# 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).

### Scientific interests

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

### Publications

- 94. "Stochastic Approaches to Deterministic Fluid Dynamics: A
Selective Review", Water 2020,
**12**(3), 864. - 93. (with A. Symeonides) "On a non-periodic modified Euler equation: well posedness and quasi-invariant measures", accepted in Potential Analysis.
- 92. (with A. Symeonides) "Invariant and quasi-invariant measures
for equations in Hydrodynamics", in
*Topics in Applied Analysis and Optimisation*, CIM Series in Math. Sc., Springer, M. Hintermuller and J.F. Rodrigues eds. (2019), pp. 101-120. - 91. "Navier-Stokes and
stochastic Navier-Stokes equations via Lagrange multipliers", J.
Geom. Mech.,
**11**(4) (2019), pp 553-560. - 90. Probabilidades e Hidrodinâmica, Bul. SPE (2018) (in portuguese).
- 89. (with M. Arnaudon, C. Léonard and J.C. Zambrini) "An entropic interpolation problem for incompressible viscid fluids", accepted in Ann. I. H. P.
- 88. (with A. Symeonides) "Invariant measures for the non-periodic
two-dimensional Euler equation"
*Bull. Sci. Math.*148 (2018) pp 33–52. - 87. (with X. Chen and T. Ratiu) "Constrained and stochastic variational principles for dissipative equations with advected quantities", arxiv:1506.05024.
- 86. (with R. Lassalle) "Weak calculus of variations for functionals
of laws of semi-martingales",
*Stoch. Proc. and their Applic.***129**(10) (2019) pp. 3585-3618. - 85. (with X. Chen and Zh. Qian) "Navier-Stokes equation and forward-backward stochastic differential system in the Besov spaces", arXiv:1305.0647.
- 84. (with D.D. Holm and T. Ratiu) "Momentum Maps and Stochastic
Clebsch Action Principles",
*Comm. Math. Phys.***357**(2) (2018) pp. 873-912. - 83. (with M. Guerra and R. Sousa) "Barrier Option Pricing under the
2-Hypergeometric Stochastic Volatility Model",
*J. of Comp. and Appl. Maths.*,**328**(2018), 197-213. - 82. (with M. Arnaudon and S. Fang) "Generalized stochastic
Lagrangian paths for the Navier-Stokes equation",
*Ann. Sc. Norm. Super. Pisa*, Cl. Sci. (5) 18, no 3 (2018) pp 1033-1060. - 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
*H*^{2}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
R
^{d}using normal charts of a Riemannian manifold",*Stoch. Processes Appl.*nº 116 (2006), pp. 1088-1095. - 49. (with X. Zhang) "
*L*gradient estimates of symmetric Markov semigroups for 1<^{p}*p*≤2",*Acta Math. Sinica V.***22**, nº 1 (2006), pp. 101-104. - 48. (with F. Cipriano) "Flows associated with irregular
R
^{d}–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
*L*^{2}-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
*L*^{2}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
R
^{d}de la classe de Sobolev W^{d,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 S. Albeverio, D. Holm)
*Stochastic Geometric Mechanics*, Springer Proceedings in Mathematics & Statistics (2017). - (with G. Ben Arous, Y. Le Jan, J. C. Zambrini)
*Stochastic Analysis in Mathematical Physics*, World Scientific (2008). - (with H. Ouerdiane, N. Obata)
*Mathematical Analysis of Random Phenomena*, World Scientific (2007). - (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).