Stochastic Modelling in Fluid Dynamics: Itô vs Stratonovich

Abstract

Suppose the observations of Lagrangian trajectories for fluid flow in some physical situation may be modelled sufficiently accurately by a spatially correlated Itô stochastic process obtained from data which is taken in fixed Eulerian space. Suppose we also want to apply Hamilton's principle to derive the stochastic fluid equations for this situation. Now, the variational calculus for applying Hamilton's principle requires the Stratonovich process, so we must transform from Itô noise in the data frame to the equivalent Stratonovich noise. However, the transformation from the Itô process in the data frame to the corresponding Stratonovich process shifts the drift velocity of the transformed Lagrangian fluid trajectory out of the data frame into a non-inertial frame obtained from the Itô correction. The question is, "Will fictitious forces arising from this transformation of reference frames make a difference in the interpretation of the solution behaviour of the resulting stochastic equations?" This issue will be resolved by elementary considerations.

Stochastic Variational Formulations of Fluid Wave-Current Interaction

“…This means that the implementation of stochasticity for the CL equations will differ in the same way. In fact, there may be a related modelling choice to be made between Itô and Stratonovich stochasticity, in choosing between Eulerian and Lagrangian implementations of stochasticity, for example, in the pursuit of uncertainty quantification [50].…”