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.
"A Probabilistic Approach to the Two-Dimensional Navier-Stokes Equations." Ann. Probab. 27 (4) 1750 - 1780, October 1999. https://doi.org/10.1214/aop/1022874814