F. Kh, B. B. Abdullaev, V. V. Baizakov, and . Konotop, Dynamics of a Bose-Einstein Condensate in Optical Trap, Nonlinearity and Disorder: Theory and Applications (Fatkhulla Abdullaev, NATO Science Series, pp.69-78, 2001.

F. Kh, J. C. Abdullaev, G. Bronski, and . Papanicolaou, Soliton perturbations and the random Kepler problem, Phys. D, vol.135, pp.3-4, 2000.

J. R. Abo-shaeer, C. Raman, J. M. Vogels, and W. Ketterle, Observation of Vortex Lattices in Bose-Einstein Condensates, Science, vol.292, issue.5516, pp.476-479, 2001.
DOI : 10.1126/science.1060182

A. Agarwal, S. De-marco, E. Gobet, and G. Liu, Rare event simulation related to financial risks: efficient estimation and sensitivity analysis, preprint, 2015.

J. O. Andersen, Theory of the weakly interacting Bose gas, Reviews of Modern Physics, vol.31, issue.2, pp.599-639, 2004.
DOI : 10.1007/s100510050905

M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Observation of Bose-Einstein Condensation in a Dilute Atomic Vapor, Science, vol.269, issue.5221, pp.198-201, 1995.
DOI : 10.1126/science.269.5221.198

X. Antoine, W. Bao, and C. Besse, Computational methods for the dynamics of the nonlinear Schr??dinger/Gross???Pitaevskii equations, Computer Physics Communications, vol.184, issue.12, pp.2621-2633, 2013.
DOI : 10.1016/j.cpc.2013.07.012

X. Antoine and R. Duboscq, GPELab, a Matlab toolbox to solve Gross???Pitaevskii equations II: Dynamics and stochastic simulations, Computer Physics Communications, vol.193, pp.95-117, 2015.
DOI : 10.1016/j.cpc.2015.03.012

URL : https://hal.archives-ouvertes.fr/hal-01095568

M. Banterle, C. Grazian, A. Lee, and C. P. Robert, Accelerating Metropolis-Hastings algorithms by delayed acceptance, 2015.

W. Bao and Y. Cai, Uniform Error Estimates of Finite Difference Methods for the Nonlinear Schr??dinger Equation with Wave Operator, SIAM Journal on Numerical Analysis, vol.50, issue.2, pp.492-521, 2012.
DOI : 10.1137/110830800

W. Bao and Q. Du, Computing the Ground State Solution of Bose--Einstein Condensates by a Normalized Gradient Flow, SIAM Journal on Scientific Computing, vol.25, issue.5, pp.1674-1697, 2004.
DOI : 10.1137/S1064827503422956

W. Bao, D. Jaksch, and P. A. Markowich, Numerical solution of the Gross???Pitaevskii equation for Bose???Einstein condensation, Journal of Computational Physics, vol.187, issue.1, pp.318-342, 2003.
DOI : 10.1016/S0021-9991(03)00102-5

W. Bao, J. Li, and . Shen, A Generalized-Laguerre???Fourier???Hermite Pseudospectral Method for Computing the Dynamics of Rotating Bose???Einstein Condensates, SIAM Journal on Scientific Computing, vol.31, issue.5, pp.3685-3711, 2009.
DOI : 10.1137/080739811

W. Bao and J. Shen, A Fourth-Order Time-Splitting Laguerre--Hermite Pseudospectral Method for Bose--Einstein Condensates, SIAM Journal on Scientific Computing, vol.26, issue.6, pp.2010-2028, 2005.
DOI : 10.1137/030601211

URL : http://arxiv.org/pdf/cond-mat/0310378

W. Bao and Y. Zhang, DYNAMICS OF THE GROUND STATE AND CENTRAL VORTEX STATES IN BOSE???EINSTEIN CONDENSATION, Mathematical Models and Methods in Applied Sciences, vol.40, issue.12, pp.1863-1896, 2005.
DOI : 10.1016/0375-9601(90)90092-3

C. Bardos, F. Golse, and N. J. Mauser, Weak coupling limit of the N -particle Schrödinger equation Cathleen Morawetz: a great mathematician, Methods Appl. Anal, vol.7, issue.2, pp.275-293, 2000.

M. D. Barrett, J. A. Sauer, and M. S. Chapman, All-Optical Formation of an Atomic Bose-Einstein Condensate, Physical Review Letters, vol.85, issue.1, p.10404, 2001.
DOI : 10.1103/PhysRevLett.85.1795

J. Basdevant and J. Dalibard, Mécanique quantique, Cours de l'Ecole Polytechnique, 2002.

R. Belaouar, A. De-bouard, and A. Debussche, Numerical analysis of the nonlinear Schrödinger equation with white noise dispersion, Stochastic Partial Differential Equations, Analysis and Computations, vol.3, issue.1, pp.103-132, 2015.

F. A. Berezin and M. A. Shubin, The Schrödinger Equation, Mathematics and its Applications (Soviet Series), 1991.

C. Besse, G. Dujardin, I. Lacroix, and . Violet, High Order Exponential Integrators for Nonlinear Schr??dinger Equations with Application to Rotating Bose--Einstein Condensates, SIAM Journal on Numerical Analysis, vol.55, issue.3, pp.1387-1411, 2017.
DOI : 10.1137/15M1029047

J. Bierkens, Non-reversible Metropolis-Hastings, Statistics and Computing, vol.240, issue.4???5, pp.1-16, 2015.
DOI : 10.1007/978-3-642-61859-8

URL : https://link.springer.com/content/pdf/10.1007%2Fs11222-015-9598-x.pdf

J. Bierkens, P. Fearnhead, and G. Roberts, The Zig-Zag process and super-efficient sampling for Bayesian analysis of Big Data, 2016.

P. B. Blakie, A. S. Bradley, M. J. Davis, R. J. Ballagh, and C. W. Gardiner, Dynamics and statistical mechanics of ultra-cold Bose gases using c-field techniques, Advances in Physics, vol.146, issue.5, pp.363-455, 2008.
DOI : 10.1137/030601211

S. Bose, P. Gesetz, and . Lichtquantenhypothese, Plancks Gesetz und Lichtquantenhypothese, Zeitschrift f???r Physik, vol.26, issue.1, pp.178-181, 1924.
DOI : 10.1007/BF01327326

N. Bou-rabee and E. Vanden-eijnden, Pathwise accuracy and ergodicity of metropolized integrators for SDEs, Communications on Pure and Applied Mathematics, vol.106, issue.3, pp.655-696, 2010.
DOI : 10.1017/CBO9780511526237

A. Bouchard-côté, S. J. Vollmer, and A. Doucet, The Bouncy Particle Sampler: A Non-Reversible Rejection-Free Markov Chain Monte Carlo Method, Journal of the American Statistical Association, vol.1, issue.1, 2017.
DOI : 10.1214/15-BA982

F. Bouchet and J. Reygner, Generalisation of the Eyring???Kramers Transition Rate Formula to Irreversible Diffusion Processes, Annales Henri Poincar??, vol.36, issue.3, pp.3499-3532, 2016.
DOI : 10.1137/0136043

URL : https://hal.archives-ouvertes.fr/ensl-01174112

A. S. Bradley, C. W. Gardiner, and M. J. Davis, Bose-Einstein condensation from a rotating thermal cloud: Vortex nucleation and lattice formation, Physical Review A, vol.329, issue.333, p.33616, 2008.
DOI : 10.1103/PhysRevLett.98.110402

URL : http://arxiv.org/pdf/0712.3436

C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet, Evidence of Bose-Einstein Condensation in an Atomic Gas with Attractive Interactions, Physical Review Letters, vol.51, issue.9, pp.75-1687, 1995.
DOI : 10.1103/PhysRevA.51.3896

C. Bréhier, M. Gazeau, L. Goudenège, T. Lelì, and M. Rousset, Unbiasedness of some generalized adaptive multilevel splitting algorithms, The Annals of Applied Probability, vol.26, issue.6, 2015.
DOI : 10.1214/16-AAP1185

V. Bretin, P. Rosenbusch, F. Chevy, G. V. Shlyapnikov, and J. Dalibard, Quadrupole Oscillation of a Single-Vortex Bose-Einstein Condensate: Evidence for Kelvin Modes, Physical Review Letters, vol.13, issue.10, pp.90-100403, 2003.
DOI : 10.1103/PhysRevLett.87.160406

URL : https://hal.archives-ouvertes.fr/hal-00001440

E. Cancès, C. L. Bris, and Y. Maday, Méthodes mathématiques en chimie quantique, Une introduction, Mathématiques & Applications (Berlin) [Mathematics & Applications], 2006.

E. Cancès, F. Legoll, and G. Stoltz, Theoretical and numerical comparison of some sampling methods for molecular dynamics, ESAIM: Mathematical Modelling and Numerical Analysis, vol.88, issue.2, pp.351-389, 2007.
DOI : 10.1103/PhysRevLett.88.100201

O. Cappé, E. Moulines, and T. Rydén, Inference in Hidden Markov Models, 2005.

F. Cérou and A. Guyader, Adaptive Multilevel Splitting for Rare Event Analysis, Stochastic Analysis and Applications, vol.23, issue.2, pp.417-443, 2007.
DOI : 10.1017/CBO9780511802256

F. Cérou, A. Guyader, T. Lelì, and D. Pommier, A multiple replica approach to simulate reactive trajectories, The Journal of Chemical Physics, vol.12, issue.5, p.54108, 2011.
DOI : 10.1137/1035046

S. Chang, S. Gustafson, K. Nakanishi, and T. Tsai, Spectra of Linearized Operators for NLS Solitary Waves, SIAM Journal on Mathematical Analysis, vol.39, issue.4, pp.1070-1111, 2007.
DOI : 10.1137/050648389

C. Chen, J. Hong, and A. Prohl, Convergence of a ?-scheme to solve the stochastic nonlinear Schrödinger equation with Stratonovich noise, Stochastic Partial Differential Equations, Analysis and Computations, pp.1-45, 2015.

F. Chen, L. Lovász, and I. Pak, Lifting Markov chains to speed up mixing, Proceedings of the thirty-first annual ACM symposium on Theory of computing , STOC '99, pp.275-281, 1999.
DOI : 10.1145/301250.301315

URL : http://www.cs.yale.edu/HTML/YALE/CS/HyPlans/lovasz/liftst99.ps

C. Cohen-tannoudji, Nobel Lecture: Manipulating atoms with photons, Reviews of Modern Physics, vol.35, issue.3, pp.707-719, 1998.
DOI : 10.1103/PhysRevA.35.198

C. Cohen-tannoudji, J. Dalibard, and F. Laloë, La condensation de Bose-Einstein dans les gaz, Einstein aujourd'hui, EDP Sciences et CNRS Editions, pp.87-127, 2005.

P. Collet, S. Martínez, and J. San-martín, Quasi-Stationary Distributions, 2012.
DOI : 10.1007/978-3-642-33131-2

URL : https://hal.archives-ouvertes.fr/hal-00666794

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Applied Mathematical Sciences, 1997.
DOI : 10.1007/978-1-4614-4942-3

F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Theory of Bose-Einstein condensation in trapped gases, Reviews of Modern Physics, vol.32, issue.3, pp.71-463, 1999.
DOI : 10.1103/PhysRevA.57.4695

J. Dalibard, Statistique de Bose?Einstein et condensation, Cours ducolì ege de France, 2016.

K. B. Davis, M. O. Mewes, M. R. Andrews, N. J. Van-druten, D. S. Durfee et al., Bose-Einstein condensation in a gas of sodium atoms, Phys. Rev. Lett, pp.75-3969, 1995.

A. , D. Bouard, and A. Debussche, A semi-discrete scheme for the stochastic nonlinear Schrödinger equation, Numerische Mathematik, vol.96, issue.4, pp.733-770, 2004.

A. De-bouard and A. Debussche, Random modulation of solitons for the stochastic Korteweg???de Vries equation, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, vol.24, issue.2, pp.251-278, 2007.
DOI : 10.1016/j.anihpc.2006.03.009

URL : https://hal.archives-ouvertes.fr/hal-00383280

A. De-bouard and R. Fukuizumi, Stochastic fluctuations in the Gross???Pitaevskii equation, Nonlinearity, vol.20, issue.12, p.2823, 2007.
DOI : 10.1088/0951-7715/20/12/005

A. De-bouard, R. Fukuizumi, and R. Poncet, Vortex solutions in Bose-Einstein condensation under a trapping potential varying randomly in time, Discrete and Continuous Dynamical Systems -Series, pp.2793-2817, 2015.

A. De-bouard and E. Gautier, Exit problems related to the persistence of solitons for the Korteweg-de Vries equation with small noise, Discrete Contin, Dyn. Syst, vol.26, issue.3, pp.857-871, 2010.

A. Del-campo, A. Retzker, and M. B. Plenio, The inhomogeneous Kibble???Zurek mechanism: vortex nucleation during Bose???Einstein condensation, New Journal of Physics, vol.13, issue.8, p.83022, 2011.
DOI : 10.1088/1367-2630/13/8/083022

M. Delfour, M. Fortin, and G. Payr, Finite-difference solutions of a non-linear Schr??dinger equation, Journal of Computational Physics, vol.44, issue.2, pp.277-288, 1981.
DOI : 10.1016/0021-9991(81)90052-8

C. Dellago and P. G. Bolhuis, Transition Path Sampling and Other Advanced Simulation Techniques for Rare Events, pp.167-233, 2009.
DOI : 10.1007/978-3-540-87706-6_3

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.621.4154

J. Denschlag, J. E. Simsarian, D. L. Feder, C. W. Clark, L. A. Collins et al., Generating Solitons by Phase Engineering of a Bose-Einstein Condensate, Science, vol.287, issue.5450, pp.97-101, 2000.
DOI : 10.1126/science.287.5450.97

G. , D. Gesu, T. Lelì-evre, D. L. Peutrec, and B. Nectoux, Jump Markov models and transition state theory: the quasi-stationary distribution approach, Faraday Discuss, vol.195, pp.469-495, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01314233

L. and D. Menza, Numerical computation of solitons for optical systems, ESAIM: Mathematical Modelling and Numerical Analysis, vol.87, issue.1, pp.173-208, 2009.
DOI : 10.1007/BF01208265

P. Diaconis, S. Holmes, and R. M. Neal, Analysis of a nonreversible Markov chain sampler, The Annals of Applied Probability, vol.10, issue.3, pp.726-752, 2000.
DOI : 10.1214/aoap/1019487508

D. Down, S. P. Meyn, and R. L. Tweedie, Exponential and Uniform Ergodicity of Markov Processes, The Annals of Probability, vol.23, issue.4, pp.1671-1691, 1995.
DOI : 10.1214/aop/1176987798

URL : http://doi.org/10.1214/aop/1176987798

J. R. Driscoll, D. M. Healy, J. , and D. N. Rockmore, Fast Discrete Polynomial Transforms with Applications to Data Analysis for Distance Transitive Graphs, SIAM Journal on Computing, vol.26, issue.4, pp.1066-1099, 1997.
DOI : 10.1137/S0097539792240121

URL : http://pascal.dartmouth.edu/~rockmore/24012.ps.gz

S. Duane, A. D. Kennedy, B. J. Pendleton, R. Duncan, and M. Carlo, Hybrid Monte Carlo, Physics Letters B, vol.195, issue.2, pp.216-222, 1987.
DOI : 10.1016/0370-2693(87)91197-X

R. Duboscq and R. Marty, Analysis of a splitting scheme for a class of random nonlinear partial differential equations, ESAIM: Probability and Statistics, vol.50, issue.40, pp.572-589, 2016.
DOI : 10.1002/9780470316658

A. B. Duncan, T. Lelì, and G. A. Pavliotis, Variance Reduction Using Nonreversible Langevin Samplers, Journal of Statistical Physics, vol.155, issue.3, pp.457-491, 2016.
DOI : 10.1007/s10955-014-0963-5

URL : https://hal.archives-ouvertes.fr/hal-01164466

M. Eichhorn, M. Mudrich, and M. Weidemüller, Optical dipole trap inside a laser resonator, Optics Letters, vol.29, issue.10, pp.1147-1149, 2004.
DOI : 10.1364/OL.29.001147

L. Erd?-os, B. Schlein, and H. Yau, Derivation of the cubic non-linear Schr??dinger equation from quantum dynamics of many-body systems, Inventiones mathematicae, vol.116, issue.4, pp.515-614, 2007.
DOI : 10.4310/ATMP.2001.v5.n6.a6

M. Fathi and G. Stoltz, Improving dynamical properties of metropolized discretizations of overdamped Langevin dynamics, Numerische Mathematik, vol.69, issue.10, pp.545-602, 2017.
DOI : 10.1063/1.436415

URL : https://hal.archives-ouvertes.fr/hal-01153573

M. Fauquembergue, Réalisation d'un dispositif de condensation de Bose?Einstein et de transport d'un echantillon cohérent d'atomes, 2004.

G. Fibich and N. Gavish, Theory of singular vortex solutions of the nonlinear Schrödinger equation, Phys, pp.2696-2730, 2008.

M. I. Freidlin and A. D. , Random Perturbations of Dynamical Systems, Fundamental Principles of Mathematical Sciences], vol.260

J. Fröhlich, S. Gustafson, B. L. Jonsson, and I. M. Sigal, Solitary Wave Dynamics in an External Potential, Communications in Mathematical Physics, vol.16, issue.3, pp.613-642, 2004.
DOI : 10.1002/cpa.3160390103

C. W. Gardiner, J. R. Anglin, and T. I. Fudge, The stochastic Gross-Pitaevskii equation, Journal of Physics B: Atomic, Molecular and Optical Physics, vol.35, issue.6, p.1555, 2002.
DOI : 10.1088/0953-4075/35/6/310

C. W. Gardiner and M. J. Davis, The stochastic Gross???Pitaevskii equation: II, Journal of Physics B: Atomic, Molecular and Optical Physics, vol.36, issue.23, p.4731, 2003.
DOI : 10.1088/0953-4075/36/23/010

J. Garnier, F. Kh, B. B. Abdullaev, and . Baizakov, Collapse of a Bose-Einstein condensate induced by fluctuations of the laser intensity, Physical Review A, vol.303, issue.5, p.53607, 2004.
DOI : 10.1103/PhysRevLett.77.2921

M. Gazeau, Analyse de modèles mathématiques pour la propagation de lalumì ere dans les fibres optiques en présence de biréfringence aléatoire Probability and pathwise order of convergence of a semidiscrete scheme for the stochastic Manakov equation, SIAM J. Numer. Anal, vol.84, issue.1, pp.52-533, 2012.

M. E. Gehm, K. M. O-'hara, T. A. Savard, and J. E. Thomas, Dynamics of noise-induced heating in atom traps, Physical Review A, vol.2, issue.5, pp.3914-3921, 1998.
DOI : 10.1364/JOSAB.2.001707

J. Ginibre and G. Velo, On the global Cauchy problem for some non linear Schr??dinger equations, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, vol.1, issue.4, pp.309-323, 1984.
DOI : 10.1016/S0294-1449(16)30425-5

E. Gobet and G. Liu, Rare Event Simulation Using Reversible Shaking Transformations, SIAM Journal on Scientific Computing, vol.37, issue.5, pp.2295-2316, 2015.
DOI : 10.1137/14098418X

URL : https://hal.archives-ouvertes.fr/hal-01058748

D. G. Greif, Evaporative cooling and Bose?Einstein condensation of Rb-87 in a moving?coil TOP trap geometry, 2010.

E. P. Gross, Structure of a quantized vortex in boson systems, Il Nuovo Cimento, vol.8, issue.3, pp.454-477, 1955.
DOI : 10.1007/BF02731494

H. Haario, E. Saksman, and J. Tamminen, An Adaptive Metropolis Algorithm, Bernoulli, vol.7, issue.2, pp.223-242, 2001.
DOI : 10.2307/3318737

URL : http://www.geo.fmi.fi/~jtammine/ampure.ps.gz

E. Hairer, C. Lubich, and G. Wanner, Geometric Numerical Integration, 2002.
DOI : 10.4171/owr/2006/14

URL : https://hal.archives-ouvertes.fr/hal-01403326

B. Helffer and F. Nier, Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach: the case with boundary, Mémoires de la Société mathématique de France, vol.1, pp.1-89, 2006.
DOI : 10.24033/msmf.417

URL : https://hal.archives-ouvertes.fr/hal-00002744

K. Hukushima and Y. Sakai, An irreversible Markov-chain Monte Carlo method with skew detailed balance conditions, Journal of Physics: Conference Series, vol.473, issue.1, p.12012, 2013.
DOI : 10.1088/1742-6596/473/1/012012

D. Hutchinson, E. Zaremba, and A. Griffin, Finite Temperature Excitations of a Trapped Bose Gas, Physical Review Letters, vol.78, issue.10, 1997.
DOI : 10.1103/PhysRevLett.78.764

C. Hwang, S. Hwang-ma, and S. Sheu, Accelerating Gaussian Diffusions, The Annals of Applied Probability, vol.3, issue.3, pp.897-913, 1993.
DOI : 10.1214/aoap/1177005371

C. Hwang, R. Normand, and S. Wu, Variance reduction for diffusions, Stochastic Processes and their Applications, pp.3522-3540, 2015.
DOI : 10.1016/j.spa.2015.03.006

URL : http://arxiv.org/pdf/1406.4657

J. Iaia and H. Warchall, Nonradial Solutions of a Semilinear Elliptic Equation in Two Dimensions, Journal of Differential Equations, vol.119, issue.2, pp.533-558, 1995.
DOI : 10.1006/jdeq.1995.1101

B. Jackson, N. P. Proukakis, C. F. Barenghi, and E. Zaremba, Finite-temperature vortex dynamics in Bose-Einstein condensates, Physical Review A, vol.109, issue.5, p.53615, 2009.
DOI : 10.1103/PhysRevLett.91.100402

URL : http://arxiv.org/pdf/0809.4400

B. Jackson and E. Zaremba, -body simulations, Physical Review A, vol.88, issue.3, p.33606, 2002.
DOI : 10.1103/PhysRevLett.88.010402

A. Jentzen and P. Kloeden, Taylor Approximations for Stochastic Partial Differential Equations, 2011.
DOI : 10.1137/1.9781611972016

D. S. Jin, M. R. Matthews, J. R. Ensher, C. E. Wieman, and E. A. Cornell, Temperature-dependent damping and frequency shifts in collective excitations of a, pp.493-496, 2012.

B. L. Jonsson, J. Fröhlich, S. Gustafson, and I. M. Sigal, Long Time Motion of NLS Solitary Waves in a Confining Potential, Annales Henri Poincar??, vol.7, issue.4, pp.621-660, 2006.
DOI : 10.1007/s00023-006-0263-y

A. D. Kennedy and B. Pendleton, Cost of the generalised hybrid Monte Carlo algorithm for free field theory, Nuclear Physics B, vol.607, issue.3, pp.456-510, 2001.
DOI : 10.1016/S0550-3213(01)00129-8

W. Ketterle, D. S. Durfee, and D. M. Stamper-kurn, Making, probing and understanding Bose?Einstein condensates, Proceedings of the international school of physics Bibliography " Enrico Fermi, p.67, 1999.

T. W. Kibble, Topology of cosmic domains and strings, Journal of Physics A: Mathematical and General, vol.9, issue.8, p.1387, 1976.
DOI : 10.1088/0305-4470/9/8/029

URL : http://iopscience.iop.org/article/10.1088/0305-4470/9/8/029/pdf

R. Kollar, Existence and stability of vortex solutions of certain nonlinear Schrödinger equations, p.147

R. Kollár and R. L. Pego, Spectral Stability of Vortices in Two-Dimensional Bose???Einstein Condensates via the Evans Function and Krein Signature, Applied Mathematics Research eXpress, vol.2012, issue.1, pp.1-46, 2012.
DOI : 10.1093/amrx/abr007

H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge Studies in Advanced Mathematics, 1997.

G. Lamporesi, S. Donadello, S. Serafini, F. Dalfovo, and G. Ferrari, Spontaneous creation of Kibble???Zurek solitons in a Bose???Einstein condensate, Nature Physics, vol.13, issue.10, pp.656-660, 2013.
DOI : 10.1103/PhysRevLett.51.1336

G. Leibon, D. N. Rockmore, W. Park, R. T. , G. S. Chirikjian et al., A fast Hermite transform, Theoretical Computer Science, vol.409, issue.2, pp.211-228, 2008.
DOI : 10.1016/j.tcs.2008.09.010

URL : https://doi.org/10.1016/j.tcs.2008.09.010

T. Lelì-evre, F. Nier, and G. A. Pavliotis, Optimal Non-reversible Linear Drift for the Convergence to Equilibrium of a Diffusion, Journal of Statistical Physics, vol.18, issue.3, pp.237-274, 2013.
DOI : 10.1137/0118063

T. Lelì-evre, M. Rousset, and G. Stoltz, Free Energy Computations: a Mathematical Perspective, 2010.

T. Lelì-evre and G. Stoltz, Partial differential equations and stochastic methods in molecular dynamics, Acta Numerica, vol.24, pp.681-880, 2016.
DOI : 10.1063/1.2996509

M. Lewin, P. T. Nam, and N. Rougerie, Derivation of Hartree??s theory for generic mean-field Bose systems, Advances in Mathematics, vol.254, pp.570-621, 2014.
DOI : 10.1016/j.aim.2013.12.010

E. H. Lieb, The Mathematics of the Bose Gas and Its Condensation, Oberwolfach Seminars, 2005.

J. Liu, Order of convergence of splitting schemes for both deterministic and stochastic nonlinear Schrödinger equations, SIAM J. Numer. Anal, vol.51, issue.4, 1911.

K. W. Madison, F. Chevy, W. Wohlleben, and J. Dalibard, Vortex Formation in a Stirred Bose-Einstein Condensate, Physical Review Letters, vol.58, issue.5, pp.806-809, 2000.
DOI : 10.1103/PhysRevA.58.3168

M. Maeda, Symmetry breaking and stability of standing waves of nonlinear Schrödinger equations, 2008.

G. Mastroianni and G. Monegato, Error Estimates for Gauss-Laguerre and Gauss-Hermite Quadrature Formulas, pp.421-434, 1994.
DOI : 10.1007/978-1-4684-7415-2_28

J. C. Mattingly, A. M. Stuart, and D. J. Higham, Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise, Stochastic Processes and their Applications, pp.185-232, 2002.
DOI : 10.1016/S0304-4149(02)00150-3

URL : https://doi.org/10.1016/s0304-4149(02)00150-3

H. J. Metcalf and P. Van-der-straten, Laser Cooling and Trapping, Graduate Texts in Contemporary Physics, 2001.

S. P. Meyn and R. L. Tweedie, Stability of Markovian processes II: continuous-time processes and sampled chains, Advances in Applied Probability, vol.VII, issue.03, pp.487-517, 1993.
DOI : 10.2307/1425932

T. Mizumachi, Vortex solitons for 2D focusing nonlinear Schrödinger equation, Differential Integral Equations, vol.18, issue.4, pp.431-450, 2005.

P. Monmarché, Hypocoercive relaxation to equilibrium for some kinetic models via a third order differential inequality, arXiv preprint arXiv:1306, p.4548, 2013.

C. A. Newell, Inelastic collisions in cold dipolar gases, 2010.

Y. Oh, Cauchy problem and Ehrenfest's law of nonlinear Schr??dinger equations with potentials, Journal of Differential Equations, vol.81, issue.2, pp.255-274, 1989.
DOI : 10.1016/0022-0396(89)90123-X

M. Ottobre, N. S. Pillai, and K. Spiliopoulos, Optimal scaling of the MALA algorithm with irreversible proposals for Gaussian targets, 2017.

S. P. Cockburn, Bose gases in and out of equilibrium within the stochastic Gross- Pitaevskii Equation, 2010.

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, vol.44, 1983.
DOI : 10.1007/978-1-4612-5561-1

R. L. Pego and H. A. , Spectrally Stable Encapsulated Vortices for Nonlinear Schr??dinger Equations, Journal of Nonlinear Science, vol.12, issue.4, pp.347-394, 2002.
DOI : 10.1007/s00332-002-0475-3

J. Pérez-ríos and A. S. Sanz, How does a magnetic trap work?, American Journal of Physics, vol.81, issue.11, pp.836-843, 2013.
DOI : 10.1119/1.4819167

E. A. Peters and G. De-with, Rejection-free Monte Carlo sampling for general potentials, Physical Review E, vol.96, issue.2, p.26703, 2012.
DOI : 10.1002/jcc.21915

L. P. Pitaevskii, Vortex lines in an imperfect Bose gas, Sov. Phys.?JETP, vol.13, p.451, 1961.

R. Poncet, Generalized and hybrid Metropolis-Hastings overdamped Langevin algorithms , working paper or preprint, 2017.

D. Potts, G. Steidl, and M. Tasche, Fast algorithms for discrete polynomial transforms, Mathematics of Computation, vol.67, issue.224, pp.1577-1590, 1998.
DOI : 10.1090/S0025-5718-98-00975-2

URL : http://www.ams.org/mcom/1998-67-224/S0025-5718-98-00975-2/S0025-5718-98-00975-2.pdf

M. Qin and W. Zhu, Volume-Preserving Schemes and Numerical Experiments, Computers & Mathematics with Applications, vol.26, issue.4, pp.33-42, 1993.
DOI : 10.1016/0898-1221(93)90032-Q

URL : https://doi.org/10.1016/0898-1221(93)90032-q

M. Quiroga-teixeiro and H. Michinel, Stable azimuthal stationary state in quintic nonlinear optical media, Journal of the Optical Society of America B, vol.14, issue.8, 1997.
DOI : 10.1364/JOSAB.14.002004

R. Duboscq, Analyse et simulation d'equations de Schrödinger déterministes et stochastiques. Applications aux condensats de Bose-Einstein en rotation, 2013.

M. Reed and B. Simon, Fourier Analysis, Self-Adjointness, 1975.

L. Rey-bellet and K. Spiliopoulos, Irreversible Langevin samplers and variance reduction: a large deviations approach [151] , Variance reduction for irreversible Langevin samplers and diffusion on graphs, Nonlinearity Electron. Commun. Probab, vol.28, issue.16, p.pp, 2015.
DOI : 10.1088/0951-7715/28/7/2081

URL : https://doi.org/10.1088/0951-7715/28/7/2081

G. O. Roberts and R. L. Tweedie, Exponential Convergence of Langevin Distributions and Their Discrete Approximations, Bernoulli, vol.2, issue.4, pp.341-363, 1996.
DOI : 10.2307/3318418

S. J. Rooney, P. B. Blakie, and A. S. Bradley, Numerical method for the stochastic projected Gross-Pitaevskii equation, Physical Review E, vol.89, issue.1, p.13302, 2014.
DOI : 10.1006/jcph.1996.5638

P. J. Rossky, J. D. Doll, and H. L. Friedman, Brownian dynamics as smart Monte Carlo simulation, The Journal of Chemical Physics, vol.69, issue.10, pp.4628-4633, 1978.
DOI : 10.1063/1.435856

T. A. Savard, K. M. , and J. E. Thomas, Laser-noise-induced heating in far-off resonance optical traps, Physical Review A, vol.21, issue.2, p.1095, 1997.
DOI : 10.1103/PhysRevA.21.1606

R. Seiringer, Gross-Pitaevskii theory of the rotating Bose gas, Communications in Mathematical Physics, vol.229, issue.3, pp.491-509, 2002.
DOI : 10.1007/s00220-002-0695-2

J. Skilling, Nested sampling for general Bayesian computation, Bayesian Analysis, vol.1, issue.4, pp.833-859, 2006.
DOI : 10.1214/06-BA127

URL : http://doi.org/10.1214/06-ba127

H. T. Stoof and M. J. Bijlsma, Dynamics of fluctuating Bose?Einstein condensates, Journal of Low Temperature Physics, vol.124, issue.3/4, pp.431-442, 2001.
DOI : 10.1023/A:1017519118408

S. K. Suslov, Dynamical invariants for variable quadratic Hamiltonians, Physica Scripta, vol.81, issue.5, p.55006, 2010.
DOI : 10.1088/0031-8949/81/05/055006

URL : http://arxiv.org/pdf/1002.0144.pdf

V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, 1984.
DOI : 10.1007/978-3-662-03359-3

L. Tierney and A. Mira, Some adaptive Monte Carlo methods for Bayesian inference, Statistics in Medicine, vol.43, issue.17-18, pp.17-18, 1999.
DOI : 10.2307/2986121

URL : http://aim.unipv.it/~anto/papers/adaptive.ps

E. C. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations. Part I, Second Edition, 1962.

J. Villen-altamirano and M. Villen-altamirano, Restart: a method for accelerating rare event simulations, Queueing, Performance and Control, pp.71-76, 1991.

M. Vucelja, Lifting???A nonreversible Markov chain Monte Carlo algorithm, American Journal of Physics, vol.84, issue.12, pp.958-968, 2016.
DOI : 10.1119/1.4961596

URL : http://arxiv.org/pdf/1412.8762

C. N. Weiler, Spontaneous formation of quantized vortices in Bose-Einstein condensates, 2008.

C. N. Weiler, T. W. Neely, D. R. Scherer, A. S. Bradley, M. J. Davis et al., Spontaneous vortices in the formation of Bose???Einstein condensates, Nature, vol.86, issue.7215, pp.948-951, 2008.
DOI : 10.1038/nature07334

M. I. Weinstein, Modulational Stability of Ground States of Nonlinear Schr??dinger Equations, SIAM Journal on Mathematical Analysis, vol.16, issue.3, pp.472-491, 1985.
DOI : 10.1137/0516034

S. Wu, C. Hwang, and M. T. Chu, Attaining the Optimal Gaussian Diffusion Acceleration, Journal of Statistical Physics, vol.2, issue.3, pp.571-590, 2014.
DOI : 10.1137/1.9780898719574

URL : http://www4.ncsu.edu/~mtchu/Research/Papers/Gauss_Acceleration05.pdf

X. Yao, H. Chen, Y. Wu, X. Liu, X. Wang et al., Observation of Coupled Vortex Lattices in a Mass-Imbalance Bose and Fermi Superfluid Mixture, Physical Review Letters, vol.5, issue.14, p.145301, 2016.
DOI : 10.1007/BF02780991

W. H. Zurek, Cosmological experiments in superfluid helium?, Nature, vol.177, issue.6037, pp.505-508, 1985.
DOI : 10.1016/0003-4916(68)90214-5

M. W. Zwierlein, J. R. Abo-shaeer, A. Schirotzek, C. H. Schunck, and W. Ketterle, Vortices and superfluidity in a strongly interacting Fermi gas, Nature, vol.88, issue.7045, pp.1047-1051, 2005.
DOI : 10.1103/PhysRevLett.88.070409

URL : http://arxiv.org/pdf/cond-mat/0505635