Stochastic modelling in fluid dynamics: Itô versus 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 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.

Wave–current interaction on a free surface

“…Actually, this separation is a general feature of wave-current interaction theories that arise from Hamilton's principle with a phase-space Lagrangian. 31…”

Wave-current interaction on a free surface

“…Remark The separation of conservation laws in (3.27) means that the two degrees of freedom do not influence each other's circulation. Actually, this separation is a general feature of wave-current interaction theories which arise from Hamilton's principle with a phase-space Lagrangian, [32].…”

“…For a discussion of the geometric mechanics underlying the deterministic case, see, e.g., [31]. For a discussion of the geometric mechanics underlying the stochastic case, see, e.g., [25,32].…”

Compressible Navier--Stokes system with transport noise

“…Mikulevicius and Rozovskii [32] introduced randomness at the Lagrangian level imposing a stochastic forcing in the equation for the streamlines X = X(t), (1.1) dX = u(t, X) dt + σ • dW, where the macroscopic velocity u is augmented by a stochastic forcing of Stratonovich type. Similar ideas were incorporated in the general theory developed by Holm [26], see also the most recent development in [1], [15], [27]. In the same spirit, a series of different models has been proposed by Cruzeiro et al [2], [11], [12], including stochastic variants of the compressible Navier-Stokes system, see Section 1.2 below.…”