2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/172833By reversing the time variable we derive a stochastic representation for backward incompressible Navier-Stokes equations in terms of stochastic Lagrangian paths, which is similar to Constantin and Iyer's forward formulations in \cite{Co-Iy}. Using this representation, a self-contained proof of local existence of solutions in Sobolev spaces are provided for incompressible Navier-Stokes equations in the whole space. In two dimensions or large viscosity, an alternative proof to the global existence is also given. Moreover, a large deviation estimate for stochastic particle trajectories is presented when the viscosity tends to zero.17 Pages. Corrected some misprints and added a proof to the global existence for large viscosity by probability approachProbabilityAnalysis of PDEs60H15, 37A25text