# Ana Bela Cruzeiro

Full professor at the Mathematics Department of Instituto Superior Técnico.

Corresponding member of the Academy of Sciences of Lisbon.

Member of the FCT Scientific Council for Exact Sciences and Engineering (2017-).

Member of the General Council of the University of Lisbon (2021-).

Member of the Scientific Council of IST (2021-).

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

- 96. (with J.C. Zambrini) "Stochastic geodesics", arxiv:2007.05291.
- 95. (with C. Oliveira and J.C. Zambrini) "Time-symmetric optimal stochastic control problems in space-time domains", arxiv:2007.02110.
- 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",
*Potential Analysis*54, 4 (2021), pp 607-626. - 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", Ann. Inst. H. Poincaré
Probab. Statist.,
**56**, 3 (2020), pp 2211-2235. - 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).