You are here: Home People Ana Bela Cruzeiro

Ana Bela Cruzeiro
Permanent member
Degree: Agregação
Av. Rovisco Pais
1049-001 Lisboa
Phone: +351 218417063
Fax: +351 218417598
External home page:
https://math.tecnico.ulisbo ...
Document Actions

Ana Bela Cruzeiro

Full professor at the Mathematics Department of Instituto Superior Técnico.

Principal investigator in the project From stochastic geometric mechanics to mass transportation problems (PTDC/MAT-­STA/0975/2014).

Principal investigator in the project "Probabilistic approach to finite and infinite dimensional dynamical systems" (PTDC/MAT/104173/2008).

Curriculum Vitæ

Scientific interests

  • Stochastic analysis
  • Infinite dimensional analysis
  • Malliavin calculus
  • Probabilistic methods in hydrodynamics
  • Mathematical physics
  • Euclidean quantum mechanics


  • 89. (with G. Liu) "A stochastic variational approach to the viscous Camassa-Holm and related equations", arxiv:1508.04064.
  • 88. (with D.D. Holm and T. Ratiu) "Momentum Maps and Stochastic Clebsch Action Principles", arXiv:1604.04554.
  • 87. (with A. Symeonides) "Gibbs-type measures for the non-periodic two-dimensional Euler equation", arXiv:1612.08587.
  • 86. (with R. Sousa and M. Guerra) "Barrier Option Pricing under the 2-Hypergeometric Stochastic Volatility Model", arXiv:1610.03230.
  • 85. (with M. Arnaudon and S. Fang) "Generalized stochastic Lagrangian paths for the Navier-Stokes equation", arXiv:1509.03491.
  • 84. (with X. Chen and T. Ratiu) "Constrained and stochastic variational principles for dissipative equations with advected quantities", arxiv:1506.05024.
  • 83. (with R. Lassalle) "Weak calculus of variations for functionals of laws of semi-martingales", arXiv:1501.05134.
  • 82. (with X. Chen and Zh. Qian) "Navier-Stokes equation and forward-backward stochastic differential system in the Besov spaces", arXiv:1305.0647.
  • 81. (with G. Liu) "A stochastic variational approach to the viscous Burgers equations", Acta Math. Sinica 32, nº 9 (2017), pp. 1027-1034.
  • 80. (with G. Liu) "A stochastic variational approach to the viscous Camassa-Holm and Leray-alpha equations", Stoch. Proc. and their Applic. 127, nº 1 (2017), pp. 1-17.
  • 79. (with R. Lassalle) "Symmetries and martingales in a stochastic model for the Navier-Stokes equation", From Particle Systems to PDEs III, ed. P. Gonçalves, A.J. Soares, Springer proc. In Math. and Stat. (2016).
  • 78. (with P.-A. Vuillermot) "Forward-backward systems of stochastic differential equations generated by Bernstein diffusions", Stochastic Analysis and Applications, 33, Issue 1 (2015), p. 91-109.
  • 77. (with M. Arnaudon) "Stochastic Lagrangian flows and the Navier-Stokes equation", in Stochastic Analysis: a series of lectures, Springer, Progress in Probability 68, ed. R.C. Dalang, M. Dozzi, F. Flandoli, F. Russo (2015).
  • 76. (with Iván Torrecilla) "On a 2D stochastic Euler equation of transport type: existence and energy transfer", Stochastic and Dynamics, 15, 1 (2015), 1450012.
  • 75. (with R. Lassalle) "On the Stochastic Least Action Principle for the Navier-Stokes Equation", Stochastic Analysis and Applications 2014, Springer Proceedings in Mathematics & Statistics, 100, ed. D. Crisan, B. Hambly, T. Zariphopoulou, 2014.
  • 74. (with M. Arnaudon and X. Chen) "Stochastic Euler-Poincaré reduction", J. Math. Physics 55 (2014), 081507.
  • 73. (with Zh. Qian) "Backward Stochastic Differential Equations Associated with the Vorticity Equations", J. Funct. Anal., 267, 3 (2014), p. 660-677.
  • 72. (with André de Oliveira Gomes and Liangquan Zhang) "Asymptotic Properties of Coupled Forward-Backward Stochastic Differential Equations", Stochastic and Dynamics, 14, 3 (2014), 1450004.
  • 71. (with X. Chen) "Stochastic geodesics and stochastic backward equations on Lie groups", Discrete and Cont. Dyn. Systems (2013).
  • 70. (with A. Antoniouk and M. Arnaudon) "Generalized stochastic flows and applications to incompressible viscous fluids", Bulletin des Sciences Mathématiques, 138, 4 (2014), p. 565-584.
  • 69. (with M. Arnaudon) "Lagrangian Navier-Stokes diffusions on manifolds: variational principle and stability", Bulletin des Sciences Mathématiques 136, 8 (2012), p. 857–881.
  • 68. (with M. Arnaudon) "Stochastic Lagrangian flows on some compact manifolds" Stochastics, 84, Issue 2-3 (April 2012), p. 367-381.
  • 67. "Stochastic calculus of variations for the diffeomorphisms group", Bulletin des Sciences Mathématiques 135, 6-7 (2011), p. 557-564.
  • 66. (with M. Arnaudon and N. Galamba) "Lagrangian Navier-Stokes flows: a stochastic model", J. Phys. A: Math. Theor. 44 (2011), 175501.
  • 65. (with E. Shamarova) "On a forward-backward stochastic system associated to the Burgers equation", arXiv:1001.3367, Stochastic Analysis and Financial Applications, A. Kohatsu-Higa, N. Privault, S. J. Sheu ed, Birkhauser p.p. 65 (2011).
  • 64. "Hydrodynamics, probability and the geometry of the diffeomorphism group", arXiv:0810.5507, Seminar on Stochastic Analysis, Random Fields and Applications IV, R. C. Dalang. M. Dozzy, F. Russo ed, Birkhauser P.P. 63 (2011).
  • 63. (with E. Shamarova) "Navier-Stokes equations and forward-backward SDEs on the group of diffeomorphisms of the torus", Stoch. Proc. and their Applic. 119 (2009), pp. 4034-4060.
  • 62. (with P. Malliavin) "Renormalized stochastic calculus of variations for a renormalized infinite-dimensional Brownian motion", Stochastics 81, 3-4 (2009), pp. 385-399.
  • 61. (with P. Malliavin) "Stochastic calculus of variations on complex line bundle and construction of unitarizing measures for the Poincaré disk", J. Funct. Anal. 256 (2009), 2, pp. 385-408.
  • 60. (with P. Malliavin) "Nonexistence of infinitesimally invariant measures on loop groups", J. Funct. Anal. 254 (2008), 2, pp. 1974-1987.
  • 59. (with P. Malliavin) "Nonergodicity of Euler fluid dynamics on tori versus positivity of the Arnold-Ricci tensor", J. Funct. Anal, 254 (2008),7, pp. 1903-1925.
  • 58. (with C. J. S. Alves) "Monte-Carlo simulation of stochastic differential systems — a geometrical approach", Stoch. Proc. and their Applic. 118 (2008), pp. 346-367.
  • 57. (with P. Malliavin), "Stochastic parallel transport on the d-dimensional torus", in Stochastic Analysis in Mathematical Physics, ed. A. B. Cruzeiro, G. Ben Arous, Y. Le Jan, J. C. Zambrini, World Scientific 2008.
  • 56. (with P. Malliavin) "Stochastic evolution of inviscid Burgers fluid", in Probability, Geometry and Integrable Systems, MSRI Publications, vol. 55, 2007.
  • 55. (with F. Cipriano) "Variational principle for diffusions on the diffeomorhism group with the H2 metric", in Mathematical Analysis of Random Phenomena, ed. A. B. Cruzeiro, H. Ouerdiane, N. Obata, World Scientific, 2007.
  • 54. (with F. Cipriano), "Navier-Stokes equation and diffusions on the group of homeomorphisms of the torus", Comm. Math. Phys. 275 (2007), pp. 255-269.
  • 53. (with F. Flandoli and P. Malliavin), "Brownian motion on volume preserving diffeomorphisms group and existence of global solutions of 2D stochastic Euler equation", J. Funct. Anal. 242 (1) (2007), pp. 304-326.
  • 52. "Malliavin Calculus", in Encyclopedia of Mathematical Physics, ed. J. P. Françoise, G. Naber, T. S. Tsun, Oxford: Elsevier (2006), v.3, p 383.
  • 51. (with X. Zhang) "Bismut type formulae for diffusion semigroups on Riemannian manifolds", Pot. Anal. 25 (2006), pp. 121-130.
  • 50. (with P. Malliavin) "Numerical approximation of diffusions in Rd using normal charts of a Riemannian manifold", Stoch. Processes Appl. nº 116 (2006), pp. 1088-1095.
  • 49. (with X. Zhang) "Lp gradient estimates of symmetric Markov semigroups for 1<p≤2", Acta Math. Sinica V. 22, nº 1  (2006), pp. 101-104.
  • 48. (with F. Cipriano) "Flows associated with irregular Rd–vector fields", J. Diff. Equations 219 (2005), pp. 183-201.
  • 47. "On some probabilistic estimates of heat kernels and applications", Publicaciones de la Real Sociedad Matemática Española, vol. 9, Proceedings of the XIII Fall Workshop on Geometry and Physics, Murcia, September 2004 (2005).
  • 46. (with X. Zhang) "Ornstein-Uhlenbeck semigroups on Riemannian path spaces", in Recent developments in Stochastic Analysis and Related Topics, Proceedings of the First Sino-German Conference on Stochastic Analysis (a satellite conference of ICM 2002), eds. S. Albeverio, Z-M Ma & M. Roeckner, World Scientific, 2004.
  • 45. "Non adapted transformations of the Wiener measure", in Stochastic Analysis and Mathematical Physics, eds. R. Rebolledo, J. Rezende, J. C. Zambrini, World Scientific (2004).
  • 44. (with X. Zhang) "A Littlewood-Paley type inequality on the path space", in Seminar on Stochastic Analysis, Random Fields and Applications IV, Birhauser P.P. nº 58, eds. R. Dalang, M. Dozzi, F. Russo (2004).
  • 43. (with P. Malliavin and A. Thalmaier) "Geometrization of Monte-Carlo numerical analysis of an elliptic operator: strong approximation", C. R. Acad. Sci., Ser I, 338 (2004), pp. 481-486.
  • 42. (with K-N. Xiang) "On metrics for tangent processes on the path space", in Stochastic Analysis and Related Topics VIII, Birhäuser P.P. nº 53, eds. U. Capar and A. S. Ustunel (2003).
  • 41. (with X. Zhang) "Finite dimensional approximation of Riemannian path space geometry", J. Funct. Anal. 205 (2003), pp. 206-270.
  • 40. (with P. Malliavin) "Riemannian Geometry on the Path Space", in Stoch. Partial Diff. Equations and Applications, Lecture Notes in Pure and Applied Math., Marcel Decker Inc. vol. 227, eds. G. da Prato, L. Tubaro (2002), pp. 133-165.
  • 39. (with P. Malliavin) "Stochastic calculus of variations and Harnack inequality on Riemannian path spaces", C. R. Acad. Sc. Paris, Ser.I, t. 335 (2002), pp. 817-820.
  • 38. (with P. Malliavin) "A class of anticipative tangent processes on the Wiener space", C. R. Acad. Sc. Paris, Ser.I, t. 333 (2001), pp. 353-358.
  • 37. (with P. Malliavin and S. Taniguchi) "Ground state estimations in Gauge theory", Bull. Sci. Math. 125, 6-7 (2001), pp. 623-640.
  • 36. (with S. Fang) "Weak Levi-Civita connection for the damped metric on the Riemannian path space and vanishing of Ricci tensor in adapted differential geometry", J. Funct. Anal. 185 (2001), pp. 681-698.
  • 35. (with S. Fang) "A Weitzenböck formula for the damped Ornstein-Uhlenbeck operator in adapted differential geometry", C. R. Acad. Sc. Paris, t. 332 (2001), pp. 447-452.
  • 34. "Construction of some processes on the Wiener space associated to second order operators", J. Korean Math. Soc. 38, nº 2 (2001), pp. 307-315.
  • 33. (with W. Liming and J. C. Zambrini) "Bernstein processes associated with a Markov process", in Stochastic Analysis and Mathematical Physics, Trends in Mathematics, Birkhäuser, ed. R. Rebolledo (2000).
  • 32. (with P. Malliavin) "Frame bundle of Riemannian Path Space and Ricci tensor in adapted differential geometry", J. Funct. Anal. 177 (2000), pp. 219-253.
  • 31. (with S. Fang and P. Malliavin) "A Probabilistic Weitzenböck formula on Riemannian Path Space", J. Anal. Math. Jerusalem, 80 (2000), pp. 87-100.
  • 30. (with P. Malliavin) "Riesz Transforms, Commutators and Stochastic Integrals", in Harmonic Analysis and Partial Diff. Equations, Chicago Lectures in Mathematics, The Univ. of Chicago Press, 1999, pp. 151-162.
  • 29. (with F. Cipriano) "Flows associated to tangent processes on the Wiener space", J. Funct. Anal. 166, 2 (1999), pp. 310-331.
  • 28. (with P. Malliavin) "Energy Identities and Estimates for Anticipative Stochastic Integrals on a Riemannian Manifold", Stochastic Analysis and Related Topics VI, Birkhäuser PP 42 (1998), pp. 221-234.
  • 27. (with P. Malliavin) "Non Perturbative Construction of Invariant Measure through Confinement by Curvature", J. Math. Pures Appl. 77 (1998), pp. 527-537.
  • 26. (with Z. Haba) "Invariant Measure for a Wave Equation on a Riemannian Manifold", Stochastic Differential and Difference Equations, Birkhäuser Progress in Systems and Control Theory 23 (1997), pp. 35-42.
  • 25. (with S. Fang) "An L2-Estimate for Riemannian Anticipative Stochastic Integrals", J. Funct. Anal. 143, 2 (1997), pp. 400-414.
  • 24. (with P. Malliavin) "Renormalized Differential Geometry and Path Space: Structural Equation, Curvature" J. Funct. Anal. 139, 1 (1996), pp. 119-181.
  • 23. (with Z. Haba and J. C. Zambrini) "Bernstein diffusions and Euclidean Quantum Field Theory", Proceedings Monte Verità Meeting, eds. E. Bolthausen, F. Russo, Birkhäuser (1995) (P.P. nº 36).
  • 22. (with A. Brandão) "Singular differential operators and associated processes", in Stochastic Processes, Physics and Geometry II, eds. S. Albeverio, U. Cattaneo, D. Merlini, World Sc. (1995).
  • 21. (with S. Fang) "Une inegalité L2 pour des integrales stochastiques anticipatives sur une variété Riemanniene", C. R. Acad. Sc. Paris, t. 321, Ser. I (1995), pp. 1245-1250.
  • 20. (with P. Malliavin) "Courbures de l’espace de Probabilité d’un Brownien Riemannien", C. R. Acad. Sc. Paris, t. 320, ser. I (1995), pp. 603-607.
  • 19. (with J. C. Zambrini) "Malliavin Calculus and Euclidean Quantum Mechanics II - Variational Principle for Infinite Dimensional Processes", J. Funct. Anal. 130, nº 2 (1995), pp. 450-476.
  • 18. (with P. Malliavin) "Repère Mobile et Géometrie Riemannienne sur les espaces des chemins", C. R. Acad. Sc. Paris, t. 319, ser. I (1994), pp. 859-864.
  • 17. (with J. C. Zambrini) "Euclidean Quantum Mechanics: an outline", Proceedings NATO ASI, Stochastic Analysis and Applications in Physics, ed. L. Streit, Kluwer-Acad. Publ. (1994), pp. 59-97.
  • 16. (with J. C. Zambrini) "Ornstein-Uhlenbeck processes as Bernstein diffusions", Proceedings of the Conference on Stochastic Analysis (Barcelona), Birkhäuser, Boston, Inc. (1993) (P.P. nº 32).
  • 15. (with J. C. Zambrini) "Feynman's functional calculus and stochastic calculus of variations", in Stochastic Analysis and Applications, eds. A. B. Cruzeiro and J. C. Zambrini, Birkhäuser (1991) (P.P. nº 26).
  • 14. (with J. C. Zambrini) "Malliavin Calculus and Euclidean Quantum Mechanics, I - Functional Calculus", J. Funct. Anal., vol. 96, nº 1 (1991), pp. 62-95.
  • 13. "Invariant measures in hydrodynamic systems with random perturbations", in Dynamics and Stochastic Processes, eds. R. Lima, L. Streit, R. Vilela Mendes, Springer-Verlag (1990).
  • 12. (with S. Albeverio) "Global flows with invariant (Gibbs) measures for Euler and Navier-Stokes two dimensional fluids" Comm. Math. Phys. 129 (1990), pp. 431-444.
  • 11. "Flows in infinite dimensions and associated transformations of Gaussian measures", in Stochastic Methods in Mathematics and Physics, eds. R. Gielerak and W. Karwowski, World Sc. (1989).
  • 10. "Invariant measures for Euler and Navier-Stokes systems", in Stochastic Analysis, Path Integration and Dynamics, eds. K. D. Elworthy and J. C. Zambrini, Pitman Res. Notes in Math. Ser. 200 (1989).
  • 9. "Solutions et mesures invariantes pour des équations d'évolution stochastiques du type Navier-Stokes", Expo. Math. 7 (1989), pp. 73-82.
  • 8. "Processus sur l’espace de Wiener associés à des opérateurs élliptiques à coefficients dans certains espaces de Sobolev", J. Funct. Anal. 72, nº 2 (1987), pp. 346-367.
  • 7. "Estimations capacitaires sur l’espace de Wiener, I", Bull. Sc. Math., ser. II, 110 (1986), pp. 139-147.
  • 6. "Diffusions sur l'espace de Wiener", C. R. Acad. Sc. Paris, t. 302, ser. I, nº 8 (1986), pp. 295-298.
  • 5. "Unicité de solutions d’équations différentielles sur l’espace de Wiener", J. Funct. Anal. 58 (1985),  pp. 335-347.
  • 4. "Equations différentielles sur l'espace de Wiener et formules de Cameron-Martin non linéaires", J. Funct. Anal. 54 (1983), pp. 206-227.
  • 3. "Equations différentielles ordinaires: non explosion et mesures quasi-invariantes", J. Funct. Anal. 54 (1983), pp. 193-205.
  • 2. "Convergence au bord pour les fonctions harmoniques dans Rd de la classe de Sobolev Wd,1", C. R. Acad. Sc. Paris, t. 294 (1982), pp. 71-74.
  • 1. "Convergence quasi partout dans des domaines paraboliques des fonctions d'intégrale de Dirichlet finie", C. R. Acad. Sc. Paris, t. 294 (1982), pp. 13-16.

Books (as editor)

  • (with J. C. Zambrini)  Stochastic Analysis and Mathematical Physics (Proceedings of the Lisbon Conference 1999), Birkhäuser, Boston, Inc., P.P. 50 (2001).
  • (with J. C. Zambrini) Stochatic Analysis and Applications (Proceedings of the Lisbon Conference 1989), Birkhäuser, Boston, Inc., P.P. 26 (1991).
  • (with H. Ouerdiane, N. Obata) Mathematical Analysis of Random Phenomena, World Scientific (2007).
  • (with G. Ben Arous, Y. Le Jan, J. C. Zambrini), Stochastic Analysis in Mathematical Physics, World Scientific (2008).