Stochastic Modelling in Fluid Dynamics: Itô vs Stratonovich

Abstract

Suppose the observations of Lagrangian trajectories for fluid flow in some physical situation can be modelled sufficiently accurately by a spatially correlated Itô stochastic process (with zero mean) 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 \emph{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 issue is, "Will non-inertial 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.

12 pages, 1 figure, 2nd version, comments welcome!

References in corpus (4)

• Global L_2-solutions of stochastic Navier-Stokes equations
• Stochastic partial differential fluid equations as a diffusive limit of deterministic Lagrangian multi-time dynamics
• Stochastic Closures for Wave--Current Interaction Dynamics
• Stochastic Variational Formulations of Fluid Wave-Current Interaction

Cited by in corpus (1)

• Stochastic Variational Formulations of Fluid Wave-Current Interaction