The Annals of Probability

A Probabilistic Approach to the Two-Dimensional Navier-Stokes Equations

Barbara Busnello

Full-text: Open access


We turn the Navier-Stokes equations for a 2-dimensional viscous incompressible fluid into a system of functional integrals in the trajectory space of a suitable diffusion process. Using probabilistic techniques as Girsanov’s transformation and Bismut-Elworthy formula, we prove the existence of a unique global solution of this system in a constructive way.

Article information

Ann. Probab., Volume 27, Number 4 (1999), 1750-1780.

First available in Project Euclid: 31 May 2002

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H10: Stochastic ordinary differential equations [See also 34F05]
Secondary: 76D05: Navier-Stokes equations [See also 35Q30] 60J60: Diffusion processes [See also 58J65] 35C15 35Q30: Navier-Stokes equations [See also 76D05, 76D07, 76N10]

Navier-Stokes equations Functional integrals Stochastic differential equations Bismut-Elworthy formula Girsanov’s transformation


Busnello, Barbara. A Probabilistic Approach to the Two-Dimensional Navier-Stokes Equations. Ann. Probab. 27 (1999), no. 4, 1750--1780. doi:10.1214/aop/1022874814.

Export citation


  • 2 BENEDETTO, D., MARCHIORO, C. and PULVIRENTI, M. 1993. On the Euler flow in. Arch. Rational Mech. Anal. 123 377 386.
  • 3 CHORIN, A. J. and MARSDEM, J. E. 1990. A Mathematical Introduction to Fluid Mechanics, 2nd ed. Springer, New York.
  • 4 ELWORTHY, K. D. and LI, X. M. 1994. Formulae for the derivatives of heat semigroups. J. Funct. Anal. 125 252 286.
  • 5 FREIDLIN, M. I. 1985. Integration and Partial Differential Equations. Princeton Univ. Press.
  • 6 GILBARG, D. and TRUDINGER, N. S. 1983. Elliptic Partial Differential Equations of Second Order, 2nd ed. Springer, New York.
  • 7 KUNITA, H. 1984. Stochastic differential equations and stochastic flows of diffeomorphisms. ´ Ecole d'ete de probabilites de Saint Flour XII. Lecture Notes in Math. 143 303. ´ ´ ´ Springer, Berlin.
  • 8 MARCHIORO, C. and PULVIRENTI, M. 1982. Hydrodinamics in two dimensions and vortex theory. Comm. Math. Phys. 84 483 503.
  • 9 REVUZ, D. and YOR, M. 1991. Continuous Martingales and Brownian Motion. Springer, New York.