Alice Delvar

Pascale De Caro

Laure Candy

Yanis Caro\

Alain Shum

Cheong Sing

Christine Raynaud

Integrated process for extraction and formulation in emulsions of active molecules from fresh passion fruits (Passifl.ora edulis Sims)

Keywor ds: Passion fruit Twln-screw extrusion Combined extractions Polyphenols Seed oil Blo-based emulslons

Intensified green processes were investigated to extract several classes of active molecules from fresh purple passion fruits, and to prepare emulsions directly by extrusion for the fruit comprehensive utilization. A thermo mechanical treatment was carried out through a discontinuous process from the edible part and through a continuous process by twin-screw extrusion using the whole fruit. Only mechanical energy was applied to the materials to generate a liquid fracti on (filtrate ) and a solid residue. Lipid contents of the obtained filtrates ranged from 0.5 to 2.6g/100 g close to the extractible lipophilic part. Polyphenols extraction was improved when using the whole fruit and with the addition of water in extrusion. In-situ emulsions with droplets si7.eS between 2 and 8 µm were obtained. Emulsions obtained after extrusion showed an enhanced stability. This way represents a green alternative method to sol vent extractions applied to the different parts of the fruit, and leads to bio-based emulsions enriched in vegetable oil, natural antioxidants and emulsifiers, such as phosphol ipids, proteins, polysaccharides, pectins and poly phenols.

Introduction

Purple passion fruit (Passi.ftora edulis Sims) is originated from Brazi1 lt is now widely cultivated in tropical regions, such as India, Australia or South Africa. On Reunion Island in 2017, 32 ha have been dedicated to the passion fruit cultivation, with a production of 500 metric tons according to the French Department of Agriculture and Food statistics. Passion fruit transformation is a growing industry, due to increasing demand for j uice in Europe, in Canada and in the United States (Von der Linden, 2007 ). The j uioe production generates large amount of agro industrial residues composed of rinds and seeds. Residue represent 76% of the weight of the raw material after fruit processing (Leao et al., 2014). Thus, the e xpansi on of this industry may create environmental issue by causing an increase in the production of wastes [START_REF] Malacrida | Yellow passion fruit seed oil (Passiflora edulis f. flavicarpa): physical and Chemical Characteristics[END_REF].

In the last decade, many studies focused on the valorization of agro industrial wastes. Indeed, the production of high added value molecules from by products is an economically and environmentally attractive solution. Fruit residues like shells and seeds are thus recognized as a valuable source of chemicals for food, cosmetic or pharmaoeutical

• Corresponding author.

E-mail address: pascale.decaro @ ensiacet.fr (P. de Caro ) .

industries ( Da Silva et al., 2014). Previous works showed that passion fruit seeds contain up to 300/4 of oil rich in unsaturated fatty acids, and especi ally in linoleic acid (700/4) [START_REF] Delvar | Semi-siccative oils and bioactive fractions isolated from reunion island fruit co-product: two case studies[END_REF]. Moreover, the different parts of passion fruit are rich in phenolic compounoe (Septembre [START_REF] Malaterre | Evaluation of nutritional and antioxidant properties of the tropical fruits banana, litchi, mango, papaya, passion fruit and pineapple cultivated in Reunion French Island[END_REF] leading to potential applications in food or cosmetic industries [START_REF] Oliveira | Valorization of passion fruit (Passiflora edulis sp.) by-products:Sustainable recovery and biological activities[END_REF][START_REF] Morais | Antioxidant activity, phenolics and UPLC-ESI(-)-MS of extracts from different tropical fruits parts and processed peels[END_REF] . Concerning the passion fruit seed oil extraction, several methods were studied like maoeration in sol vents of different polarities [START_REF] Oliveira | Valorization of passion fruit (Passiflora edulis sp.) by-products:Sustainable recovery and biological activities[END_REF], supercritical carbon dioxide extraction ( SC C0 2 ) [START_REF] Zahedi | Optimization of supercritical carbon dioxide extraction of Passiflora seed oil[END_REF], ultrasound assisted or Soxhlet extractions (De [START_REF] De Oliveira | The extraction of passion fruit oil with green solvents[END_REF]. In these cases, it is usually necessary to separate the seeds from the residual pulp and to dry the fruit materials. More over, common extraction organic solvents may have potential en vironmental and health impacts, and may alter the quality of the ex tracts. lt is also necessary to consider recycling of the solvents or recovery steps. A supercritical fluidic extraction from passion fruit seeds gives a high quality extract (vegetable oil) but requires a high pressure equipment [START_REF] Zahedi | Optimization of supercritical carbon dioxide extraction of Passiflora seed oil[END_REF].

Twin screw technology, as a mechanical treatment adapted to ve getable raw materials, has the advantages to combine compression and https://doi.org/10.1016/ j . j foodeng.2019.07.014 shearing effects, to pa-form liquid solid extraction and separation [START_REF] Evon | Direct extraction of oil from sunflower seeds by twin-screw extruder according to an aqueous extraction process: feasibility study and influence of operating conditions[END_REF]. The type of screw elements, their length and their dis position determine the screw profile according to the materials. New valorization ways for agro industrial by products have been reoently developed with this method. For instance, polyphenols were extracted with water from wood residues [START_REF] Celhay | Aqueous extraction of polyphenols and antiradicals from wood by-products by a twin-screw extractor: feasibility study[END_REF] and hydro xicinnamic acids were obtained from hemp by products [START_REF] Candy | Thermo-mechano-chemical extraction of hydroxycinnamic acids from industrial hemp by-products using a twinscrew extruder[END_REF]. Many studies also focused on the extraction of seed oil by ex trusion, by expression and by aqueous extraction [START_REF] Evon | Direct extraction of oil from sunflower seeds by twin-screw extruder according to an aqueous extraction process: feasibility study and influence of operating conditions[END_REF][START_REF] Kartika | Extraction of sunflower oil by twin screw extruder: screw configuration and operating condition effects[END_REF]; Lacaze Du fa ure et al., 1999; [START_REF] Magimel | Procédé d'obtention d'émulsion stable à partir de graines entières oleo-protagineuses, l'émulsion obtenue et les compositions cosmétiques contenant ladite emulsion[END_REF]. The aim of this work was to implement new extraction prooesses of molecules of interest from purple passion fruit (whole fruit or co pro ducts) and to prepare oil in water emulsion (which is the target pro duct) directly by extrusion for the fruit comprehensive utilization. If extractions are usually performed from the seeds after fractioning the fruit co product, in this study, the objective was to develop a one step prooess to extract and formulate at the same time several hydrophilic and lipophilic active compounds in oil in water emulsions. To our knowledge, no work mentions the possibility to prepare emulsions from fresh fruit directly by extrusion. Furthermore, most of works have fo cused in the recovery of the pure seed oil after the extrusion process. So, a stage of centrifugation is usually necessary in order to obtain the pure oil. The new approach proposed here should save processing costs and fruit processing time because the final product, the crude oil in water emulsion, could be used directly in natural formulations which required lipophilic compounds emulsified in water, such as in natural paint, cosmetic, or food formulations.

The strategy of combining the extraction of active molecules (such as vegetable oil, phospholipids, polyphenols and proteins) and their formulation in oil in water emulsion was tested with two methods. The feasibility has been first evaluated in a batch process at laboratory scale, before using the continuous twin screw extrusion at pilot scale. Twin screw extruder was supplied with the whole fruit, including the shell rich in cellulosic fibers, for the fruit comprehensive utilization. In order to prepare enriched oil in water emulsions, the media were characterized in connection with their possible applications in industry (food, cosmetic, paint, etc.).

Material and methods

Plant material

Plant materials consisted of fresh ripe purple passion fruit (Passiftora edulis Sims). They were collected from a local industry (Vergers Law Yat) in Reunion Wand. The fruits were frozen and stored at -2o•c until use.

Chemi.cal reagents

Gallic acid, Folin Ciocalteu reagent, DPPH (2,2 diphenyl 1 pi crylhydrazyl) and L fucose were purchased from Sigma Aldrich.

Feedinput

Solvents were analytical grade and were obtained from VWR International.

Raw materia1 characterizatin

The dry matter was measured by drying the sample at 103 •c for 24 h. The minerai content was determined after calcination at 550 •c for 6 h. The following analyses were pa-formed from dried materials. The cellulose, hemicellulose and lignin contents were estimated using the Van Soest and Wine method. The nitrogen content was determined by the Kjeldahl method (Tecator Kjehec �. Denmark) according to the NF V18 100 standard and the protein content was calculated by applying a 6.25 multiplication factor. Lipidic extractives were esti mated by Soxhlet extraction in hexane for 4 h. Hydrophilic extractives were obtained after extraction with water for 1 h in a Fibertec appa ratus (Tecator M1017, Denmark).

Mechanical pressing of passion fruit seeds

Mechanical pressed seed oil was obtained by pressing dried passion fruit seeds (300 g) in a screw press (Komet, Germany) with a maximum oil flow rate of 1 kg/h, powered by an electric motor (0.75 kW). The press cylinder was heated before extraction to a temperature comprised between 80 and 100 •c. The press had a nozzle diameter of 5 mm, a screw length of 18 cm, and a screw speed of 40 rpm.

Discontinuous process: batch extraction

The batch extraction prooess consisted of two ways depending on the raw material used. In both ways, the edible part of 15 passion fruits was separated from the shell. For the way 1, ail the edible part (iuice, pulp, seeds) was used. For the way 2, the juice (223 g) was replaced by the same amount of water (223 g). For both ways, the material was grinded with a blender (Gründig) at maximum speed for 15 s (3 times) with manual homogenization between each grinding and then grinded with a laboratory mixer (Silverson, France) for 2 min. The mixture was filtered on a nylon tissue (lOOµm) to obtain Filtrate 1 (way 1) and Filtrate 2 (way 2). The filtrates were emulsified with an Ultraturax apparatus (T25 digital, IKA, Germany) for 2min at 18000 rpm to pre pare emulsions lA and 2A (ways 1 and 2, respectively). Enriched emulsions 1B and 2B containing 100/4 of oil were prepared by adding the required quantity of mechanical pressed passion fruit seed oil to the Fi ltrates 1 and 2 before emulsification. flowrate of 6 kg/h (Q RM ). Fruit juice acted as a solvent. The screw ro tation speed was set to 150 rpm. The raw liquid fraction and the solid fraction (cake) were recovered.

Th.'În screw extrusion of whole passion fruit

A second extraction was performed by feeding the solid cake of the first extraction (1 kg) in the extruder at a flowrate of 6 kg/h. The screw configuration was modified by changing the pitch of the reversed double thread screw in module 7 (Fig. 2). Water was injected in the module 2 at a flowrate of 4.8 kg/h with a piston pump giving a liquid to solid ratio of 2.8 on a dry matter basis, calculated according to the following formula:

= × × - + × ratio Q DM Q D M Q (100 ) 100 S L S S S S W /
where Q S (kg/h) is the fed extrudate flowrate, DM S (g/100 g) is the extrudate dry matter and Q W (kg/h) is the water flowrate.

The particles contained in the raw liquid fractions were filtered on nylon tissue (100 μm) to obtain Filtrate 3 (first extraction) and Filtrate 4 (second extraction). Enriched emulsions 3A and 3B containing 2.5% and 10% of oil respectively were prepared by adding the required quantity of mechanical pressed passion fruit seed oil to Filtrate 3 before emulsification for 2 min at 18000 rpm.

The solid fraction (extrudate), filtrates and filtration residue (par ticles) were weighed and their dry matters (g/100 g) were determined (DM S , DM F , and DM P respectively). The solid (Q S ), filtrate (Q F ) and particle (Q P ) flow rates were measured. The yield (g/100 g DM) in solid fraction (Y S ), filtrate (Y F ), and particles (Y P ) were calculated according to the formula:

= × × × Y Q DM Q DM 100 X X X RM RM
where Q X (kg/h) is the considered fraction flowrate, DM X (g/100 g) the considered fraction dry matter, Q RM (kg/h) is the raw materials flow rate, and DM RM (g/100 g) is the raw materials dry matter.

The oil yield Y O (g/100 g DM), protein yield Y N (g/100 g DM) and polyphenols yield Y PP (g/100 g DM) were determined according to the formula:

= × × × Y Q DM R Q DM x F F x RM RM
where R x (g/100 g DM) is the oil, protein or polyphenol content in the filtrate.

2.1.6. Filtrates composition 2.1.6.1. Liquid liquid oil extraction with cyclohexane. A liquid liquid extraction with cyclohexane was performed on the filtrates. 150 g of filtrate and 100 mL of cyclohexane were mixed and were centrifuged at 10 4 g for 10 min (Sigma 6 16K). Aqueous phases and organic phases were recovered. The centrifugation pellets were then suspended in 100 mL of cyclohexane and were centrifuged in the same conditions. The aqueous phase was washed with 50 mL of cyclohexane. The organic phase was washed with 50 mL of distilled water and was dried with anhydrous magnesium sulfate. The solvent was then evaporated to recover the oil.

2.1.6.2. Total polysaccharides content. Polysaccharides of the filtrates were analyzed by ion exchange chromatography using a CarboPac PA1 column (Dionex, France). The eluent was potassium hydroxide with a 1 mL/min flowrate. A gradient of concentration was used starting at 2 mM for 39 min then increased to 10 mM for 2 min, 100 mM for 8 min and 100 mM for 3 min. The temperature was set to 25 °C. The injection volume was 25 μL. External standards were used for the polysaccharides identification. Fucose was used as internal standard for the polysaccharides quantification.

2.1.6.3. Total phenolics content. Total phenolic contents of the filtrates were determined using Folin Ciocalteu method as follows: 20 μL of filtrates, 10 μL of Folin Ciocalteu reagent and 170 μL of sodium carbonate solution (2.36%) were added in a 96 well microplate. After 45 min of incubation at 45 °C, the absorbance was measured at 760 nm with a spectrophotometer (Spectrostar, BMG Labtech). Standard solutions of gallic acid were used to build the calibration curve. Each sample was analyzed in 4 wells. Results were expressed as gram of gallic acid equivalent per 100 g of sample dry basis (GAE/100 g d.b.).

2.1.7. Oil in water emulsions characterization 2.1.7.1. Optical microscopy. The microstructure of filtrates and freshly prepared emulsions 1A, 2A and 3A was observed by optical microscopy with a Nikon Eclipse E600 with a ×40 magnification. The images acquisition was done with a camera Nikon DS Fi2. A drop of sample was placed on a microscope slide and was covered by a cover slip.

Images were captured at several location in order to be representative of the sample.

2.1.7.2. Droplet size distribution. Filtrates 1 and 2 and freshly prepared oil in water emulsions 1A, 1B, 2A and 2B of edible part of passion fruit were analyzed for the droplet size distribution using a Mastersizer 2000 laser diffraction particle size analyzer (Malvern). The refractive index of water (1.33) was used for the droplet size distribution calculation.

Results are shown as the droplet size distribution profile corresponding to the volume fraction (%) vs. the droplet size (μm).

2.1.7.3. Rheology. A rheometer MCR 302 (Anton Paar) with a cone plate geometry of 50 mm diameter and 2°angle was used to perform rheological analysis of the samples (Filtrates 1, 2, 3, and emulsions 1A, 1B, 2A, 2B, 3A, 3B). All experiments were carried out at a temperature of 25 °C regulated with a bath thermostat (Huber). The viscosity and shear stress were studied by increasing exponentially the shear strain from 0 to 50 s -1 . Then, in order to determine the viscoelastic region of the emulsions, amplitude sweeps were done by increasing shear strain (0.01% 100%) at a frequency of 1 Hz. Storage module G', loss module G" and tangent δ were obtained. All experiments were performed in triplicates.

2.1.7.4. Oil in water emulsion stability. The emulsion stability was assessed by the determination of the creaming index (CI). Freshly prepared emulsions (16 g) were transferred into transparent glass tubes (22 mm internal diameter, 45 mm height) sealed by plastic caps. The tubes were kept at room temperature (25 °C) for 14 days. The total heights of the emulsions in the tubes (H T ) were noted. The emulsion stability was followed by measuring the height of the transparent serum layer (H S ) formed below the opaque "cream" layer. The creaming index (CI) was then calculated with the following formula:

= × CI H H (%) 100 S T

Results and discussion

Raw materials characterization

The chemical compositions of the different parts (pulp, seeds, and shells) of the whole purple passion fruit were determined (Table 1). Cellulose (25.6 g/100 g DM whole fruit) was the main structural com ponent found in all parts of the fruit. The whole fruit presented a high content in minerals (14%) resulting mainly from the shells. As ex pected, the lipids are mainly concentrated in fruit seeds (23.6 g of lipids per 100 g of dry seeds). The edible part of the fruit, i.e., the pulp and the seeds, are also rich in polyphenols with 3.1 and 6.7 g GAE/100 g DM in their hydrophilic fractions, respectively.

Discontinuous process: batch extraction

The discontinuous process extraction from edible part of passion fruit was implemented according to two ways to obtain filtrates, in situ emulsions and enriched emulsions (Fig. 3).

The results are presented in Table 2. As expected, the starting ma terials dry matter is lower for the way 2 due to the removal of the juice. Similar quantities of filtrates are obtained for both ways of the batch process (81.1 83.7 g per 100 g of fresh fruit). The dry matter yield showed that the same proportion of initial dry matter is obtained with both ways. The oil is efficiently extracted with a 75% recovery of the lipids of the seeds contained in the edible part. The fruit juice does not influence the oil extraction yield. The protein yield is higher for the way 1, thanks to the juice proteins extraction. Nearly 75% of the proteins of the edible part are recovered in the Filtrate 1. The polyphenols yield is also higher for the way 1, indicating that polyphenols are present in the passion fruit juice. The way 1 allows the recovery of 43% of the ex tractable polyphenols of the edible part.

Twin screw extrusion of whole passion fruit

A two step extraction was also performed by twin screw extrusion on the passion fruit (Fig. 4). In the first step, the extrusion was con ducted with the whole fruit (shells, pulp, seeds) using the water con tained in the fresh fruit as solvent. The step two was performed with the residual solid cake of the first step. The screw profile was modified and a flow rate of water was added to the extrudate giving a liquid to solid ratio of 2.8 on a dry matter basis. The particles contained in the raw liquid fractions were filtered to obtain Filtrate 3 (first step) and Filtrate 4 (second step).

The operating conditions implemented with passion fruits generated the formation of a fibrous dynamic plug without clogging. The yields of extrudate (Y E ) and particles (Y P ) were used as indicators for the raw materials defibration. The extrudate yield (53%) enables to check the efficiency of the fruit destructuring during the first step (Table 3). The solid/liquid separation was also effective, as reflected by the filtrate flow rate (Q F ) and the extrudate dry matter (DM E ). For the second step of extraction with twin screw extrusion, the decrease of the pitch of the reversed double thread screw in module 7 and the addition of water enabled a further defibrating as shown by the extrudate and particles yields.

From the inlet and filtrates flow rates obtained, extraction yields in lipids, proteins and polyphenols were calculated (Table 4).

The oil extraction yield of from passion fruits (246.2 mg/100 g of fresh whole fruit) corresponds to 18% of the lipids of the seeds in the fresh whole fruit (1,4 g/100 g). The global yield obtained after the aqueous extraction of the extrudate 1 (268.9 mg/100 g of fresh whole fruit) indicates that part of the released lipophilic molecules is still retained in the solid residue. The results from other studies, obtained for vegetable oil extraction from sunflower seeds [START_REF] Evon | Aqueous extraction of residual oil from sunflower press cake using a twin-screw extruder: feasibility study[END_REF] and from coriander fruits [START_REF] Sriti | Extraction of coriander oil by twin-screw extruder: screw configuration and operating conditions effect[END_REF], seem to indicate that it should be possible to improve this extraction yield by modifying the screw configuration and the operating conditions. For oil extraction, the batch process achieves higher yields than the continuous process. The use of the high shear mixer therefore promotes the extraction of lipids in comparison to the extruder. These results could be explained in terms of grinding. The seeds crushed with the mixer led to thin particles, which made the release of the vegetable oil easier compared to extru sion. The first step of the continuous process enables the recovery of 28% of the proteins of the whole fresh fruit (1.4 g/100 g). The aqueous extraction of the extrudate 1 during the second step makes it possible to extract a portion of the proteins contained in extrudate 1 and to achieve a global protein yield of 0.7 g/100 g of fruit, i.e. 38% of the potential. The higher yields obtained for the batch process seem to indicate that the proteins of the edible part are more accessible than those of the shells, and that their extraction is favored by the use of the mixer and the application of a greater shear of the material. The proteins of the edible part are mainly from the seeds (68%), which are ground more finely in the batch process.

The polyphenol extraction yield of the first step corresponds to 51% of the extractable polyphenol content of the whole fruit (117.8 mg GAE/100 g of fruit). Thanks to the remarkable increase of this yield (+21%) after the aqueous extraction of the extrudate 1, the continuous process makes it possible to recover 64.5% of the extractable poly phenols from the whole fruit. Extraction of polyphenols is thus im proved in extrusion compared to the batch process, particularly through the addition of water.

Emulsions characterization

Chemical composition

The chemical compositions of the filtrates obtained with the high shear mixer (Filtrates 1 and 2) and with the twin screw extrusion (Filtrates 3 and 4) are presented in Table 5.

Filtrate 1 from the continuous process showed the highest dry matter content (17.2 g/100 g). The continuous process gives Filtrate 3 with lower dry matter content and a lower dry matter yield (Tables 2 and3). The liquid solid separation taking place in the twin screw Table 1 Chemical composition of the raw materials expressed in g/100 g DM (n = 3).

extruder and the difference in fruit material treatment between the two processes can explain this result.

The filtrates 1 and 2 had higher oil and protein contents than the filtrates 3 and 4, in accordance with the higher yields obtained. The efficient grinding of the seeds performed by the high shear mixer al lowed an easier release of these compounds compared to the twin screw extruder. On the other hand, the other compounds extraction (poly saccharides and polyphenols) is enhanced in the continuous process. Filtrates obtained from the extrusion (Filtrates 3 and 4) show higher content in polysaccharides and polyphenols. The Filtrates 4 contents indicate that the aqueous extraction of extrudate 1 allow a further ex traction of these molecules. Polyphenol contents in filtrates 3 and 4 were twice as high as those of filtrates 1 and 2, due to the extraction of additional polyphenols contained in the shells. According to the raw materials characterization (Table 1), shells represents in fact half of the extractible polyphenols of the whole fruit.

Emulsions characterization

Filtrates 1 and 2 were used to prepare emulsion 1A and 2A con taining 2.6% and 2.3% of oil, respectively. Additional mechanical pressed passion fruit seed oil was added to the filtrates 1, 2 and 3 to

Table 2

Yields in filtrate and in active molecules for the discontinuous process.

Way 1

Way 2

Input

Dry matter (g/100 g fresh edible part) 25.4 14.0 Oil content of the seeds (g/100 g fresh edible part)

2.8

Protein content (g/100 g fresh edible part)

2.1 TPC of hydrophilic fraction (mg GAE/ 100 g fresh edible part) 129.5

Filtrate

Filtrate yield (g/100 g fresh edible part) 81.1 ± 4.0 83.7 ± 4.4 Dry matter yield (g DM/100 g DM) 54.4 ± 0.1 54.6 ± 0.9 Oil yield (g/100 g fresh edible part)

2.1 ± 0.2 1.9 ± 0.2 Protein yield (g/100 g fresh edible part)

1.6 ± 0.1 1.1 ± 0.1 Polyphenols yield (mg GAE/100 g fresh edible part) 55.2 ± 1.1 33.4 ± 0.8 Fig. 4. Continuous extraction by twin-screw extrusion from whole passion fruits.

Table 3 Solid and liquid flowrates for the two-step passion fruit twin-screw extrusion (Clextral BC21).

Step 1

Step 2 prepare enriched oil in water emulsions: emulsion 3A with 2.5 g oil/ 100 g filtrate and emulsions 1B, 2B and 3B with 10 g oil/100 g filtrate.

Input Q RM (
In the filtrates, the dispersion was carried out by the mixer or in the twin screw extruder, whereas a further dispersion with Ultraturax™ was applied to the filtrates and led to emulsions. Micrographs of filtrates (Fig. 5) showed the lipidic phase dispersion in the aqueous matrix, whatever the process used to put in contact the active molecules (mixer for filtrates 1 and 2 or extrusion for filtrates 3). Thus, the filtrates (F1, F2) and the emulsions (1A, 2A) from dis continuous process presented droplets diameters of less than 10 μm. The continuous process generated a filtrate F3 and an emulsion 3A having a fine dispersion of lipids droplets of sizes 1 5 μm. For all en riched emulsions (B), the addition of vegetable oil resulted in an abundant population of larger droplets, with diameter between 8 and 15 μm.

Droplet size distributions of the samples obtained with the batch process with both ways are shown in Fig. 6A and6B. No coalescence process was observed as the droplet sizes remained constant throughout the experiments. Filtrates and emulsions B had very similar size dis tributions exhibiting three main groups of droplets. The most abundant size was around 8 μm. The other probable sizes were about 1.5 μm and 80 μm. For emulsions A, the population of droplets with a size between 1.5 and 6 μm was more important compared to emulsions B and fil trates. The analyses confirm that the emulsification tends to generate a higher number of small droplets.

Rheology of emulsion

The flow behavior of the filtrates and the emulsions were analyzed. The apparent viscosity was plotted against shear rate in Fig. 7. The cold pressed passion fruit oil was used as a standard behaving like a New tonian fluid. The filtrates and the emulsions showed a rheofluidifying behavior, as their viscosities decreased with the increase of the shear rate.

Samples obtained with the way 1 of the batch process (Filtrate 1, Emulsions 1A and 1B) had the highest viscosities, followed by the ex trusion samples (Filtrate 3, Emulsions 3A and 3B) whereas samples of the way 2 had the lowest (delta = 13 mPa s). Samples 1 (filtrate and emulsions) had higher dry matter (DM F1 = 17.2 g/100 g) compared to samples 2 (DM F2 = 8.7 g/100 g), which can partly explain the viscosity difference. Furthermore, samples 1 and samples 3 obtained from the continuous process (Filtrate 3 and Emulsions 3A and 3B) were richer in polysaccharides coming from the passion fruit juice and thus con tributing to increase the viscosity of aqueous media [START_REF] Ozturk | Progress in natural emulsifiers for utilization in food emulsions[END_REF]. Unlike samples 2, samples 1 and 3 contain natural emulsifiers extracted from the juice or the shells, such as proteins, phospholipids and pectins, thereby accounting for their higher viscos ities [START_REF] Ozturk | Progress in natural emulsifiers for utilization in food emulsions[END_REF][START_REF] Schmidt | Effect of molecular weight reduction, acetylation and esterification on the emulsification properties of citrus pectin[END_REF].

Moreover, we notice that emulsions showed higher viscosities compared to their corresponding filtrates, showing the effect of the emulsification step on the viscosity (delta = 3 13 mPa s). As expected, the oil enrichment performed from filtrates led to emulsions B with higher viscosities than emulsions A.

The viscoelastic properties of the samples were studied by per forming small amplitude oscillatory shear measurements. The storage or elastic modulus G′ is plotted against the deformation in Fig. 8. Rheological data have been collected in Table 6 to characterize the linear viscoelastic region (G' LVR , G" LVR and γ 90% ) and the crossover points of G′ and G" (G' = G" and γ G' G" ). The end point of the linear viscoelastic region (LVR) was determined at the point corresponding to the value of G′ reduced by 10%.

For all the samples, the storage module G′ is 2 3 times greater than the loss modulus G" in the LVR, thus reflecting the dominant elastic behavior of the filtrates and emulsions in this region. The length of the LVR is however different depending on the type of samples. The com parison of the critical strains (γ C ) (Table 6) shows that the LVR of the emulsions is longer than that of their corresponding filtrate. The emulsions, thanks to their increased viscosity, have a better resistance to deformation than the filtrates and will not flow until a larger de formation (γ C ). The observation of the deformation corresponding to the intersection point of both modules confirms this analysis: the values of γ G ' G " are higher for the emulsions than for the filtrates.

Samples 1 showed the longest linear viscoelastic regions, while samples 2 have the shortest. Thus it appears that the factors favoring a higher viscosity (emulsions versus their filtrates, or samples 1 compared to samples 2) confer a better elasticity to the medium. For instance, the higher content of polysaccharides (identified as one of the factors en hancing the viscosity) may also be responsible for the improved stabi lity of samples 1, by inhibiting the droplets movement [START_REF] Ozturk | Progress in natural emulsifiers for utilization in food emulsions[END_REF]. Moreover, it was highlighted that the proteins can participate in the shear resistance after their adsorption at the droplets surface, which generates electrostatic and steric repulsion between the droplets (Bouyer et al., 2012). So, when keeping the passion fruit juice and its natural emulsifying molecules within the formulation, the pre formulated samples have better deformation capacities, resulting in a higher flow threshold.

The shorter LVR observed for emulsions B compared to the corre sponding emulsions A, could be explained by the larger size droplet population (8 15 μm) in emulsions B (Figs. 5 and6). Larger droplets promote rearrangement between droplets leading to lower resistance to

Step 1

Step

Table 5

Chemical composition of the filtrates obtained from the edible part of the passion fruit with a high shear mixer at laboratory scale (filtrates 1 and 2) and from the whole fruit by twin-screw extrusion (filtrates 3 and 4). 29.2 ± 1.4 40.5 ± 0.6 52.4 ± 1.9 45.9 ± 1.2 Polyphenols (g GAE/L filtrate) 0.7 ± 0.0 0.4 ± 0.1 1.2 ± 0.1 0.5 ± 0.0 (g/100g DM F ) 0.4 ± 0.0 0.4 ± 0.1 0.9 ± 0.0 1.0 ± 0.1 GAE: gallic acid equivalent; DM F : Filtrate dry matter.

Table 4

Yields in active molecules in the Filtrates 3 (Step 1) and Filtrate 4 (Step 2).

Emulsion stnbility

• • • i • • 1 t • • • • • • • • a a a • • • t. 1 i 1 • • • • • • • • • •• • • •es • • • •• • • • • • • • a• a • • • • • • • 1 1 1 1 • • • • 1 a , 1 • • • • a • • , • , 1 a o ; • • • • • • • • • • • • • • • • • • • • • • • , a •• . •al, • • • • • • 0.1 • ElA • ElB • Fl 1.0 • • •== = • • •::i-: :••• = = :• • • • • • • • • • • • • Shear rate (1/s)
. . . : . . . . --•' The vegetable oil dispersion in the resulting media was highlighted through analyses by optical microscopy and granulometry of filtrates and emulsions (droplets mean diameter around 3 µm). The pseudo plastic behavior observed for the tested emulsions is of interest for an eventual use of these emulsions for food applications. Indeed, the low viscosity at high shear facilitates the implementation of production processes like pumping and filling, whereas the higher viscosity at low shear improve the taste sensation (Amid and Mirhosseini, 2013). Emulsions lA and 1B from the discontinuous process had the highest viscosities, also corresponding to high levels of protein and dry matter. In addition, these emulsions have the highest flow thresholds, which is interesting for paint applications.

2

VI - § 1.5 'U 0 • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • -� VI .,, 1 •• • • • • • • • • • � . . . . . . . . . . . . . . • ♦ • ♦ • • • ♦ • • • • • • • • • . . . . . . . � •. 0.5 • • • • • • • • • • • .. • • • • .. ♦ •• • • • • • • • • • • • • • • • • • • • 0 0.10% •• • • • • • • � . . ..... • • • • • • • 1.00% -� .. • • • • • • • • • • .. ........ ♦. • • • 10.00% Strain {%) • • • • • • • • 1 i s 100.00%

AFig. 1 .

 1 Fig. 1. Schematic modular barrel of the BC21 twin-screw extruder.

Fig. 2 .

 2 Fig. 2. Screw configuration for the passion fruit extractions.

Fig. 3 .

 3 Fig. 3. Discontinuous extraction of edible part of passion fruit.

Fig

  Fig. 5. Optical micrographs ofemulsion lA (S.A), emulsion 2A (5.B) and emulsion 3A (5.C) (scale bar= l0µm, magnification x40).

  The stability of emulsions was assessed by the creaming index, calculated from the serum layer formation. Results are exposed in Fig. 9. After 14 days, the emulsion 3A is the most stable, resulting in a creaming index of Jess than 200/4. This emulsion prepared from the whole fruit is more stable than the lA and 2A emulsions (CI: 32%) prepared without the fruit shells, indicating the probable extraction of natural emulsifiers (phospholipids) from the shells that contribute to the emulsion stability. Polyphenols, present in greater amounts in the emulsion 3A than in the emulsions lA and 2A, can also play a role in the stabilization of the emulsion. Indeed, the work of Di Mattia et al. (2009) and Sabouri et al. (2015) showed that some polyphenols can concentrate on the surface of oil droplets by binding to the emulsifying molecules present at the interface via hydrogen bonds or hydrophobie interactions (Di Mattia et al., 2009; Sabouri et al., 2015). This layer of polyphenols improves the degree of dispersion of the oil and enhances the resistance of the droplets surface to dilation. The addition of ca techin and quercetin thus improved the oil in water emulsion stability over a period of 10 days (Sabouri et al., 2015). However, it is difficult to predict these properties, which depend in particular on the nature of the emulsifying molecules and the pH of the solution. Conoerning our media, the combination of polyphenols and natural emulsifying mole cules extracted during the extrusion seems more favorable to the sta bilization of the emulsions than the compounds extracted by the batch prooess.Oil enrichment resulted in emulsions B having lower stabilities than the corresponding emulsion A. Thus, emulsion 2B which does not contain emulsifying molecules either from the shells or the juioe (water based emulsion) logically has the highest creaming index. The en hanced stability of emulsions A relative to emulsions B is in agreement with the smaller droplet sizes (Fig.5 and Table 6) and the larger LVR (Fig.8) obtained for emulsions A.

Fig. 7 .

 7 Fig. 7. Flow characterization of filtra tes (F), emulsions (E) and contrai (passion fruit seed oil) -Mean values c:i triplicate, SD < 5%.

Fig. 8 .Fig. 9 .

 89 Fig. 8. Evolution of the elastic modulus G' as a function of the strain -Mean values of triplicate, SD < 5%.

  These features are important to en sure adequate consistency in the final formulation. Emulsions prepared from filtrates after the extrusion of the whole fruit (3A emulsion) proved to be the most stable over time. The coextracted polyphenols, polysaccharides and other natural emulsifiers probably work in favor of this physical stability. Ail the filtrates obtained can constitute basic media to prepare oil enriched emulsions. Their polyphenols contents could contribute to the conservation of these natural emulsions. For twin screw extrusion, a supply of 100 kg would lead to the production of 78.5 kg of pre formulated emulsion. If the same amount of fruit is used for juice production, the co product (seeds and pulp) resulting from this transformation could be treated with the dis continuous process (way 2) to obtain of 41.9kg of pre formulated emulsion. The use of thermo mechanical processes to up grade the output gap or the co products of passion fruit would help to minimize energy, solvent costs and extraction steps, thus limiting the environmental impacts. At the same time, the extrudate could be reused for the pro duction of agromaterials by thermo pressing as suggested by the work of Evon et al. (2014).

Acknowledgments

The authors thank Derivery SAS for their financial and technical support, as well as National Agency for Research and Technology (ANRT) for funding the program.


Anne De Bouard

Arnaud Debussche

Reika Fukuizumi

Romain Poncet

Fluctuations and Temperature Effects in Bose-Einstein Condensation

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Introduction

The first experimental realisations of Bose-Einstein condensates of ultra cold gases in 1995 (see [START_REF] Anderson | Observation of bose-einstein condensation in a dilute atomic vapor[END_REF][START_REF] Bradley | Evidence of bose-einstein condensation in an atomic gas with attractive interactions[END_REF][START_REF] Davis | Bose-einstein condensation in a gas of sodium atoms[END_REF]) has been the starting point of experimental studies of those objects whose behaviour close to superfluidity was predicted a long time ago. But it has also led to a large increase of the theoretical studies, as the comparisons with the experiments were henceforth made feasible.

Mean field techniques have then shown their power in the sense that the Gross-Pitaevskii equation, which is the mean field limit equation for the macroscopic wave function of the condensate, was proven to be able to capture a large part of the physical phenomena for zero-temperature condensates in harmonic traps, especially concerning the energy and density distributions (see [START_REF] Dalfovo | Theory of bose-einstein condensation in trapped gases[END_REF]). The predictions are in particular in very good agreement with the experiments. An important feature of the system is the role played by the two-body interactions. Indeed, at temperature close to the absolute zero, due to its inhomogeneity, and despite the very dilute nature of the gases, the atom-atom interactions, when taken into account in the model, can lead to density in the center of the condensate much smaller than the density predicted for an ideal gas in the same trap (see [START_REF] Dalfovo | Theory of bose-einstein condensation in trapped gases[END_REF]). A consequence is that the nonlinearity in the Gross-Pitaevskii equation is essential and cannot be neglected.

When the temperature of a gas is decreased below a critical temperature, the wavelength of the atomic cloud becomes equal to the typical distance between the particles, wich leads to the necessity to take account of quantum effects, so that the statistical nature of the particles becomes essential. In a Boson gas, at a temperature close to the asbolute zero, the wave function of the system of N particles x 1 , . . . , x N ∈ R 3 may be written with the use of a single wave functionψ : ψ(x 1 , . . . , x N ) = Π N j=1 ψ(x j ).

Denoting by V the external potential allowing to confine the system, and taking account of two-body interactions through the symetric interaction potential U , the Hamiltonian of the system may be written as

H = N j=1 - 2 2m ∆ xj + V (x j ) + 1≤j,k≤N U (x j -x k ).
This induces the following expression for the energy of the system, after rescaling the wave function ψ by a factor 1 √ N and assuming N is large :

E(ψ) = R 3 2 2m |∇ψ(t, x)| 2 + V (x)|ψ(t, x)| 2 + 1 2 R 3 U (x -x)|ψ(t, x )| 2 dx |ψ(t, x)| 2 dx,
and the Schrödinger evolution i ∂ t ψ(t, x) = ∇ ψ E(ψ)(t, x). Now, when the distance between the particle is less than the de Broglie wavelength λ T and for weak interactions, the particles do not see the details of the interaction potential and this later may be replaced with the effective potential U eff (x) = 4π 2 as m δ 0 (x). Here a s is the atomic diffusion length, a constant that may be positive or negative. We then finally end up with the evolution equation

i ∂ t ψ = - 2m ∆ψ(t, x) + V (x)ψ(t, x) + 4π 2 a s m |ψ| 2 ψ(t, x),
usually called the Gross-Pitaevskii (GP) equation (see [START_REF] Antoine | Modeling and computation of Bose-Einstein condensates: stationary states, nucleation, dynamics, stochasticity[END_REF][START_REF] Pethick | Bose-Einstein condensation in dilute gases[END_REF]). One may also consult [START_REF] Lewin | Mean-field limit of bose systems: rigorous results[END_REF], for rigorous results on this derivation. Many effects may be added in the system, which can still be handled by the above mean field procedure. The effects of rotation of the condensate -which, above a certain critical speed leads the nucleation of vortices, a signature of the superfluid beviour -leasds to the addition of a first order term in the (GP) equation (see [START_REF] Antoine | Modeling and computation of Bose-Einstein condensates: stationary states, nucleation, dynamics, stochasticity[END_REF]). Dipolar interations may also be taken into account, for the modeling of condensates of atoms with strong magnetic moments, like e.g. 52 Cr, and leads to additional nonlocal terms in (GP). Multicomponents BEC are also widely studied nowadays in the mean field approximation.

Our interest here lies in the case where additional fluctuations are taken into account and modeled thanks to the addition of stochastic terms. This is the case e.g. when fluctuations of the parameters of the laser used for the confinement are taken into account. It leads to a (GP) equation in which the confining potential is perturbed by a random stationary process, that can be approximated with white noise (see [START_REF] Abdullaev | Dynamics of a bose-einstein condensate in optical trap[END_REF][START_REF] Abdullaev | Dynamics of a trapped 2d bose-einstein condensate with periodically and randomly varying atomic scattering length[END_REF][START_REF] Garnier | Collapse of a bose-einstein condensate induced by fluctuations of the laser intensity[END_REF][START_REF] Gehm | Dynamics of noise-induced heating in atom traps[END_REF][START_REF] Savard | Laser-noise-induced heating in far-off resonance optical traps[END_REF]). We will review in Section 1 the mathematical and numerical results on the corresponding (GP) equation).

Another situation in which stochastic effects are included in the model is the case where the temperature is close to the critical temperature of condensation. In this partially condensed situation, interactions of the condensate with the "thermal cloud" formed by non-condensed atoms need to be taken into account. Based on the remark that a consistent description of those interactions should preserve the principles of the fluctuationdissipation theorem, thus leading (formally) to the relaxation of the system to the expected physical equilibrium, both dissipation and stochastic fluctuations are included in the (GP) equation (see [START_REF] Blakie | Dynamics and statistical mechanics of ultra-cold bose gases using c-field techniques[END_REF][START_REF] Duine | Stochastic dynamics of a trapped bose-einstein condensate[END_REF][START_REF] Gardiner | The stochastic grosspitaevskii equation: Ii[END_REF]). A review of some mathematical and numerical results on the corresponding model -so called stochastic (projected) Gross-Pitaevskii equation -is presented in Section 2. Note that the model has proved to be particularly efficient in reproducing the experimentally observed formation of defects in the condensation process, consisting in decreasing the gas temperature from T above the critical temperature T c to T < T c , while in the same time increasing the chemical potential µ (see [START_REF] Weiler | Spontaneous vortices in the formation of bose-einstein condensates[END_REF], and Section 2).

Random variations of the confining potential

In the mean field theory of BEC based on the Gross-Pitaevskii (GP) equation, fluctuations of the laser field intensity can be regarded as modulations of the harmonic trap potential. Such a model has been considered e.g. in [START_REF] Abdullaev | Dynamics of a bose-einstein condensate in optical trap[END_REF] where moments methods were used to discuss possible collapse of the (attractive) condensate, in space dimension 2, for a radially symmetric condensate. Fluctuations were then modeled by a delta correlated Gaussian process, that is a white noise in time. The same model, but with a more general time-dependent stationary process was also considered in the three-dimensional case in [START_REF] Garnier | Collapse of a bose-einstein condensate induced by fluctuations of the laser intensity[END_REF], and considerations about the noise power spectrum may be found in [START_REF] Savard | Laser-noise-induced heating in far-off resonance optical traps[END_REF].

In a far-off resonance trap created by red detuned lasers, one may write the effective trapping potential in the form V (t, x) = -1 4 α|E(t, x)| 2 , where sα is the atomic polarizability and E(t, x) is the field amplitude (see [START_REF] Garnier | Collapse of a bose-einstein condensate induced by fluctuations of the laser intensity[END_REF][START_REF] Savard | Laser-noise-induced heating in far-off resonance optical traps[END_REF]). Then, if the dynamics of the trapped atoms is described by the hamiltonian

H = p 2 2m + 1 2 mω 2 tr (1 + ε(t))x 2 ,
where ω tr = k0 m is the mean-square trap oscillation frequency (which is proportional to the time-averaged laser intensity I 0 ) and ε(t) represents the fluctuations in the spring constant k, with ε(t) = I(t)-I0 I0 , one end up with the Gross-Pitaevskii equation with stochastic potential

i ∂ t ψ = - 2 2m ∆ψ + V (t, x)ψ + g|ψ| 2 ψ. (1)
Here, the nonlinear coefficient is g = 4π 2 as m , and in the harmonic situation described above, V has the expression

V (t, x) = mω 2 tr 2 x 2 (1 + ε(t)).
In general, the fluctuation process ε(t) is taken stationary, zero mean, and with sufficiently decaying correlation function c(t -s) = E(ε(t)ε(s)).

Introducing the variables t = ω tr t, x = x r0 with r 0 = mωtr and ψ = 4π|a s |r 2 0 ψ, ε (t ) = ε(t /ω tr ) and dropping the primes, allows to write (1) in dimensionless form

i∂ t ψ = - 1 2 ∆ψ + 1 2 x 2 (1 + ε(t))ψ + λ|ψ| 2 ψ, (2)
where λ = sign(a s ) = ±1. A variational approach was used in [START_REF] Garnier | Collapse of a bose-einstein condensate induced by fluctuations of the laser intensity[END_REF], assuming a Gaussian profile for the wave function of the condensate of the form

ψ(t, x) = A(t) exp - |x| 2 2a(t) + i 2 b(t)|x| 2 + iθ(t) ;
moreover, a diffusion-approximation theorem was used on the system rewritten in terms of action-angle variables, showing that the dynamics of the action variable is well approximated by a Markov diffusion process, with a generator depending on some integral in time of the correlation function c. This allowed the authors to predict the collapse time (in the attractive cas λ = -1) and compare it with numerical simulations. Such considerations on the collapse had also previously been considered in the two-dimensional case in [START_REF] Abdullaev | Dynamics of a bose-einstein condensate in optical trap[END_REF].

From a mathematical point of view, considering a delta-correlated fluctuation process ε (see also Theorem 2 below), it is convenient to write the equation as a Stratonovich stochastic partial differential equation, with the help of a real valued Brownian motion W on a probability space (Ω, F, P) endowed with a filtration (F t ) t≥0 associated with W and writing formally ε(t) = σ 0 dW dt . We then find

idψ = - 1 2 (∆ψ -V (x)ψ)dt + λ|ψ| 2 ψdt + σ 0 2 V (x)ψ • dW, (3)
where

V (x) = |x| 2 , σ 0 ∈ R, λ = ±1 and
• stands for a Stratonovich product in the right hand side of (3). Note that the choice of a Stratonovich product is natural at least for two reasons : first, as explained above, this delta-correlated situation is the limiting case (for vanishing correlation length) of a more general situation where the fluctuations are modeled with a stationary process; secondly, such a noise allows the preservation of the total number of atoms, N = |ψ| 2 L 2 , almost surely (see below) which is also a natural assumption. As is classical, we may rewrite equation ( 3) as a Itô equation :

idψ = - 1 2 (∆ψ -V (x)ψ)dt + λ|ψ| 2 ψdt + σ 0 2 V (x)ψdW -i σ 2 0 4 V 2 (x)ψdt. ( 4
)
This equation has been studied in the series of papers [START_REF] Bouard | Stochastic fluctuations in the gross-pitaevskii equation[END_REF][START_REF]Modulation analysis for a stochastic nls equation arising in bose-einstein condensation[END_REF][START_REF]Representation formula for stochastic schrödinger evolution equations and applications[END_REF][START_REF] De Bouard | Vortex solutions in bose-einstein condensation under a trapping potential varying randomly in time[END_REF] and [START_REF] Poncet | Numerical analysis of the stochastic gross-pitaevskii equation[END_REF]. The question of interest were in particular the local and global existence of solutions (and the possible collapse, in the attractive cas λ = -1, which was previously discussed in the physics papers), the dynamics of the solutions close to a multiple vortex, numerical simulations and the numerical analysis of the Crank-Nicolson scheme. We describe in more details the corresponding results in the following subsections.

Existence results and diffusion-approximation

Note that in equation ( 3), the noise term σ0 2 V (x)ψ • dW , although linear in ψ, occurs at the highest order of the operator -∆ + V (x) of the linear evolution (this was also the case for the (NLS) equation with white noise dispersion considered e.g. in [START_REF]The nonlinear schrödinger equation with white noise dispersion[END_REF][START_REF] Debussche | 1d quintic nonlinear schrödinger equation with white noise dispersion[END_REF]). A consequence is that the noise term cannot be treated as a perturbation of the deterministic equation.The strategy in [START_REF]Representation formula for stochastic schrödinger evolution equations and applications[END_REF] is rather to use the Gauge transform

ψ(t, x) = e -iG(t,x) u(t, x), (5)
with

G(t, x) = 1 2 V (x)(t + W (t))
. The new wave function u then satisfies the following Schrödinger equation with a random magnetic field:

i∂ t u = - 1 2 ∇ -iA(t, x) 2 u + λ|u| 2σ u. (6)
with

A = ∇G(t, x) = 1 2 ∇V (x)(t + W (t)
). In order to obtain local existence results for equation [START_REF] Blakie | Dynamics and statistical mechanics of ultra-cold bose gases using c-field techniques[END_REF], we need dispersive estimates on the linear evolution, deduced from "good" properties of the integral kernel of the linear evolution propagator (with λ = 0), which can be expressed in terms of classical orbits, as is often used in semiclassical analysis. Using these classical paths, we can write the propagator as an oscillatory integral operator associated to the action integral. Such oscillatory integral operators have been studied by many authors in the context of deterministic Schrödinger equations (see e.g. [START_REF] Fujiwara | A construction of the fundamental solution for the schrödinger equation[END_REF][START_REF] Kumano-Go | Feynman path integrals as analysis on path space by time slicing approximation[END_REF][START_REF] Yajima | Schrödinger evolution equations with magnetic fields[END_REF]). The theory of [START_REF] Yajima | Schrödinger evolution equations with magnetic fields[END_REF] does not apply directly to equation [START_REF] Blakie | Dynamics and statistical mechanics of ultra-cold bose gases using c-field techniques[END_REF], since it requires that the time derivative of the vector potential A(t, x) is uniformly bounded, while this time derivative only exist as a distribution in our case, since Ẇ (t) is a white noise. We were however able to prove, making use of the almost sure C α regularity of the Brownian motion, with 0 < α < 1/2, that the estimates in [START_REF] Yajima | Schrödinger evolution equations with magnetic fields[END_REF] can be generalized to our case. Actually, in our study, W (t) could be replaced by any real valued C α function of the time variable, with α > 0. The analysis was generalized in [START_REF] Duboscq | A construction of the fundamental solution of the schrödinger equation with a perturbed quadratic hamiltonian[END_REF] to the case of more general Hamiltonian, allowing to consider the situation where rotation of the condensate is taken into account.

Using those dispersive estimates, we obtained in [START_REF]Representation formula for stochastic schrödinger evolution equations and applications[END_REF] the following proposition concerning the local solutions of equation [START_REF] Blakie | Dynamics and statistical mechanics of ultra-cold bose gases using c-field techniques[END_REF]. Here Σ denotes the energy space

Σ = v ∈ L 2 (R d ), |∇v| 2 L 2 + |xv| 2 L 2 = |v| 2 Σ < +∞ . Proposition 1. Assume d = 1, 2 or 3, λ = ±1 and r = 8/d. (i) Let d = 1 and u 0 ∈ L 2 (R).
Then there exists a unique global solution u of ( 6), adapted to

(F t ) t≥0 , almost surely in C([0, T 0 ]; L 2 (R))∩L r (0, T 0 ; L 4 (R)
) for any T 0 > 0. Moreover, the L 2 norm is conserved:

|u(t)| L 2 = |u(0)| L 2 , a.s. in ω, for all t ≥ 0,
and u depends continuously on the initial data u 0 in the following sense: if u 0,n → u 0 in L 2 (R), and if u n denotes the solution of ( 6) with u 0 replaced by u 0,n , then u n → u in L ∞ (0, T 0 ; L 2 ). (ii) Let d = 1 and u 0 ∈ Σ. Then there exists a unique global adapted solution u of ( 6) almost surely in C(R + ; Σ). (iii) Let u 0 ∈ Σ, and d = 2 or 3. Then there exists a maximal time T * = T * u0,ω > 0 such that there exists a unique adapted solution u(t) of ( 6) almost surely in C([0, T * ); Σ), and the following alternative holds:

T * = +∞ or T * < +∞ and lim t↑T * |u(t)| Σ = +∞.
We actually considered in [START_REF]Representation formula for stochastic schrödinger evolution equations and applications[END_REF] more general power nonlinearities, and the relation between the power |u| 2σ u and the space dimension d for local existence in L 2 (R d ) is the usual condition σ < 2/d. An application of the inverse of the Gauge transform given by ( 5) immediately leads to the corresponding local existence and uniqueness result for equation (3) (or equivalently (4)), and the preservation of the L 2 -norm of ψ, almost surely. Global results are deduced from the evolution of the Hamiltonian

H(ψ) = 1 4 |∇ψ| 2 L 2 + 1 4 R d V (x)|ψ(x)| 2 dx + λ 4 |ψ| 4 L 4 . (7)
Indeed, an application of the Itô formula to equation (4) allows to show that for any stopping time τ ≤ T * ∧ T :

H(ψ(τ )) = H(ψ 0 ) + σ 2 0 16 τ 0 R d |∇V (x)| 2 |ψ(x)| 2 dxdt - σ 0 4 Im τ 0 R d ∇V (x) • ∇ψ(x) ψ(x) dx dW, a.s (8)
and

R d V (x)|ψ(τ, x)| 2 dx = R d V (x)|ψ 0 | 2 dx + Im R d x ψ • ∇ψdx, for any τ ≤ T * ∧ T, a.s. (9)
The attractive case λ = -1 was already proved to possibly lead to collapse of the wave function ψ in [14] and we end up with the following existence and uniqueness results for equation (3). (i) Let d = 1 and ψ 0 ∈ L 2 (R). Then there exists a unique global solution ψ(t) of (3), adapted to

(F t ) t≥0 with ψ(0) = ψ 0 , which is almost surely in C(R + ; L 2 (R)) ∩ L r loc (R + ; L 4 ).
Moreover, the L 2 -norm is conserved by the time evolution, that is,

|ψ 0 | L 2 = |ψ(t)| L 2 , a.s., for all t ≥ 0.
If moreover ψ 0 ∈ Σ, then ψ is almost surely in C(R + ; Σ). (ii) Let d = 2 or 3, and ψ 0 ∈ Σ. Then there exist a stopping time τ * = τ * ψ0,ω > 0 and a unique solution ψ(t) of (3), adapted to (F t ) t≥0 with ψ(0) = ψ 0 , almost surely in C([0, τ * ); Σ). In fact, τ * = T * , defined in Proposition 1 (iii). If λ = 1, then T * = +∞. If λ = -1 and there exists a deterministic time t > 0 such that

R d V (x)|ψ 0 (x)| 2 dx + 4 t Im R d ψ0 (x)
x.∇ψ 0 (x) dx + 8H(ψ 0 ) t2 < 0, then P(T * ψ0 ≤ t) > 0, i.e. collapse occurs with a positive probabilty. As a byproduct of the representation formula, and in particular of the continuous dependence of the solution u given by Proposition 1 with respect to the trajectories of the Brownian motion, that can be proved thanks to this representation formula in the L 2 -subcritical case (which implies d = 1), a diffusion-approximation result was obtained in [START_REF]Representation formula for stochastic schrödinger evolution equations and applications[END_REF], which justifies in particular the choice of time white noise in some scaling. Indeed, consider a process m(t) which is either a stationary ergodic process satisfying some mixing conditions, or a Markov process with a unique ergodic invariant measure, so that by the Donsker functional central limit theorem, the process t → ε σ0 t/ε 2 0 m(s)ds converges in distribution in C([0, T ]) to a standard real valued Brownian motion. Then the following holds.

Theorem 2. Let d = 1, λ = ±1 and (m(t)) t≥0 as above. Then, for any ε > 0 and ψ 0 ∈ L 2 (R d ) there exists a unique solution ϕ ε , with continuous paths on R + with values in L 2 (R d ), of the following equation:

i∂ t ϕ = 1 2 (-∆ + V (x))ϕ + λ|ϕ| 2 ϕ + 1 2ε m t ε 2 V (x)ϕ, ϕ(0) = ψ 0 . ( 10
)
Moreover the process ϕ ε converges in distribution in C([0, T ]; L 2 (R d )) as ε tends to zero, to the solution ψ of (3), for any positive T .

Dynamics of solutions close to a multiple vortex

In dimension two, some specific solutions of the deterministic equation (that is when σ 0 = 0), and for repulsive condensates, are given by vortices, which are solutions of the form u(t, r, θ) = e -iµt e imθ ψ µ,m (r), [START_REF] Bouard | A semi-discrete scheme for the stochastic nonlinear schrödinger equation[END_REF] where r, θ are polar coordinates, m is the vortex degree, µ is the chemical potential and ψ µ,m (r) is the radial positive vortex profile. We show in figures 1 and 2 some profiles ψ µ,m , for different values of m and µ. The influence of white noise perturbations of the confining potential on the dynamics of those solutions has been studied in [START_REF] De Bouard | Vortex solutions in bose-einstein condensation under a trapping potential varying randomly in time[END_REF] (and in [START_REF]Modulation analysis for a stochastic nls equation arising in bose-einstein condensation[END_REF] in the case m = 0), in the small noise asymptotics. More precisely, consider the following stochastic Gross-Pitaevskii equation

idu + (∆u -|x| 2 u -|u| 2σ u)dt = ε|x| 2 u • dW, (12)
where • stands for a Stratonovich product in the right hand side of ( 12), σ > 0, ε > 0. Here, as before, W (t) is a standard real valued Brownian motion on R + . We recall that Σ is the energy space consisting of square integrable functions f such that ∇f ∈ L 2 (R 2 ) and xf ∈ L 2 (R 2 ). Let then X m be the closed subspace of Σ that has the same symmetry, i.e., m-equivariant symmetry as the vortex solutions φ µ,m (x) = e imθ ψ µ,m (r):

X m := {v ∈ Σ, e -imθ v(x)
does not depend on θ}.

One may check (see [START_REF] De Bouard | Vortex solutions in bose-einstein condensation under a trapping potential varying randomly in time[END_REF] for details) that the noise in the right hand side of ( 12) preserves the equivariance structure, or in other terms that the solution u(t) of ( 12) belongs to X m if the initial data u(0) = u 0 belongs to X m . Defining then the action functional S µ as

S µ (u) = 1 2 |∇u| 2 L 2 + 1 2 |xu| 2 L 2 + 1 2σ + 2 |u| 2σ+2 L 2σ+2 - µ 2 |u| 2 L 2 , (13)
which is a conserved quantity of the deterministic equation, the profile φ µ,m = e imθ ψ(r) of the vortex solution (11) may be characterized as the unique global minimizer of S µ on X m , as soon as µ > λ m = 2m + 1. A straighforward consequence of this is the stability of the vortex solution [START_REF] Bouard | A semi-discrete scheme for the stochastic nonlinear schrödinger equation[END_REF] for the deterministic equation ( 12) with ε = 0. Using this stability property, the following result was proved in [START_REF] De Bouard | Vortex solutions in bose-einstein condensation under a trapping potential varying randomly in time[END_REF] concerning the solution u ε (t, x) of equation ( 12) with initial data u ε (0, x) = φ µ0,m (x) : it says that we can decompose u ε in X m as the sum of a modulated vortex solution and a remainder with small Σ norm, for t less than some stopping time τ ε , and that this τ ε goes to infinity in probability as ε goes to zero. This decomposition is in the form

u ε (t, x) = e -iξ ε (t) (φ µ0,m (x) + εη ε (t, x)) ( 14
)
for a semi-martingale process ξ ε (t) with values in R, and η ε with value in X m . Here, we use the following orthogonality condition Re(η ε , iφ µ0,m ) L 2 (R 2 ) = 0, a.s., t ≤ τ ε , (15) for a suitable stopping time τ ε , which corresponds to the exit time of εη ε from a small neighborhood of 0 in Σ. More precisely, we have the following result. Theorem 3. Let 1/2 ≤ σ < ∞ and µ 0 > λ m be fixed. For ε > 0, let u ε (t, x) be the solution of [START_REF]The nonlinear schrödinger equation with white noise dispersion[END_REF] with u(0, x) = φ µ0,m (x). Then there exists α 0 > 0 such that, for each α, 0 < α ≤ α 0 , there is a stopping time τ ε α ∈ (0, ∞) a.s., and there is a semi-martingale process ξ ε (t), defined a.s. for t ≤ τ ε α , with values in R, so that if we set εη ε (t, x) = e iξ ε (t) u ε (t, x) -φ µ0,m (x), then (15) holds. Moreover, a.s. for t ≤ τ ε α ,

|εη ε (t)| Σ ≤ α. ( 16
)
In addition, there is a constant C = C α,µ0 > 0, such that for any T > 0 and any α ≤ α 0 , there is an ε 0 > 0, such that for each ε < ε 0 ,

P(τ ε α ≤ T ) ≤ exp - C ε 2 T . ( 17
)
The modulation parameter ξ ε may be written, for t ≤ τ ε α , as

dξ ε = µ 0 dt + εy ε dt + εz ε dW,
for some adapted processes y ε , z ε with values in R satisfying: as ε goes to zero, y ε , z ε converge respectively to some real valued processes y, z in probability in C([0, T ]).

Numerical simulations were also performed in [START_REF] De Bouard | Vortex solutions in bose-einstein condensation under a trapping potential varying randomly in time[END_REF], which showed numerical evidences of the optimality of the estimate [START_REF] De Bouard | Vortex solutions in bose-einstein condensation under a trapping potential varying randomly in time[END_REF], in terms of rates in time and ε. Taking into account the m-equivariant symmetry, Equation ( 12) is only solved in the radial component and reduces to a one dimensional equation. Numerical simulations were performed with a numerical scheme based on a Crank-Nicolson discretization in time, which is natural to consistently approximate the Stratonovich integral (see also section 1.3 below). The nonlinearity is not computed by a fixed point method, but by a relaxation method which amounts to compute an explicit extrapolation of the nonlinearity. The space approximation is based on finite differences. The radial profile of the vortex solutions (which are used as Cauchy condition) were computed by a shooting method. Figure 3 and4 the time evolution of the radial profile of a solution with a vortex solution as initial Cauchy condition with vortex degree m = 1 and m = 2. We can observe that the wave function keeps the same structure over time and that it oscillates with an almost periodic rhythm.

display respectively

The computation of the processes ξ ε (t) and η ε (t, x) are straightforward using the orthogonality condition 15 and are given almost surely for t ≤ τ ε by,

ξ ε (t) = -arg u ε (t), φ µ0,m L 2 (R 2 ) , and εη ε (t, x) = u ε (t, x)e iξ ε (t) -φ µ0,m (x).
An approximation of the law of the stopping time τ ε α with ε = 9 • 10 -3 was computed using a classical Monte-Carlo method, and is given in Figure 5 and6. We can observe that the slopes therein (for ε small enough) are in good agreement with estimate 17.

Numerical analysis of the Crank-Nicolson scheme

The use of a Crank-Nicolson scheme to discretize equation ( 3) is natural since mid-point discretization is consistent with the Stratonovich product. It also preserves the conservation of L 2 norm. Such a discretization has been used in [START_REF] Belaouar | Numerical analysis of the nonlinear schrödinger equation with white noise dispersion[END_REF][START_REF] Bouard | A semi-discrete scheme for the stochastic nonlinear schrödinger equation[END_REF][START_REF] Gazeau | Probability and pathwise order of convergence of a semidiscrete scheme for the stochastic manakov equation[END_REF] for other types of stochastic nonlinear Schrödinger equations. Even if, in terms of computational costs, it is often convenient to consider its linearly implicit derivative given by the relaxation scheme (see the above simulations), the numerical analysis of this later scheme is closely related to the numerical analysis of the Crank-Nicolson scheme, which is in some sense a prerequisite. This numerical analysis has been carried out in [START_REF] Poncet | Numerical analysis of the stochastic gross-pitaevskii equation[END_REF], where the convergence at order one of the time-semi-discrete scheme was proved. The naive Crank-Nicolson discretization of ( 3) is given by

i ψ n+1 -ψ n δt = 1 2 A + χ n+1 √ δt |x| 2 ψ n + ψ n+1 2 + λg(ψ n , ψ n+1 ), (18)
where A = -∆ + |x| 2 , ψ n is an approximation of ψ(nδt), (χ n ) n≥0 is a sequence of independent normal random variables and g is the approximation of the nonlinearity given by

g(ψ n , ψ n+1 ) = 1 2 (|ψ n | 2 + |ψ n+1 | 2 ) ψ n + ψ n+1 2 .
There are several difficulties in this analysis. The first one is the non Lipschitz behavior of the nonlinearity, which is in particular an obstacle to the global existence and uniqueness of a solution ψ n of ( 18), and requires the use of a cut-off for the nonlinear term. Secondly, although the above scheme allows the conservation of the L 2 norm, giving immediately the L 2 -stability, it does not allow to get stable estimates on higher order derivatives, which are needed to study the order of convergence of the scheme. The argument is again to use a cut-off in the highest modes of the operator A = -∆ + |x| 2 which would correspond to the use of a spectral space discretization. The result may be stated as follows. Denote, for k ∈ N,

Σ k = {v ∈ L 2 (R d ), |α|+|β|≤k |x β ∂ α v| 2 L 2 := |v| 2 Σ k < +∞}.
We consider, for a fixed time step δt a solution (ψ n K,L,δt ) nδt≤T of [START_REF] Debussche | 1d quintic nonlinear schrödinger equation with white noise dispersion[END_REF] where A is replaced by A k = P K A, P K being the projection on the K first eigenmodes of A, and g is replaced by a Lipschitz approximation with Lipschitz constant L (see [START_REF] Poncet | Numerical analysis of the stochastic gross-pitaevskii equation[END_REF] for details). Then the following convergence result holds.

Theorem 4. For all T > 0, k ∈ N, ψ 0 ∈ Σ k+12 (R d ), C > 0 and α < 1, there is a choice of the cut-off constants K(δt), L(δt) such that

lim δt→0 sup nδt≤T P |ψ n K(δt),L(δt),δt -ψ(t n )| Σ k ≥ C(δt) α = 0.
2. A model with positive temperature : the projected stochastic Gross-Pitaevskii equation

When temperature is close to the critical temperature of condensation, one has to take into account the non-condensate particle contributions, which are negligible for temperature close to the absolute zero. Even though a unified approach does not seem to be yet available in this situation, the use of a "truncated Wigner theory" (see [START_REF] Blakie | Dynamics and statistical mechanics of ultra-cold bose gases using c-field techniques[END_REF][START_REF] Gardiner | The stochastic gross-pitaevskii equation[END_REF][START_REF] Gardiner | The stochastic grosspitaevskii equation: Ii[END_REF]), leads to a model for the wave function which, in a simplified form taking account only of the growth terms, may be written as

dψ = P c - i L GP ψdt + γ k B T (µ -L GP )ψdt + dW γ (t, x) (19)
where

L GP = - 2 2m ∆ + V (x) + g|ψ(t, x)| 2 ,
m is the atomic mass, g = 4π 2 a m with a the (positive) s-wave scattering length, V (x) = m 2 ω 2 x 2 , µ is the chemical potential, and P c is a projection operator on the lowest energy modes (the non-condensate band). Moreover, the noise term dW corresponds here to a space-time white noise, that is a Gaussian field which is δ-correlated both in space and time :

E dW * γ (t, x)dW γ (t , x ) = 2γδ(t -t )δ(x -x )dt.
The methodology in [START_REF] Gardiner | The stochastic grosspitaevskii equation: Ii[END_REF] consisted in first considering that the modes with wavenumbers k larger than ∆, where 2π/∆ is the order of magnitude of the range of the interatomic potential have no occupation and can be eliminated. Next, the remaining components of the wave function can be divided into a low momentum component (the condensate band) which is treated fully quantum-mechanically, and a high momentum component (the non-condensate band) which is considered to be a bath of thermalized atoms.

The same model was obtained previously in [START_REF] Duine | Stochastic dynamics of a trapped bose-einstein condensate[END_REF] with different methods, in particular a functional integral formulation of the Keldysh method, in which the elimination of the highest energy modes is achieved in the action functional. Moreover, the model is in very good qualitative agreement with the experiments aiming to simulate the dynamics of the condensation process (see [START_REF] Weiler | Spontaneous vortices in the formation of bose-einstein condensates[END_REF], and the simulations below).

At zero temperature, one recovers the standard (GP) equation for the projected wave function ψ c on the lowest energy modes, for which the hamiltonian

H(ψ c ) = 2 2m |∇ψ c | 2 L 2 + |V (x)ψ c | 2 L 2 -µ|ψ c | 2 L 2 + g 4 |ψ c | 4 L 4
is conserved. Note that a symmetry breaking occurs for the ground state of H at µ = ω, since for small values of µ the minimum of H is equal to zero. At positive temperature, it may easily be formally shown that the Gibbs measure given by

π T (dψ c ) = α c exp - H(ψ c ) k B T dψ c ,
for some normalizing constant α c , is invariant under the transition semi-group associated with [START_REF] Duboscq | A construction of the fundamental solution of the schrödinger equation with a perturbed quadratic hamiltonian[END_REF], and the convergence to equilibrium of ψ c with a geometric rate may be proved (see [START_REF] Roberts | Exponential convergence of langevin distributions and their discrete approximations[END_REF]). This model has been used to observe the spontaneous nucleation of vortices during the condensation process (see [START_REF] Weiler | Spontaneous vortices in the formation of bose-einstein condensates[END_REF]). Indeed, the appearance of topological defects during a phase transition is predicted by the Kibble-Zurek mechanism, and can be modeled by Equation [START_REF] Duboscq | A construction of the fundamental solution of the schrödinger equation with a perturbed quadratic hamiltonian[END_REF] in the case of Bose-Einstein condensation. In this model, the phase transition is initiated by a positive value of the chemical potential µ. Thus, starting from a thermalized state with low chemical potential and high temperature, a sudden quench to a higher chemical potential and a lower temperature will lead to the formation of vortices during the growth process of the condensate. This dynamics is shown in Figure 2 We can observe a fast appearance of vortices after the instantaneous quench, followed by a slow decay of the vortices. The numerical scheme used to solve equation ( 19) is an implicit/explicit scheme for the time discretization, and a Hermite spectral discretization in space, which is natural to take precisely into account the cut-off operator P c .

From a mathematical point of view, the role of the cut-off operator P c is also crucial as equation ( 19) is not expected to possess solutions if P c is equal to the identity operator, in space dimension three. However, the equation has been studied in space dimension one, without the cut-off operator, in [START_REF] De Bouard | Long time behavior of a gross-pitaevskii equation at positive temperature[END_REF] and the existence of a global solution of the equation, for a.s. all initial data with respect to the invariant measure of the standard (GP) equation was proved. This later invariant measure was indeed studied in [START_REF] Burq | Long time dynamics for the one dimensional non linear schrödinger equation[END_REF] and its support was shown to contain L p (R) for any p larger than 2. The convergence to equilibrium was also obtained in [START_REF] De Bouard | Long time behavior of a gross-pitaevskii equation at positive temperature[END_REF], thanks to a Poincaré inequality.

More precisely, consider the equation in dimensionless form

dψ = (i + γ) ∆ψ -x 2 ψ + µψ -|ψ| 2 ψ dt + √ γdW (20)
where ψ(t, x) is the wave function, x ∈ R, γ > 0, µ ≥ 0, and W is a cylindrical Wiener process on L 2 (R) that may be modeled as follows : let (h n ) n∈N be a real valued complete orthonormal system in L 2 (R) such that (-∆ + x 2 )h n = λ 2 h n , with λ 2 n = 2n + 1, so that (h n ) n is a sequence of Hermite functions; let (β n ) n∈N be a sequence of independent normalized complex-valued Brownian motions, and write W (t, x) = k∈N β k (t)h k (x). Then denoting

S(ψ) = 1 2 R |∇ψ| 2 + |xψ| 2 dx - µ 2 R |ψ| 2 dx + 1 4 R |ψ| 4 dx,
the Gibbs measure may now be formally defined by ρ(dψ) = e -S(ψ) dψ. It may actually be rigorously defined -at least for µ = 0 -as an absolutely continuous measure with respect to the Gaussian measure on L p (R) for p > 2, defined as the law of the random function series

ϕ(ω, x) = ∞ n=0 √ 2 λ n g n (ω)h n (x),
where (g n ) n is a sequence of independent complex valued N C (0, 1). Indeed, the series converges in L p (R) for any p > 2 (see [START_REF] Burq | Long time dynamics for the one dimensional non linear schrödinger equation[END_REF]). Let ν(dψ) denote the Gaussian measure defined by the law of ϕ in L p (R), p > 2 (which is also the invariant measure of the flow corresponding to the linear evolution in [START_REF] Duine | Stochastic dynamics of a trapped bose-einstein condensate[END_REF] for µ = 0). We may then define the Gibbs measure ρ as ρ(dψ) = Γ -1 e -1 4 |ψ| 4 L 4 ν(dψ), for some normalizing constant Γ. For µ > 0, the rigorous definition of the measure ρ requires a more subtle decomposition (see [START_REF] De Bouard | Long time behavior of a gross-pitaevskii equation at positive temperature[END_REF]).

We can then solve equation [START_REF] Duine | Stochastic dynamics of a trapped bose-einstein condensate[END_REF] locally in time in L p (R), and use the invariance of ρ for the evolution to prove the following theorem.

Theorem 5. Let γ > 0, µ ≥ 0, λ = 1 and p ≥ 3. There exists a ρ-measurable set O ⊂ L p (R) such that ρ(O) = 1, and such that for ψ 0 ∈ O there exists a unique solution of (20), ψ(•) ∈ C([0, ∞), L p (R)) a.s.

Let (P t ) t≥0 be the transition semi-group associated with Equation ( 20) and defined by P t Φ(ψ 0 ) = E(Φ(ψ(t, ψ 0 ))), where ψ(•, ψ 0 ) is the solution of (20) with ψ(0) = ψ 0 , so that P t is well defined in L 2 (L p , dρ), for any p ≥ 3, and any t > 0. The following convergence result holds. Theorem 6. Let γ > 0, µ ≥ 0, λ = 1 and p ≥ 3. Let Φ ∈ L 2 ((L p , dρ); R), and Φ = L p Φ(y)ρ(dy). Then u(t, •) := P t Φ(•) converges exponentially to Φ in L 2 ((L p , dρ), R), as t → ∞; more precisely, Note that the results of Theorem 5 and Theorem 6 are still true for more general power nonlinearities |ψ| 2σ ψ if p ≥ 2σ + 1 and p > 2. Those results represent a first step in the understanding of Equation [START_REF] Duine | Stochastic dynamics of a trapped bose-einstein condensate[END_REF]. The main difference with the more classical Ginzburg-Landau equation (without the confining potential, possibly posed on a bounded domain D) is the growth of the eigenvalues λ n which makes the problem critical, even though the space dimension is only one. The study of larger dimensions, as well as the qualitative study of the dynamics, will require a more refined analysis.

Theorem 1 .

 1 Assume d = 1, 2 or 3, λ = ±1 and r = 8/d.

Figure 1 .

 1 Figure 1. Amplitudes of vortices |ψµ,m| for µ close to λm, for different values of m

Figure 2 .

 2 Figure 2. Amplitudes of vortices |ψµ,m| for m = 1 for different values of µ

Figure 3 .

 3 Figure 3. A trajectory of |u (r, t)| for m = 1 and σ = 1

Figure 4 .

 4 Figure 4. A trajectory of |u (r, t)| for m = 2 and σ = 1

Figure 5 .Figure 6 .

 56 Figure 5. Estimation of P(τ ε α ≤ t) by a Monte-Carlo method, with respect to

Figure 7 .

 7 Figure 7. Time evolution of a BEC submitted to an instantaneous quench from µ = 1 to µ = 25 at t = 0.

  L p |u(t, y) -Φ| 2 ρ(dy) ≤ e -γt L p |Φ(y) -Φ| 2 ρ(dy).

* This work was supported by the ANR through project ANR-12-MONU-0007 BECASIM, the LabEx LMH and the JSPS KAKENHI Grant Number 15K04944.


Giuseppe Da Prato
email: giuseppe.daprato@sns.it

Arnaud Debussche
email: arnaud.debussche@ens-rennes.fr

Estimate for P t D for the stochastic Burgers equation

We consider the Burgers equation on H = L 2 (0, 1) perturbed by white noise and the corresponding transition semigroup P t . We prove a new formula for P t Dϕ (where ϕ : H → R is bounded and Borel) which depends on ϕ but not on its derivative. Then we deduce some consequences for the invariant measure ν of P t as its Fomin differentiability and an integration by parts formula which generalises the classical one for gaussian measures.

Introduction

We consider the following stochastic Burgers equation in the interval [0, 1] with Dirichlet boundary conditions,    dX(t, ξ) = (∂ 2 ξ X(t, ξ) + ∂ ξ (X 2 (t, ξ)))dt + dW (t, ξ), t > 0, ξ ∈ (0, 1), X(t, 0) = X(t, 1) = 0, t > 0, X(0, ξ) = x(ξ), ξ ∈ (0, 1).

(1)

The unknown X is a real valued process depending on ξ ∈ [0, 1] and t ≥ 0 and dW/dt is a space-time white noise on [0, 1] × [0, ∞). This equation has been studied by several authors (see [START_REF] Bertini | The stochastic Burgers equation[END_REF], [START_REF] Da Prato | Stochastic Burgers equation[END_REF], [START_REF] Da Prato | Gatarek Stochastic Burgers equation with correlated noise[END_REF], [START_REF] Gyongy | Existence and uniqueness results for semilinear stochastic partial differential equations in Hilbert space[END_REF]) and it is known that there exists a unique solution with paths in C([0, T ]; L p (0, 1)) if the initial data x ∈ L p (0, 1), p ≥ 2. In this article, we want to prove new properties on the transition semigroup associated to (1).

We rewrite (1) as an abstract differential equation in the Hilbert space H = L 2 (0, 1),

   dX = (AX + b(X))dt + dW t , X(0) = x.
(2)

As usual A = ∂ ξξ with Dirichlet boundary conditions, on the domain D(A) = H 2 (0, 1) ∩ H 1 0 (0, 1), b(x) = ∂ ξ (x 2 ). Here and below, for s ≥ 0, H s (0, 1) is the standard L 2 (0, 1) based Sobolev space. Also, W is a cylindrical Wiener process on H. We denote by X(t, x) the solution.

We denote by (P t ) t≥0 the transition semigroup associated to equation (2) on B b (H), the space of all real bounded and Borel functions on H endowed with the norm ϕ 0 = sup

x∈H |ϕ(x)|, ∀ ϕ ∈ B b (H).
We know that P t possesses a unique invariant measure ν so that P t is uniquely extendible to a strongly continuous semigroup of contractions on L 2 (H, ν) (still denoted by P t ) whose infinitesimal generator we shall denote by L . Let E A (H) be the linear span of real parts of all ϕ of the form ϕ h (x) := e i h,x , x ∈ D(A).

We have proved in [START_REF] Da Prato | m-dissipativity of Kolmogorov operators corresponding to Burgers equations with space-time white noise[END_REF] that E A (H) is a core for L and that

L ϕ = 1 2 Tr [QD 2 x ϕ] + Ax + b(x), D x ϕ , ∀ ϕ ∈ E A (H). (3)
Here and below, D x denotes the differential with respect to x ∈ H. When ϕ is a real valued function, we often identify D x ϕ with its gradient. Similarly, D 2 x is the second differential and for a real valued function D 2

x ϕ can be identify with the Hessian. In this paper, we use a formula for P t D x ϕ which depends on ϕ but not on its derivative. To our knowledge, this formula is new.

For a finite dimensional stochastic equation a formula for P t D x can be obtained, under suitable assumptions, using the Malliavin calculus and it is the key tool for proving the existence of a density of the law of X(t, x) with respect to the Lebesgue measure, see [START_REF] Malliavin | Stochastic analysis[END_REF]. Concerning SPDEs, several results are available for densities of finite dimensional projections of the law of the solutions, see [START_REF] Solé | Malliavin Calculus with Applications to Stochastic Partial Differential Equations[END_REF] and the references therein. For these results, Malliavin calculus is used on a finite dimensional random variable. Malliavin calculus is difficult to generalize to a true infinite dimensional setting and it does not seem useful to give estimate on P t D x ϕ in terms of ϕ. The formula we use allows a completely different approach. It relates P t D x to D x P t . In the recent years several formulae for D x P t ϕ independent on D x ϕ have been proved thanks to suitable generalizations of the Bismut-Elworthy-Li formula (BEL). Thus, combining our formula to estimates obtained on D x P t implies useful information on P t D x . As we shall show, these can be used to extend to the measure ν a basic integration by parts identity well known for Gaussian measures.

Let us explain the main ideas. Let u(t, x) = P t ϕ(x), under suitable conditions, it is a solution of the Kolmogorov equation

     D t u(t, x) = 1 2 Tr [QD 2 x u(t, x)] + Ax + b(x), D x u(t, x) , u(0, x) = ϕ(x). (4)
Set v h (t, x) = D x u(t, x) • h, the differential of u with respect to x in the direction h.

Then formally v h is a solution to the equation

   D t v h (t, x) = L v h (t, x) + Ah + b (x)h, D x u(t, x) , v h (0, x) = D x ϕ(x) • h. (5)
We notice that formal computations may be made rigorous by approximating ϕ with elements from E A (H). By (5) and variation of constants, it follows that

v h (t, x) = P t (D x ϕ(x) • h) + t 0 P t-s ( Ah + b (x)h, D x u(s, x) )ds, (6)
which implies

P t (D x ϕ(x) • h) = D x P t ϕ(x) • h - t 0 P t-s ( Ah + b (x)h, D x u(s, x) )ds. ( 7
)
This formula allows to obtain the following estimate: For all ϕ ∈ B b (H), δ > 0 and all h ∈ H 1+δ (0, 1), we have

|P t (Dϕ • h)(x)| ≤ ce ct (1 + t -1/2 )(1 + |x| L 4 ) 8 ϕ 0 |h| 1+δ , (8)
where | • | 1+δ is the norm in H 1+δ (0, 1). We will not prove this formula here, it could be proved by similar arguments as in section 3.

Integrating with respect to ν over H and taking into account the invariance of ν, yields

H (D x ϕ(x) • h)dν = H (D x P t ϕ(x) • h) dν - t 0 H ( Ah + b (x)h, D x P s ϕ(x) )dν ds. (9)
Using identity (9) we arrive at the main result of the paper, proved in Section 2.

Theorem 1. For any p > 1, δ > 0, there exists C > 0 such that for all ϕ ∈ B b (H) and all h ∈ H 1+δ (0, 1), we have

H D x ϕ(x) • h ν(dx) ≤ C ϕ L p (H,ν) |h| 1+δ , ( 10
)
for t > 0, where | • | 1+δ is the norm in H 1+δ (0, 1).
For a gaussian measure, it is easy to obtain such estimate. In fact, if µ is the invariant measure of the stochastic heat equation on (0, 1), i.e. equation (2) without the nonlinear term, then the same formula holds with δ = 0. Thus our result is not totally optimal and we except that it can be extended to δ = 0. Also, identity ( 9) is general and we believe that it can be used in many other situations. For instance, we will investigate the generalization of our results to other SPDEs such as reaction-diffusion and 2D-Navier-Stokes equations. This will be the object of a future work.

In section 2, we show that Theorem 1 can be used to derive an integration by part formula for the measure ν. Theorem 1 is proved in section 3.

We end this section with some notations. We shall denote by (e k ) an orthonormal basis in H and by (α k ) a sequence of positive numbers such that

Ae k = -α k e k , k ∈ N.
For any k ∈ N, D k will represent the directional derivative in the direction of e k .

The norm of L 2 (0, 1) is denoted by | • |. For p ≥ 1, | • | L p is the norm of L p (0, 1). The operator A is self-adjoint negative. For any α ∈ R, (-A) α denotes the α power of the operator -A and | • | α is the norm of D((-A) α/2 ) which is equivalent to the norm of the Sobolev space H α (0, 1). We have

| • | 0 = | • | = | • | L 2 . We shall use the interpolatory estimate |x| β ≤ |x| γ-β γ-α α |x| β-α γ-α γ , α < β < γ, (11)
and the Agmon's inequality

|x| L ∞ ≤ |x| 1 2 |x| 1 2 1 . ( 12
)
Acknowledgement A. Debussche is partially supported by the french government thanks to the ANR program Stosymap. He also benefits from the support of the french government "Investissements d'Avenir" program ANR-11-LABX-0020-01.

Integration by part formula for ν

This section is devoted to some consequences of Theorem 1. Here we take p = 2 for simplicity. In this case (10) can be rewritten as

H ((-A) -α D x ϕ(x) • h) ν(dx) ≤ C ϕ L 2 (H,ν) |h|, ∀ h ∈ H, (13)
where α = 1+δ 2 . Proposition 2. Let α > 1 2 , then for any h ∈ H the linear operator

ϕ ∈ C 1 b (H) → ((-A) -α D x ϕ(x) • h) ∈ C b (H) is closable in L 2 (H, ν). Proof. Let (ϕ n ) ⊂ C 1 b (H) and f ∈ L 2 (H, ν) such that ϕ n → 0 in L 2 (H, ν), ((-A) -α D x ϕ n (x) • h) → f in L 2 (H, ν). Let ψ ∈ C 1 b (H), then by (13) it follows that H [ψ(x)((-A) -α D x ϕ n (x) • h) + ϕ n (x)((-A) -α D x ψ(x) • h)] ν(dx) ≤ ϕ n ψ L 2 (H,ν) |h| H ≤ ψ 0 ϕ n L 2 (H,ν) |h| H .
Letting n → ∞, yields

H ψ(x)f (x) ν(dx) = 0,
which yields f = 0 by the arbitrariness of ψ.

We can now define the Sobolev space W 1,2 α (H, ν). First we improve Proposition 2.

Corollary 3. Let α > 1 2 , then the linear operator

ϕ ∈ C 1 b (H) → (-A) -α D x ϕ ∈ C b (H; H) is closable in L 2 (H, ν).
Proof. By Proposition 2 taking h = e k we see that D k is a closed operator on L 2 (H, ν)

for any k ∈ N. Set (-A) -α D x ϕ(x) = ∞ k=1 α -α k D k ϕ(x) e k , ∀ h ∈ H, the series being convergent in L 2 (H, ν). Then |(-A) -α D x ϕ(x)| 2 = ∞ k=1 α -2α k |D k ϕ(x)| 2 Let (ϕ n ) ⊂ C 1 b (H) and F ∈ L 2 (H, ν; H) such that ϕ n → 0 in L 2 (H, ν), (-A) -α D x ϕ → F in L 2 (H, ν; H).
We have to show that F = 0. Now for any k ∈ N we have

D k ϕ n (x) → α α k F (x), e k in L 2 (H, ν)
. So, F, e k = 0 and the conclusion follows.

Let us denote by W 1,2 α (H, ν) the domain of the closure of (-A) -α D x . Then if M * denotes the adjoint of (-A) -α D x we have

H ((-A) -α D x ϕ(x) • F (x)) ν(dx) = H ϕ(x) M * (F )(x) ν(dx). (14)
Set now F h (x) = h where h ∈ H. By Theorem 1 F h belongs to the domain of M * . Setting M * (F h ) = v h we obtain the following integration by part formula.

Proposition 4. Let α > 1 2 , then for any h ∈ H there exists a function

v h ∈ L 2 (H, ν) such that H ((-A) -α D x ϕ(x) • h) ν(dx) = H ϕ(x) v h (x) ν(dx), (15)
for any ϕ ∈ W 1,2 α (H, ν).

By (15) it follows that the measure ν possesses the Fomin derivative in all directions (-A) -α h for h ∈ H, see e.g. [START_REF] Pugachev | Surface measures in infinite-dimensional spaces[END_REF].

If, in (2), b = 0 then the gaussian measure µ = N Q , where

Q = -1 2 A -1 , is the invariant measure and v h (x) = √ 2 Q -1/2
x, h . Then (15) reduces to the usual integration by parts formula for the Gaussian measure µ. Note that it follows that, as already mentionned, Theorem 1 is true with δ = 0 in this case.

We recall the importance of formula (15) for different topics as Malliavin calculus [START_REF] Malliavin | Stochastic analysis[END_REF], definition of integral on infinite dimensional surfaces of H [AiMa88], [START_REF] Feyel | La Pradelle: Hausdorff measures on the Wiener space[END_REF], [START_REF]Bogachev Gaussian Measures[END_REF], definition of BV functions in abstract Wiener spaces [START_REF] Ambrosio | BV functions in abstract Wiener spaces[END_REF], infinite dimensional generalization of DiPerna-Lions theory [AmFi09], [START_REF] Da Prato | Uniqueness for continuity equations in Hilbert spaces with weakly differentiable drift[END_REF] and so on.

We think that Theorem 1 open the possibility to study these topics in the more general situations of non Gaussian measures.

Proof of Theorem 1

For h ∈ H, η h (t, x) is the differential of X(t, x) in the direction h and (η h (t, x)) ≥0 satisfies the equation

     dη h (t, x) dt = Aη h (t, x) + b (X(t, x))η h (t, x), η h (0, x) = h. ( 16
)
Note that this equation as well as the computations below are done at a formal level. They could easily be justified rigorously by an approximation argument, such as Galerkin approximation for instance. The following result is proved in [START_REF] Da Prato | m-dissipativity of Kolmogorov operators corresponding to Burgers equations with space-time white noise[END_REF], see Proposition 2.2.

Lemma 5. For any α ∈ [-1, 0] there exists c = c(α) > 0 such that

e -c t 0 |X(s,x)| 8 3 L 4 ds |η h (s, x)| 2 α + t 0 e -c t s |X(τ,x)| 8 3 L 4 dτ |η h (s, x)| 2 1+α ds ≤ |h| 2 α . (17)
We introduce the following Feynman-Kac semigroup

S t ϕ(x) = E ϕ(X(t, x))e -K t 0 |X(s,x)| 4 L 4 ds
Next lemma is a slight generalization of Lemma 3.2 in [START_REF] Da Prato | m-dissipativity of Kolmogorov operators corresponding to Burgers equations with space-time white noise[END_REF].

Lemma 6. For any α ∈ [0, 1] and p > 1, if K is chosen large enough then for any ϕ Borel and bounded we have

|D x S t ϕ(x)| α ≤ c e ct (1 + t -1+α 2 )(1 + |x| 3 L 6 ) [E (ϕ p (X(t, x))] 1/p , (18)
where c depends on p, K, α.

Proof. It is clearly suficient to prove the result for p ≤ 2. We proceed as in [START_REF] Da Prato | m-dissipativity of Kolmogorov operators corresponding to Burgers equations with space-time white noise[END_REF] and write

D x S t ϕ(x) • h = I 1 + I 2 .
where

I 1 = 1 t E e -K t 0 |X(s,x)| 4 L 4 ds ϕ(X(t, x)) t 0 (η h (s, x), dW (s))
and

I 2 = -4KE e -K t 0 |X(s,x)| 4 L 4 ds ϕ(X(t, x)) t 0 (X 3 (s, x), η h (s, x))ds .
For I 1 we have with 1 p + 1 q = 1:

I 1 ≤ 1 t [E (ϕ p (X(t, x))] 1/p E e -Kq t 0 |X(s,x)| 4 L 4 ds t 0 (η h (s, x), dW (s) q 1/q Using Itô's formula for |z(t)| q = e -Kq t 0 |X(s,x)| 4 L 4 ds t 0 (η h (s, x), dW (s)) q
, we get:

|z(t)| q = -4Kq t 0 |X(s, x)| 4 L 4 |z(s)| q ds +q t 0 e -K s 0 |X(s,x)| 4 L 4 ds |z(s)| q-2 z(s)(η h (s, x), dW (s)) + 1 2 q(q -1) t 0 e -2K t 0 |X(s,x)| 4 L 4 ds |z(s)| q-2 |η h (s, x)| 2 ds.
We deduce:

E sup r∈[0,t] |z(r)| q ≤ qE sup r∈[0,t] r 0 e -K s 0 |X(s,x)| 4 L 4 ds |z(s)| q-2 z(s)(η h (s, x), dW (s)) + 1 2 q(q -1)E t 0 e -2K t 0 |X(s,x)| 4 L 4 ds |z(s)| q-2 |η h (s, x)| 2 ds = A 1 + A 2 .
By a standard martingale inequality, (11) and Lemma 5, we have

A 1 ≤ 3qE t 0 e -2K s 0 |X(s,x)| 4 L 4 ds |z(s)| 2(q-1) |η h (s, x)| 2 ds 1/2 ≤ 3qE sup r∈[0,t] |z(r)| q-1 t 0 e -2K s 0 |X(s,x)| 4 L 4 ds |η h (s, x)| 2 ds 1/2 ≤ 3qE sup r∈[0,t] |z(r)| q-1 t 0 e -2K s 0 |X(s,x)| 4 L 4 ds |η h (s, x)| 2(1-α) -α |η h (s, x)| 2α 1-α ds 1/2 ≤ 3qt 1-α 2 |h| -α E sup r∈[0,t] |z(r)| q-1 ≤ 3qt 1-α 2 |h| -α E sup r∈[0,t] |z(r)| q (q-1)/q ≤ 1 4 E sup r∈[0,t] |z(r)| q + ct q(1-α) 2 |h| q -α .
Similarly:

A 2 ≤ 1 2 q(q -1)E sup r∈[0,t] |z(r)| q-2 t 0 e -2K s 0 |X(s,x)| 4 L 4 ds |η h (s, x)| 2 ds ≤ 1 2 q(q -1)E sup r∈[0,t] |z(r)| q-2 t 0 e -2K t 0 |X(s,x)| 4 L 4 ds |η h (s, x)| 2(1-α) -α |η h (s, x)| 2α 1-α ds ≤ 1 2 q(q -1)t 1-α |h| 2 -α E sup r∈[0,t] |z(r)| q-2 ≤ 1 2 q(q -1)t 1-α |h| 2 -α E sup r∈[0,t] |z(r)| q (q-2)/q ≤ 1 4 E sup r∈[0,t] |z(r)| q + ct q(1-α) 2 |h| q -α .
We deduce:

I 1 ≤ ct -1+α 2 |h| -α [E (ϕ p (X(t, x))] 1/p
For I 2 we write

I 2 = 4KE e -K t 0 |X(s,x)| 4 L 4 ds ϕ(X(t, x)) t 0 (X 3 (s, x), η h (s, x))ds ≤ 4K [E (ϕ p (X(t, x))]
I 2 ≤ c q (1 + |x| 3 L 6 ) [E (ϕ p (X(t, x))] 1/p |h| -1 .
Gathering the estimates on I 1 and I 2 gives the result.

Lemma 7. For any α ∈ [0, 1), p > 1, q > 1 satisfying 1 p + 1 q < 1, if K is chosen large enough there exists a constants c depending on α, p, q such that for any ϕ Borel bounded and h :

H → D((-A) -α/2 ) Borel such that H |h(x)| q -α ν(dx) < ∞ we have H D x P t ϕ(x) • h(x)ν(dx) ≤ ce ct (1+t -1+α 2 ) ϕ L p (H,ν) H |h(x)| q -α ν(dx) 1/q . ( 19
)
Proof. We first prove a similar estimate for S t . Using Lemma 6 we have by Hölder inequality

H D x S t ϕ(x) • h(x) ν(dx) ≤ c e ct (1 + t -1+α 2 ) H (1 + |x| 3 L 6 ) [E (ϕ p (X(t, x))] 1/p |h(x)| -α ν(dx) ≤ c e ct (1 + t -1+α 2 ) H (1 + |x| 3 L 6 ) r ν(dx) 1/r × H E (ϕ p (X(t, x)) ν(dx) 1/p H |h(x)| q -α ν(dx) 1/q
. with 1 p + 1 q + 1 r = 1. Thus by Proposition 2.3 in [START_REF] Da Prato | m-dissipativity of Kolmogorov operators corresponding to Burgers equations with space-time white noise[END_REF] and the invariance of ν:

H D x S t ϕ(x) • h(x) ν(dx) ≤ ce ct (1 + t -1+α 2 ) ϕ L p (H,ν) H |h(x)| q -α ν(dx) 1/q .
We then proceed as in [START_REF] Da Prato | m-dissipativity of Kolmogorov operators corresponding to Burgers equations with space-time white noise[END_REF] to get a similar estimate on P t . We write

P t ϕ(x) = S t ϕ(x) + K t 0 S t-s (|x| 4 L 4 P s ϕ)ds.
It follows that, using the estimate above with p > p > 1 such that 1 p + 1 q < 1:

H D x P t ϕ(x) • h(x)ν(dx) ≤ ce ct (1 + t -1+α 2 ) ϕ L p (H,ν) H |h(x)| q -α ν(dx) 1/q +K t 0 ce c(t-s) (1 + (t -s) -1+α 2 ) H |x| 4 L 4 |P s ϕ(x)| pdν(dx) 1/p × H |h(x)| q -α ν(dx) 1/q ds.
The result follows by Hölder inequality and the invariance of ν.

Theorem 1 follows directly from the following result thanks to the invariance of ν and taking for instance t = 1. Proposition 8. For all p > 1, δ > 0, there exists a constant such that for ϕ Borel bounded, and all h ∈ H 1+δ (0, 1), we have

H P t (D x ϕ • h)(x) ν(dx) ≤ ce ct (1 + t -1/2 ) ϕ L p (H,ν) |h| 1+δ (20)
Proof. By Poincaré inequality, it is no loss of generality to assume δ < min{2(1 -

1 p ), 1 2 }.
We start by integrating (7) on H:

H P t (D x ϕ • h)(x)ν(dx) = H (D x P t ϕ(x) • h) ν(dx) - t 0 H P t-s [( Ah + b (x)h, D x P s ϕ(x) )]ds ν(dx). (21)
Then by Lemma 7 we deduce

H P t (D x ϕ • h)(x) ν(dx) ≤ ce ct (1 + t -1 2 ) ϕ L p (H,ν) |h| + H t 0 P t-s [(Ah + b (•) • h, D x P s ϕ)]ds ν(dx) .
By the invariance of ν:

H t 0 P t-s [(Ah + b (•) • h, D x P s ϕ)]ds ν(dx) = H t 0 (Ah + b (•) • h, D x P s ϕ)ds ν(dx).
Therefore, by Lemma 7 with α = 1 -δ and q = 2 δ : Integrating with respect to ν and using Proposition 2.3 in [START_REF] Da Prato | m-dissipativity of Kolmogorov operators corresponding to Burgers equations with space-time white noise[END_REF] we deduce:

H |x| δ/2 δ ν(dx) ≤ c δ
Then (20) follows.

H t 0 PP|Yδ

 0 t-s [(Ah + b (•) • h, D x P s ϕ)]ds ν(dx)≤ t 0 ce c(t-s) (1 + s -1+ δ 2 ) ϕ L p (H,ν) H |Ah + b (•) • h| δ/2 -1+δ ν(dx) δ/2 . Note that |b (x) • h| -1+δ = |∂ ξ (xh)| -1+δ ≤ c |xh| δThen, we have:|xh| ≤ c|x| |h| 1 by the embedding H 1 ⊂ L ∞ and |xh| 1 ≤ c|x| 1 |h| 1 ,since H 1 is an algebra. We deduce by interpolation|xh| δ ≤ c|x| δ |h| 1 . t-s [(Ah + b (•) • h, D x P s ϕ)]ds ν(dx) ≤ c δ e ct ϕ L p (H,ν)We need to estimateH |x| δ/2 δ ν(dx). We use the notation of [DaDe07, Proposition 2.2] |X(s, x)| δ ≤ |Y (s, x)| δ + |z α (s)| δ ≤ |Y (s, x)| 1-δ |Y (s, x)| δ 1 + |z α (s)| δ . Using computation in [DaDe07, Proposition 2.2], we obtain sup t∈[0,1] |Y (t, x)| 2 + 0 |Y (s, x)| 2 1 ds ≤ c(|x| 2 + κ)where κ is a random variable with all moments finite. It follows by (11): (s, x)| 2(1-δ)/δ |Y (s, x)| 2 1 ds ≤ c(|x| 2 + 1) 1/δGenelarizing slightly Proposition 2.1 in[START_REF] Da Prato | m-dissipativity of Kolmogorov operators corresponding to Burgers equations with space-time white noise[END_REF], we have:E (|z α (t)| p δ ) ≤ c δ,pfor t ∈ [0, 1], δ < 1/2, α ≥ 1, p ≥ 1. We deduce: ds ≤ c(|x| 2 + 1) 1/δ .


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