changed its meaning: it disturbs not only the legibility of the object but also its iconography.

Although we see now two eyes placed horizontally (one of zinc white and the other one, probably original, of orpiment), they were probably originally placed vertically as traditionally, turned towards to the deity (arrows in the As map, figure 7). This is confirmed by the position of the original eyebrows, indicated by the two Cu dots seen in the XRF maps: common iconography represents the snow lion with blue or green eyebrows (thangkas 16512 and 5160 from the Guimet museum [START_REF] Béguin | [END_REF] ).

Construction and colour codes

Before drawing the artist had to prepare the ground of the thangka on which the main lines of orientation were drawn. The central vertical axis is the most important because it allows the centre of the painting to be located, around which the artist would plan the rest of the composition. The correct establishment of the vertical and horizontal axes was also important from a religious standpoint. Thangkas are an expression of religious ideals through art; their figures (the ideal bodily forms of enlightened beings) have to be on a particular and precise position with respect to the central axis. Once the painting surface had been determined and its central axis established; the painter began the actual work of sketching. Traditional thangkas also exhibit colour codes written with charcoal on the ground, written by the main artist as a guide for other artists and apprentices, as previously observed by Duffy and Elgar [START_REF] Duffy | Historical painting techniques, materials and studio practice, Prepints of a symposium[END_REF] .

Infrared reflectography is a key technique to investigate thangkas revealing these underdrawings as part of the history of the thangka. As already observed in the context of paintings, near infrared HSI can be used as "improved reflectography" [26] ; it revealed here the beauty of the underlying drawing (Figure 8), barely visible today due to the state of degradation of the work of art. The diagonal lines drawn by the author can be seen under the main drawing in figure 8a. Colour notations do not seem to have been used extensively for this thangka; we found only one sign that may be a colour notation. However, it was not possible to confirm the nature of this sign, in relation to the existing literature [START_REF] Béguin | Les peintures du bouddhisme tibétain[END_REF][START_REF] Jackson | Tibetan Thangka painting: methods & materials[END_REF][START_REF] Duffy | Historical painting techniques, materials and studio practice, Prepints of a symposium[END_REF][START_REF] Tabasso | DIPINTI TIBETANI dalle spedizioni di Giuseppe TucciMateriali e tecniche alla luce delle indagini non invasive[END_REF] . As it is located in greenish foliage, we may relate it to the notation 'pa' (English spelling) abbreviation for the Tibetan word malachite, as observed by Duffy and Elgar [START_REF] Duffy | Historical painting techniques, materials and studio practice, Prepints of a symposium[END_REF] in rock crags, garments, jewels, and foliage of all the Shambhala paintings they analysed. The corresponding pigment was found to be a mixture of malachite and brochantite. Here the greenish colour is not related to the presence of malachite but to a mixture of azurite, antlerite and brochantite. It is thus difficult to reach a conclusion, which emphasizes the necessity of carrying out further scientific investigations on thangkas to better be able to define the pigments used, and how these can be described by Tibetan colour notations.

b) d) c) e) a)

Figure 8: SWIR images of the thangka (from 1650 to 1750 nm), witha "zoom" configuration from b) to e) (see Experimental). In e) the red circle indicates a sign, possibly a colour notation.

Discussion: advantages and limits of the methodology for the analyses of thangkas

The complementary use of XRF mapping, visible and near-infrared HSI techniques allow one to identify and map all the pigments listed as traditional in Tibetan paintings. The main limitation concerning the identification of pigments was here the state of conservation of the thangka analysed, which did not allow the identification of organic colourants.

Concerning imaging, we presented here "improved infrared reflectography" images, obtained thanks to near infrared HSI, and distribution maps of pigments obtained thanks to MA-XRF.

However as pigments are employed mainly pure in thangkas, one can use classification methods for the hyperspectral data treatment, when the state of degradation of the artwork allows it. It was here possible to provide distribution maps of pigments but only on very restricted area. The exact procedure for obtaining these images (spectral range, use of the derivative data…) has to be defined for each pigment according to its absorption characteristics. Different procedures are shown in figure 9: vermilion, minium and orpiment maps obtained from XRF and visible HSI, and azurite map obtained from SWIR imaging. It should be noted that both XRF and visible HSI distribution maps are not exactly the same (particularly in the case of orpiment). This is mainly due to the choice of endmembers used for the SAM algorithm (Spectral Angle Mapper, algorithm used to obtain the characteristic images of the distribution of the pigment): only pure pigments were used, whereas the horse and the flesh of the horseman are made up of a mixture of orpiment and minium, clearly observable in the XRF map. It should also be noted that the SAM algorithm provides images in which the intensity of the signal cannot be linked to the pigment concentration, which is the case for XRF maps. One must also emphasize the difference in penetration of the different techniques: in the visible range, only the pigments at the surface are detected, whereas X-rays penetrate further depending on the chemical composition of the material. Under-layers also become visible in the near-infrared data. This can be a problem for spectral interpretation, making it valuable from an imaging point of view as it can reveal underlying drawings and compositional changes, which are important steps in the composition. Near-infrared HSI, not commonly used until now on thangkas, provided invaluable information especially for the identification of white and blue/green pigments. Unusual Cu-compounds (chlorides or sulfates) were identified in blue/green areas. It would be interesting to go further, and see whether these pigments can be identified in more thangkas, and specify their origin and use (natural ore or alteration products).

This study emphasizes the importance of increasing the available information on the material aspects of thangkas, carrying out more scientific investigations to better define the pigments used, their possible alterations, and how Tibetan colour notations can describe them.

Figure 2 :

 2 Figure 2: VNIR reflectance spectra of the main reference pigments (pellets).

Figure 4 :

 4 Figure 4: MA-XRF elemental distribution images of the main deity of the thangka with a brighter hue indicating a stronger signal. 1 st line, 1 st column: corresponding photograph of the zone; 1 st line, 2 nd column: overlay of As (green), Hg-L (red) and Pb-L (blue) respectively indicating the distribution of orpiment, vermilion and minium.

Figure 5 :

 5 Figure 5: Macrophotographs of selected zones in the thangka under investigation: a) and b) golden dots on the deity; c) and d) Cu-based pigment showing various shades and appearance. In c) a green colour is visible at the edge of the paint indicating a possible mixture with malachite. In d), from the Lotus flowers, two different appearances, probably resulting from different paint formulations.

Figure 9 :

 9 Figure 9:Comparison of the three imaging techniques on a selected zone: a) visible image; b) elemental maps acquired by XRF: Hg in red (vermilion), As in green (orpiment) and Pb in blue (minium); c) hyperspectral image in the near infrared (R, G, B respectively at 1000, 1450 and 2350 nm): azurite appears in green and carbon black in dark black; d) image obtained from the derivative of the visible hyperspectral data (R,G, B respectively at 564, 593 and 717 nm) ; e) SAM (Spectral Angle Mapper) image obtained on the derivative data in the range 420-700 nm (tolerance angle 0.2 rad), from pre-defined end members shown in f: in red vermilion, in green orpiment, in blue minium.

Table 1 :

 1 Cameras and optics characteristics, with acquisition parameters (italics).

VNIR (400-1000 nm) SWIR (1000-2500 nm)

	Spectral sampling	2.8 nm	12 nm
	Integration time	85ms	13ms
	Translation frequency	4 Hz	30 Hz
	Translation speed	1 mm/s	5 mm/s
	Optics	Type	OL50	OLES30	OLES56
	(Specim)	Focal length	50 mm	30 mm	56 mm
		Distance	950 mm	1.0 m	630 mm
	Spatial size		1600 pixels	380 pixels
	Spectral size	1200 pixels	280 pixels

Table 2 :

 2 Main pigments and their characteristics Inflection point, b Apparent absorbance maximum, c No clear spectral feature detectable

		Chemical	XRF -elements	VNIR -band	SWIR -bands
		composition	detected	position (nm) and	positions
				characteristics
	Orpiment	As 2 S 3	As, S	480 a	c
	Malachite	Cu 2 CO 3 (OH) 2	Cu	800 b	2220, 2265, 2355,

a

4.2 Results from the Vaisravana thangka

Table 3 :

 3 Identified pigments on the Vaisravanathangka.

	Colour	XRF -	VNIR	SWIR	Pigment
		elements	identification	identification
		detected
	Yellow	As, S	Orpiment	*	Orpiment
	Gold	Au, As, S	Gold ***	*	Gold+Orpiment
	Orange	Pb	Minium	*	Minium
	Red / Pink	Hg, S	Vermilion	*	Vermilion
	Flesh	Pb, As	Minium +	*	Minium + orpiment
			orpiment
	Blue/Dark Blue	Cu (Mn)	**	Azurite	Azurite
	Green	Cu (Mn)	*	Mixture azurite+	Mixture	azurite+
				antlerite+	antlerite+
				brochantite	brochantite
	Silver-metallic blue	Ag (Cl)	*	*	? (restoration)
	Preparation / white	Si, Ti, Ca, Fe	*	Mixture of clay	Mixture of clay and
		(K, Cu)		and Ca-	Ca-compound
				compounds

Acknowledgments

The authors thank the infrared spectrometry platform of the MNHN analytical platform (and in particular X. Gallet for the database). We are also grateful to H. Glanville for her careful reading of the manuscript.


Charles - Edouard Bréhier
email: brehier@math.univ-lyon1.fr

Approximation of the invariant distribution for a class of ergodic SDEs with one-sided Lipschitz continuous drift coefficient using an explicit tamed Euler scheme

Keywords: 1991 Mathematics Subject Classification. 60H35, 65C30, Key words and phrases. Stochastic differential equations,tamed scheme,invariant distribution

Approximation of the invariant distribution for a class of ergodic SDEs with one-sided Lipschitz continuous drift coefficient using an explicit tamed Euler scheme

Charles-Edouard Bréhier

Introduction

The long-time behavior of deterministic and stochastic processes and of their discretetime approximations has been an active research area, with applications in all field of science. In this manuscript, we consider Stochastic Differential Equations (SDEs) of the type

dXptq " f pXptqqdt `K ÿ k"1 σ k dβ k ptq,
where Xptq P R d , σ k P R d , β k are independent standard real-valued Wiener processes. The drift f : R d Ñ R is assumed to be non-globally Lipschitz continuous, instead it has polynomial growth and satisfies a one-sided Lipschitz continuous condition, see Assumptions 1 and 2. Under these conditions, the SDE is globally well-posed and admits a unique invariant probability µ ‹ , such that

ErϕpXpT qqs Ñ T Ñ8 ż ϕdµ ‹ ,
exponentially fast, for any initial condition Xp0q and any real-valued Lipschitz continuous function ϕ. In general, no explicit expression for f is known. We study the question of approximating the invariant distribution µ ‹ using a numerical scheme. The main novelty of this article is to show that an explicit scheme can be used, without loss of computational efficiency, even if the nonlinearity f is not globally Lipschitz continuous. Applying an explicit scheme for SDEs with non-globally Lipschitz continuous coefficients may lead to important issues due to the lack of moment bounds, see for instance [START_REF] Hutzenthaler | Strong and weak divergence in finite time of Euler's method for stochastic differential equations with non-globally Lipschitz continuous coefficients[END_REF]. In the last two decades, many strategies have been explored: see for instance the monograph [START_REF] Hutzenthaler | Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients[END_REF] and references therein, and the articles [START_REF] Hutzenthaler | Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients[END_REF], [START_REF] Sabanis | A note on tamed Euler approximations[END_REF], [START_REF] Kelly | Adaptive time-stepping strategies for nonlinear stochastic systems[END_REF], [START_REF] Milstein | Computing ergodic limits for Langevin equations[END_REF], among many other contributions.

In this article, we apply the tamed Euler-Maruyama method (see Equation 6)

X n`1 " X n `∆tf pX n q 1 `∆t}f pX n q} `K ÿ k"1 σ k ∆β k n ,
for which for every T , one has moment bounds of the type

sup 0ďn∆tďT `Er}X n } m s ˘1 m ă 8.
To the best of our knowledge, the question of the dependence with respect to T , and the possibility to have uniform in time moment bounds, has not been studied in the literature yet.

Our first contribution (see Theorem 3.1) is to prove that, under appropriate assumptions, one has

sup 0ďn∆tďT `Er}X n } m s ˘1 m ď Cp}x 0 }qp1 `T M q,
where X 0 " x 0 , for some M ě 1. As a consequence, the growth with respect to T is at most polynomial. We only proved upper bounds, and the question whether uniform moment bounds (with M " 0) remains open.

Our second contribution is to apply these upper bounds for the analysis of the error between ErϕpX N qs and ş ϕdµ ‹ , see Theorem 3.2. Under appropriate assumptions, one has

ˇˇErϕpX N qs ´ż ϕdµ ‹ ˇˇď Cp}x 0 }, ϕq ´expp´γN ∆tq ``1 `pN ∆tq M ˘∆t ¯,
where γ ą 0. Contrary to existing works in the literature concerning the numerical approximation of the invariant distribution for SDEs (this is a classical problem, see for instance [START_REF] Mattingly | Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise[END_REF], [START_REF] Mattingly | Convergence of numerical timeaveraging and stationary measures via Poisson equations[END_REF], [START_REF] Talay | Second order discretization schemes of stochastic differential systems for the computation of the invariant law[END_REF]), the weak error |ErϕpX N qs ´ErϕpXpN ∆tqqs is not of size ∆t uniformly in time, and one cannot take the limit T " N ∆t Ñ 8 in the weak error estimate. Nevertheless, the fact that the dependence with respect to T " N ∆t is at most polynomial is striking, and analyzing the computational cost (see Corollary 3.3) shows that, to have an error less than ε, the cost is of size

Cpεq ď Cε ´1| logpεq| 1`M .
Passing from the case with uniform in time moment bounds(M " 0) to the case with polynomial dependence (M ě 1) does not substantially increase the computational cost and the method is effective. In the numerical experiments that have been performed (and are not reported here), it has not been possible to exhibit the polynomial growth with respect to time in the moment bounds or in the weak error estimate.

One of the objectives of this article is to present in a simplified framework the strategy and the arguments used in a companion paper devoted to the case of semilinear parabolic Stochastic Partial Differential Equations. The presentation is intended to be pedagogical. This manuscript is organized as follows. The setting is presented in Section 2, including the statement of precise assumptions (Assumptions 1 and 2), properties of the solutions of the SDE (3), and the definition of the numerical scheme (Equation ( 6)). The main results are stated and discussed in Section 3. The proof of Theorem 3.1 is provided in Section 4, whereas Section 5 presents the proof of Theorem 3.2.

Setting

Let d P N. The standard inner product and norm in the space R d are denoted by x¨, ¨y and } ¨} respectively.

Let us first state assumptions concerning the nonlinear drift coefficient f : it is assumed to be of class C 2 , with at most polynomial growth (Assumption 1, and to satisfy a one-sided Lipschitz continuity condition (Assumption 2).

Assumption 1. Let f : R d Ñ R d be of class C 2 , with at most polynomial growth in the following sense: there exists q P N such that (1) N q pf q :" sup

xPR d }f pxq} `ř2 j"1 sup }h 1 },...,}h j }ď1
}D j f pxq.ph 1 , . . . , h j q}

1 `|x| q ă 8.
The notation Df pxq.h and D 2 f pxq.ph 1 , h 2 q is used to denote first and second order derivatives of f at point x, in directions h, h 1 , h 2 P R d . Assumption 2. The following condition is satisfied: there exists γ ą 0 such that for all x 1 , x 2 P R d one has [START_REF] Hutzenthaler | Strong and weak divergence in finite time of Euler's method for stochastic differential equations with non-globally Lipschitz continuous coefficients[END_REF] xf px 2 q ´f px 1 q, x 2 ´x1 y ď ´γ|x 2 ´x1 | 2 .

Let us now describe the noise. Let σ 1 , . . . , σ K P R d and let `β1 ptq ˘tě0 , . . . `βK ptq ˘tě0 be independent standard real-valued Wiener processes, defined on a probability space pΩ, F, Pq satisfying the usual conditions. In the sequel, the notation

σdBptq " K ÿ k"1 σ k dβ k ptq is used.
In this work, we consider the following SDE with values in R d :

(3) dXptq " f pXptqqdt `σdBptq.

Owing to the locally Lipschitz and the one-sided Lipschitz properties of the function f , it is straigthforward to check that for any x 0 P R d , there exists a unique global solution `Xx 0 ptq ˘tě0 to (3) with X x 0 p0q " x 0 . Moreover, for every m P r1, 8q, there exists a polynomial function P m : R Ñ R such that (4) sup tě0 Er}X x 0 ptq} m s ď P m p}x 0 }q.

Finally, there exists a unique invariant distribution µ ‹ , such that any Lipschitz continuous function ϕ : R d Ñ R, one has for all T P p0, 8q and x 0 P R d

(5) ˇˇErϕpX x 0 pT qqs ´ż ϕdµ ‹ ˇˇď e ´γT Lippϕqp1 `}x 0 }q.

Moment bounds (4) are obtained by an application of the Itô formula, and the use of the one-sided Lipschitz continuity condition [START_REF] Hutzenthaler | Strong and weak divergence in finite time of Euler's method for stochastic differential equations with non-globally Lipschitz continuous coefficients[END_REF]. Applying the Krylov-Bogoliubov criterion, the moment bounds (4) yield existence of invariant distributions, and uniqueness follows from a straightforward coupling argument: if x 1 0 , x 2 0 P R d are two initial conditions, then almost surely for all t ě 0 one has }X x 2 0 ptq ´Xx 1 0 ptq} ď e ´γt }x 2 0 ´x1 0 }, using the one-sided Lipschitz condition [START_REF] Hutzenthaler | Strong and weak divergence in finite time of Euler's method for stochastic differential equations with non-globally Lipschitz continuous coefficients[END_REF], where the processes X x 1 0 and X x 2 0 are driven by the same noise process.

In the sequel, to simplify notation the dependence with respect to x 0 is omitted: Xptq " X x 0 ptq for all t ě 0.

The objective is to define estimators of ş ϕdµ ‹ , using an explicit integrator for the discretization of the SDE (3). Note that in general, no explicit expression of the invariant distribution µ ‹ is known, and if dimension d is large quadrature rules are not efficient and Monte-Carlo methods are usually employed.

Let ∆t denote the time-step size, and without loss of generality assume that ∆t P p0, ∆t 0 s for some arbitrarily fixed ∆t 0 ą 0. Let t n " n∆t, and define the Wiener increments as ∆β k n " β k pt n`1 q ´βk pt n q, and σ∆B n " σ k ∆tβ k n , for all n ě 0. In this work, we consider the explicit tamed Euler-Maruyama method

(6) X n`1 " X n `∆t 1 `α∆t}f pX n q} f pX n q `σ∆B n ,
where α P p0, 8q is a given parameter and X 0 " x 0 . The parameter α does not play an important role in the sequel, and the notation M n " α}f pX n q} is used. Introduce a continuous-time auxiliary process `Xptq ˘tě0 as follows: for all t ě 0, set

(7) Xptq " x 0 `ż t 0 1 1 `∆tM psq f pX psq qds `ż t 0 σdBpsq
where ptq " n if and only if t n ď t ă t n`1 , and `Xn ˘ně0 is defined by the tamed explicit Euler-Maruyama scheme [START_REF] Mattingly | Convergence of numerical timeaveraging and stationary measures via Poisson equations[END_REF]. Observe that one has Xpt n q " X n for all n ě 0.

The process `Xn ˘ně0 and `Xptq ˘tě0 depend on the time-step size ∆t and on the initial condition x 0 , however the dependence is omitted to simplify notation.

Remark 2.1. The explicit tamed Euler-Maruyama scheme (6) can be interpreted as coming from the application of the standard explicit Euler-Maruyama scheme for the SDE dX ∆t ptq " f ∆t pX ∆t ptqqdt `σdBptq with modified drift coefficient

f ∆t pxq " f pxq 1 `α}f pxq} .
However, the function f ∆t does not satisfy a one-sided Lipschitz continuity condition as in Assumption 2, with γ ą 0. Indeed, let d " 1 and assume that f is a polynomial function, for instance f pxq " ´x ´x3 , then one checks that

f 1 ∆t pxq Ñ xÑ˘8 0.
As a consequence, the long-time behavior (uniform in time moment bounds, existence and uniqueness of invariant distriutions) of the process X ∆t may be non trivial. It may even be the case that sup xPR f 1 ∆t pxq ą 0, in which case uniform in time moment bounds are not expected to hold. In turn no information on the behavior of the scheme (6) is obtained by using the interpretation of the tamed scheme.

Main results

The objective of this section is to state the two main results of this article: Theorem 3.1 concerning moment bounds and Theorem 3.2 concering weak error estimates. The most striking feature is the polynomial dependence with respect to the time T (and with respect to the norm of the initial condition). The consequences in terms of computational cost for the approximation of the invariant distributions are also discussed, see Corollary 3.3. Theorem 3.1. Let Assumptions 1 and 2 be satisfied. For every m P r1, 8q, there exists a polynomial function P m : R 2 Ñ R, such that for all T P p0, 8q and x 0 P R d , one has where `Xn ˘ně0 is defined by the tamed Euler-Maruyama scheme [START_REF] Mattingly | Convergence of numerical timeaveraging and stationary measures via Poisson equations[END_REF].

In Theorem 3.2, the function ϕ is allowed to have at most polynomial growth (as well as its derivatives).

To simplify the presentation, the degrees of the polynomial functions P m in Theorem 3.1 and P r and Q r in Theorem 3.2 are not indicated, however they could be identified by a close inspection of the proofs. This is left to the interested readers. Since we want to insist on the polynomial dependence with respect to time T or N ∆t, the exact value of the degrees does not matter.

Let us now draw consequences of Theorem 3.2 in terms of computational cost. In practice, the expectation ErϕpX n qs needs to be approximated using Monte-Carlo averages. The total computational cost may be reduced for instance using the Multilevel Monte-Carlo method. Since those aspects are not specific to the situation studied in this article, we only consider the cost per realization. The cost to sample a realization of X N is proportional to N . Then one has the following result. There exists a constant C " Cpx 0 , ϕq P p0, 8q, such that for all ε P p0, 1q, the error satisfies ˇˇErϕpX N qs ´ż ϕdµ ‹ ˇˇď ε with a computational cost satisfying

Cpεq ď Cε ´1| logpεq| 1`R
for some R P N.

Since | logpεq| R " ε 1 R
for all R P N, one has Cpεq ď C α ε ´1 α , for all α P p0, 1q. If in Theorems 3.1 and 3.2 the upper bounds were uniform in time, one could choose R " 0 in Corollary 3.3. The polynomial growth with respect to T only introduces a polynomial factor in | logpεq|, which does not significantly increase the computational cost. Note that if the dependence with respect to T in Theorems 3.1 and 3.2 had been exponential, an additional factor which would have been a power of ε ´1 would have appeared in Corollary 3.3, and in turn the computational cost would have significantly increased.

Proof of Corollary 3.3. The error estimate of Theorem 3.2 is rewritten as follows: there exists R P N such that

ˇˇErϕpX N qs ´ż ϕdµ ‹ ˇˇď C `expp´γN ∆tq `p1 `pN ∆tq R q∆t ˘,
where C is a constant (depending on x 0 and ϕ). Note that R may depend on ϕ (more precisely on the value of r such that N r pϕq ă 8.

The parameters N and ∆t are chosen such that N ∆t " C| logpεq| and pN ∆tq R ∆t " Cε. This leads to ∆t " Cε| logpεq| ´R, and finally N " C| logpεq|∆t ´1 " Cε ´1| logpεq| 1`R . This concludes the proof of Corollary 3.3. Remark 3.4. In [START_REF] Milstein | Computing ergodic limits for Langevin equations[END_REF], the authors propose to use the so-called rejecting exploding trajectories technique to approximate ergordic averages ş ϕdµ ‹ , for SDEs with non-globally Lipschitz coefficients. This technique requires to introduce an auxiliary truncation parameter. However, even if in practice it is effective, this technique does not lead to a clean analysis of the cost as in Corollary 3.3.

Remark 3.5. The results can be generalized, under appropriate assumptions, to SDEs driven by multiplicative noise dXptq " f pXptqqdt `σpXptqqdBptq, where σpxqdBptq " ř K k"1 σ k pxqdβ k ptq. Slight modifications of the proof of Theorem 3.1 are sufficient if the functions σ k , k P t1, . . . , Ku, are assumed to be bounded. These functions are also to be Lipschitz continuous. The one-sided Lipschitz condition (2) from Assumption 2 needs to be modified to ensure the uniqueness of the invariant distribution by the coupling argument providing [START_REF] Mattingly | Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise[END_REF]. The condition (2) also needs to be modified in order to be able to prove Theorem 3.2, in particular to prove exponential decrease in time of derivatives of the solution of the associated Kolmogorov equation.

These modifications and the statements of appropriate conditions would only make the presentation more complex, so for pedagogical reasons the details are only provided for the additive noise case.

Proof of Theorem 3.1

Recall that M n " α}f pX n q}, and that the auxiliary process `Xptq ˘tě0 is defined by [START_REF] Milstein | Computing ergodic limits for Langevin equations[END_REF]. Introduce an auxiliary parameter R " ∆t ´κ, where κ P p0, 1 2q q and q is given by Assumption 1.

For every n ě 0, let Ω R,tn " t sup 0ď ďn }X } ď Ru, and to simplify notation let χ n " 1 Ω R,tn denote the indicator function of the set Ω R,tn . Let also χ ´1 " 1.

To prove Theorem 3.1, it suffices to prove the following two auxiliary results.

Lemma 4.1. For every m P r1, 8q, there exists a polynomial function P 1 m : R 2 Ñ R, such that for all T P p0, 8q and x 0 P R d , one has Since γ ą 0, using Young's inequality and Gronwall's lemma, one obtains

}Y ptq} 2 ď C ż t 0 }f pZpsqq} 2 ds.
Multiplying by χ n´1 and using Minkowski's inequality, for every m P r1, 8q, one obtains

(12) `Erχ n´1 }Y pt n q} 2m s ˘1 m ď C ż t 0 `Erχ n´1 }f pZpsqq} 2m s ˘1 m ds.
Since f has at most polymomial growth (Assumption 1), oo obtain (10), it is thus sufficient to prove that for every m P r1, 8q, one has an estimate of the type (13) sup 0ďtďtnďT Erχ n´1 }Zptq} m s ď P m pT, }x 0 }q.

By definition of the auxiliary process Z, one has for all t ě 0

Zptq " Xptq ´ż t 0 f p Xpsqqds " X 0 `ż t 0 σdBpsq ´ż t 0 ∆tM ps 1 `∆tM psq f pX psq qds `ż t 0 " f pX psq q ´f p Xpsqq ‰ ds ": Z 0 ptq `Z1 ptq `Z2 ptq.
First, for all 0 ď t ď t n ď T , one has

`Erχ n´1 }Z 0 ptq} m s ˘1 m ď `ErZ 0 ptq} m ˘1 m ď }x 0 } `Cm T 1 2 ,
for some C m P p0, 8q. Second, recall that M " α}f pX q}. Since f has at most polynomial growth (Assumption 1), one obtains

`Erχ n´1 }Z 1 ptq} m s ˘1 m ď α∆t ż t 0 `Erχ n´1 }f pX psq q} 2m s ˘1 m ds ď C∆tp1 `R2q q ď Cp∆t 0 `∆t 1´2qκ 0 q,
where we recall that R " ∆t ´κ with 2qκ ă 1.

It remains to deal with the term Erχ n´1 }Z 2 ptq} m s. Using the fact that the derivative of f has at most polynomial growth (Assumption 1) and Minkowski and Cauchy-Schwarz inequalities, one obtains for all t ě 0

`Erχ n´1 }Z 2 ptq} m s ˘1 m ď ż t 0 `χn´1 }f pX psq ´f p Xpsq} m ˘1 m ds ď C ż t 0 `Erχ n´1 }X psq ´Xpsq} 2m s ˘1 2m `1 `Erχ n´1 }X psq } 2mq s `Erχ n´1 } Xpsq} 2mq s ˘1 2m ds.
The first factor in the integrand above is treated as follows: for s ď t ă t n , one obtains

χ n´1 }X psq ´Xpsq} ď χ n´1 |s ´t psq | 1 `∆tM psq }f pX psq } `C}σ `Bpsq ´Bpt psq q ˘} ď C∆tp1 `Rq q `C}Bpsq ´Bt psq }.
As a consequence, one obtains

`Erχ n´1 }X psq ´Xpsq} 2p s ˘1 2p ď C p `∆tp1 `Rq q `∆t 1 2 ˘ď C∆t 1 2 p∆t 1 2 ´κq 0 `1q,
for all ∆t P p0, ∆t 0 s, using the definition R " ∆t ´κ with 2qκ ă 1.

To treat the second factor of the integrand above, it suffices to write

Erχ n´1 }X psq } 2mq s `Erχ n´1 } Xpsq} 2mq ď CErχ n´1 }X psq } 2mq s `CErχ n´1 } Xpsq ´X psq } 2mq ,
and to use the estimate on the first factor above and the inequality Erχ n´1 }X psq } 2mq s ď R 2mq .

Finally, using again the inequality ∆t

1 2 R q ď ∆t 1 2 ´qκ 0
for all ∆t P p0, ∆t 0 s, one obtains

`Erχ n´1 }Z 2 ptq} m s ˘1 m ď Cp∆t 0 qT.
Gathering the estimates then yields (13). Inserting (13) in the inequality (12) then yields Erχ n´1 }Y pt n q} m s ď P m pT, }x 0 }q, if t n ď T and ∆t P p0, ∆t 0 s.

Since X n " Xpt n q " Y pt n q `Zpt n q, this concludes the proof of Lemma 4.1.

Proof of Lemma 4.2. Recall that χ n " 1 Ω R,tn , with Ω R,tn " t sup 0ď ďn }X } ď Ru and χ ´1 " 1. As a consequence, one has

1 ´χn " 1 Ω c R,tn " 1 Ω c R,t n´1 `1Ω R,t n´1 1 }Xn}ąR " 1 ´χn´1 `χn´1 1 }Xn}ąR .
One thus obtains the equality

1 ´χn " n ÿ "0 χ ´11 }X }ąR .
Let m P N. Using Minkowksi, Cauchy-Schwarz and Markov inequalities, one obtains

`Erp1 ´χn q}X n } m s ˘1 m ď n ÿ "0 `Erχ ´11 }X }ąR }X n } m s ˘1 m ď n ÿ "0 `Er}X n } 2m s ˘1 2m `Erχ ´1 }X } θ R θ s ˘1 2m ,
where θ P N is chosen below.

On the one hand, by construction of the tamed Euler scheme, one has

} Xptq} ď }x 0 } `T ∆t `} ż t 0 σp XpsqqdBpsq}, thus `Er}X n } 2m s ˘1 2m ď Cp}x 0 } `T 1 2 `T ∆t q.
On the other hand, applying Lemma 4.1 yields

Erχ ´1}X } θ s ď P 1 θ pT, }x 0 }q. Gathering the estimates yields

`Erp1 ´χn q}X n } m s ˘1 m ď CP 1 θ pT, }x 0 }q 1 2m p1 `T ∆t qp}x 0 } `T 1 2 `T ∆t qR ´θ 2m .
Since R " ∆t ´κ, it suffices to choose θ such that θκ 2m ą 2 in order to obtain (11). This concludes the proof of Lemma 4.2.

Proof of Theorem 3.2

The objective of this section is to prove Theorem 3.2. In Section 5.1, some auxiliary results concerning the solution of the associated Kolmogorov equation. Even if this type of results is standard, a proof is provided for completeness. Then Theorem 3.2 follows from the weak error analysis of Section 5.2.

Like in Section 4, the values of constants C P p0, 8q and of polynomial functions P r or P may change from line to line. with initial condition up0, ¨q " ϕ. Set upt, ¨q ´ş ϕdµ ‹ . The objective of this section is to prove the following lemma.

Lemma 5.1. Let Assumptions 1 and 2 be satisfied. For any r P N, there exists R P N such that the following holds: for all ϕ : R d Ñ R of class C 2 which satisfies N r pϕq ă 8q, for all γ 1 P r0, γq, one has

sup tě0 N R pupt, ¨qqe γ 1 t ă 8.
In other words, the function u and its first and second order spatial derivatives Dupt, ¨q and Dupt, ¨q have at most polynomial growth, and N R pupt, ¨qq decreases exponentially fast when t Ñ 8. This type of result and the strategy of the proof are standard. A specific feature of the approach considered in this article is that a weaker result is employed: since we do not expect to prove uniform in time weak error estimates, and to have polynomial in time dependence instead, it is sufficient to use uniform in time upper bounds for the derivatives Dupt, ¨q and Dupt, ¨q. Thus in the analysis of the weak error due to temporal discretization, one will use estimates with γ 1 " 0. Keeping γ 1 ą 0 would only allow to reduce the degree of the polynomial dependence with respect to T , but this does not qualitatively change the analysis of the cost. The fact that γ 1 ą 0 has a role only to estimate the error upt, xq " upt, xq ´ş ϕdµ ‹ , and as will be clear below for this contribution one can take γ 1 " γ (see also [START_REF] Mattingly | Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise[END_REF]).

Proof. Recall that, for every x P R d , the solution of (3) with initial condition Xp0q " x is denoted by `Xx ptq ˘tě0 .

First, let x 1 , x 2 P R d , then the one-sided Lipschitz condition (2) implies Using the one-sided Lipschitz condition (2), the at most polynomial growth of f and its derivatives (Assumption 1) and Young's inequality, one has

1 2
d}ζ h 1 ,h 2 ptq} 2 dt " xDf pX x ptqq.ζ h 1 ,h 2 ptq, ζ h 1 ,h 2 ptqy `xD 2 f pX x ptqq.pη h 1 ptq, η h 2 ptqq, ζ h 1 ,h 2 ptqy ď ´pγ ` q}ζ h 1 ,h 2 ptq} 2 `1 4 N q pf q 2 p1 `}X x ptq} 2q qe ´4γt }h 1 } 2 }h 2 } 2 .

8 ,

 8 sup ∆tPp0,∆t 0 s sup 0ďn∆tďT Er}X m } p s ď P m pT, }x 0 }q. Theorem 3.2. Let Assumptions 1 and 2 be satisfied. For every r P N, there exists polynomial functions P r : R 2 Ñ R and Q r : R Ñ R, such that the following holds: if ϕ : R d Ñ R is a function of class C 2 such that N r pϕq " sup xPR d |ϕpxq| `ř2 j"1 sup }h 1 },...,}h j }ď1 |D j f pxq.ph 1 , . . . , h j q| 1 `|x| r ă then for all N P N, x 0 P R d and ∆t P p0, ∆t 0 s, one has (9) ˇˇErϕpX N qs ´ż ϕdµ ‹ ˇˇď N r pϕq ´expp´γN ∆tqQ r p}x 0 }q `∆tP r pN ∆t, }x 0 }q ¯,

Corollary 3 . 3 .

 33 Let x 0 P R d and ϕ : R d Ñ R of class C 2 such that N r pϕq ă 8.

0ďn∆tďTLemma 4 . 2 . 2 . 0 f p Xpsqqds " ż t 0 f 2

 422002 Erχ n´1 }X n } m s ď P 1 m pT, }x 0 }q. For every m P r1, 8q, there exists a polynomial function P 2 m : R 2 Ñ R, such that for all T P p0, 8q and x 0 P R d , one has (11) sup ∆tPp0,∆t 0 s sup 0ďn∆tďT Erp1 ´χn q}X n } m s ď P 2 m pT, }x 0 }q. Theorem 3.1 is then a straightforward consequence of Lemma 4.1 and Lemma 4.Proof of Theorem 3.1. Since Ω R,tn Ă Ω R,t n´1 , one has χ n ď χ n´1 . Writing Er}X n } m s " Erχ n }X n } m s `Erp1 ´χn q}X n } m s ď Erχ n´1 }X n } m s `Erp1 ´χn q}X n } m s, combining the auxiliary moment bounds (10) and (11) then concludes the proof. In the proofs, values of constants C, C m P p0, 8q and polynomial functions P m may change from line to line. Proof of Lemma 4.1. Introduce two auxiliary processes `Y ptq ˘tě0 and `Zptq ˘tě0 as follows: for all t ě 0, Zptq :" Xptq ´ż t 0 f p Xpsqqds Y ptq :" Xptq ´Zptq. One obtains the following equality: for all t ě 0 Y ptq " ż t pY psq `Zpsqqds, thus Y solves the differential equation dY ptq dt " f pY ptq `Zptqq. Using the one-sided Lipschitz condition (2) satisfied by f (Assumption 2), one obtains 1 d}Y ptq} 2 dt " xY ptq, f pY ptq `Zptqy ď xY ptq, f pY ptq `Zptqq ´f pZptqqy `}Y ptq}}f pZptqq} ď ´γ}Y ptq} 2 `}Y ptq}f pZptqq}.

5. 1 . 2 K ÿ k" 1 D 2

 1212 Auxiliary result: Kolmogorov equation. Let upt, xq " E x rϕpX t qs, for all t ě 0 and x P R d . Then u is the solution of the Kolmogorov equation B t upt, xq " Lupt, xq " Dupt, xq.f pxq `1 upt, xq.pσ k , σ k q

  d}X x 2 ptq ´Xx 1 ptq} 2 dt " xf pX x 2 ptqq ´f pX x 1 ptqq, X x 2 ptq ´Xx 1 ptqy ď ´γ}X x 2 ptq ´Xx 1 ptq} 2 , thus }X x 2 ptq ´Xx 1 ptq} ď e ´γt }x 2 ´x1 } for all t ě 0. One obtains for all t ě 0ˇˇErϕpX x 2 ptqqs ´ErϕpX x 1 ptqqs ˇˇď CN r pϕqE "`1 `}X x 1 ptq} r `}X x 2 ptq} ˘}X x 2 ptq ´Xx 1 ptq} ‰ ď CN r pϕq `1 `Pr p}x 1 }q `Pr p}x 2 }q ˘e´γt }x 2 ´x1 }.Choosing x 2 " x and integrating with respect to dµ ‹ px 2 q then yields |upt, xq| ď CN r pϕqe ´γt p1 `}x}P r p}x}qq, using the moment bound (4) for the exact solution.Next, the first-order derivative of upt, ¨q satisfiesDupt, xq.h " ErDϕpX x ptqq.η h ptqs,where one has dη h ptq dt " Df pX x ptqq.η h ptq , η h p0qh.d}η h ptq} 2 dt " xDf pX x ptqq, η h ptqy ď ´γ}η h ptq} 2 , thus Gronwall's lemma yields the inequality }η h ptq} ď e ´γt }h} for all t ě 0. One obtains |Dupt, xq.h| ď e ´γt }h}N r pϕqEr1 `}X x ptq} r s ď e ´γt }h}N r pϕq `1 `Pr p}x}q ˘, using the moment bound (4) for the exact solution.Finally, the second-order derivative of upt, ¨q satisfiesD 2 upt, xq.ph 1 , h 2 q " ErDϕpX x ptqq.ζ h 1 ,h 2 ptqs `ErD 2 ϕpX x ptqq.pη h 1 ptq, η h 2 ptqqs,where one has dζ h 1 ,h 2 ptq dt " Df pX x ptqq.ζ h 1 ,h 2 ptq `D2 f pX x ptqq.pη h 1 ptq, η h 2 ptqq , ζ h 1 ,h 2 p0q " 0.

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	Using the one-sided Lipschitz condition (2) implies
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Using Gronwall's lemma and the moment bound (4) for the exact solution, one obtains Er}ζ h 1 ,h 2 ptq} 2 s ď Ce ´2γ 1 t }h 1 } 2 }h 2 } 2 `1 `P2q p}x}q ˘.

Finally, this gives the inequalities ˇˇD 2 upt, xq.ph 1 , h 2 q ˇˇď N r pϕq `Erp1 `}X x ptq} r q 2 s ˘1 2 `E}ζ h 1 ,h 2 ptq} 2 ˘1 2 `Nr pϕqErp1 `}X x ptq}q r qse ´2γt }h 1 }}h 2 } ď Ce ´γ1 t }h 1 }}h 2 }N r pϕqp1 `P2r p}x}qq.

Gathering the estimates then concludes the proof.

5.2. Weak error analysis. By a linearity argument, without loss of generality one assumes that N r pϕq ď 1.

Proof of Theorem 3.2. The weak error can be written as

ErϕpX N qs ´ż ϕdµ ‹ " Erup0, X N qs " Erup0, X N qs ´ErupN ∆t, X 0 qs `upN ∆t, x 0 q. Using Lemma 5.1, one has (14) ˇˇupN ∆t, x 0 q ˇˇď e ´γN ∆t N r pϕqQ r p}x 0 }q, for some polynomial function Q r .

Using successively a telescoping sum argument, Itô's formula and the fact that u solves the Kolmogorov equation, one obtains Erup0, X N qs ´ErupN ∆t, X 0 qs "

where t n " n∆t, and the auxiliary process `Xptq ˘is defined by [START_REF] Milstein | Computing ergodic limits for Langevin equations[END_REF] and satisfies Xpt n q " X n , with M n " α}f pX n q}.

Introduce the following decomposition:

Er `Dupt N ´t, Xptqq ´Dupt N ´t, X n q ˘.`f pX n q ´f p Xptqq ˘sdt.

First, using Lemma 5.1 and Theorem 3.1, one obtains

ď C∆tPpt N , }x 0 }q, where P is a polynomial function.

Note that the following result has been used above: Theorem 3.1 gives moment bounds for X n , with 0 ď n∆t ď t N , and it is straightforward to deduce moment bounds of the same type for Xptq, with 0 ď t ď T , i.e. sup 0ďtďT Er} Xptq} m s ď P m pT, }x 0 }q.

Second, applying Itô's formula and a conditional expectation argument gives for all t P rt n , t n`1 s ErDupt N ´t, X n q. `f pX n q ´f p Xptqq ˘sdt " ErDupt N ´t, X n q. `ż t tn Df p Xpsqq.

This yields

Using the Cauchy-Schwarz inequality and moment bounds for X n and Xpsq, one then obtains

where P is a polynomial function.

Finally, using a Taylor expansion, one has ˇˇErup0, X N qs ´ErupN ∆t, X 0 qs ď ∆tPpN ∆t, }x 0 }q.

Combining the inequalities ( 14) and (15) then concludes the proof of Theorem 3.2.

Acknoledgments

This work is partially supported by the SIMALIN project ANR-19-CE40-0016 of the French National Research Agency.