Stochastic Modelling in Fluid Dynamics: It\^o vs Stratonovich

TL;DR

This paper examines the impact of transforming Itô stochastic processes to Stratonovich form in fluid dynamics, focusing on how non-inertial forces affect the interpretation of stochastic fluid equations derived via Hamilton's principle.

Contribution

It clarifies the effects of Itô to Stratonovich transformation on drift velocities and non-inertial forces in stochastic fluid models, linking stochastic calculus with variational principles.

Findings

Transformation shifts drift into a non-inertial frame.

Non-inertial forces influence solution behavior interpretation.

Elementary considerations resolve the impact of frame transformation.

Abstract

Suppose the observations of Lagrangian trajectories for fluid flow in some physical situation can be modelled sufficiently accurately by a spatially correlated It\^o 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 It\^o noise in the \emph{data frame} to the equivalent Stratonovich noise. However, the transformation from the It\^o 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 It\^o 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.