Lagrangian averaging of singular stochastic actions for fluid dynamics

We construct sub-grid scale models of incompressible fluids by considering expectations of semi-martingale Lagrangian particle trajectories. Our construction is based on the Lagrangian decomposition of flow maps into mean and fluctuation parts, and it is separated into the following steps. First, through Magnus expansion, the fluid velocity field is expressed in terms of fluctuation vector fields whose dynamics are assumed to be stochastic. Second, we use Malliavin calculus to give a regularised interpretation of the product of white noise when inserting the stochastic velocity field into the Lagrangian for Euler's fluid. Lastly, we consider closures of the mean velocity by making stochastic analogues of Talyor's frozen-in turbulence hypothesis to derive a version of the anisotropic Lagrangian averaged Euler equation.