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In this paper we establish a mathematical framework which may be used todesign Monte-Carlo simulations for a class of time irreversible dynamicsystems, such as incompressible fluid flows, including turbulent flows inwall-bounded regions, and some other (non-linear) dynamic systems. Pathintegral representations for solutions of forward parabolic equations areobtained, and, in combining with the vorticity transport equations,probabilistic formulations for solutions of the Navier-Stokes equations aretherefore derived in terms of (forward) McKean-Vlasov stochastic differentialequations (SDEs), which provides us with the mathematical framework forMonte-Carlo simulations of wall-bounded turbulent flows.

Stochastic formulation of incompressible fluid flows in wall bounded regions