Implications of Kunita–Itô–Wentzell Formula for k-Forms in Stochastic Fluid Dynamics

Abstract

We extend the Itô-Wentzell formula for the evolution of a time-dependent stochastic field along a semimartingale to k-form-valued stochastic processes. The result is the Kunita-Itô-Wentzell (KIW) formula for k-forms. We also establish a correspondence between the KIW formula for k-forms derived here and a certain class of stochastic fluid dynamics models which preserve the geometric structure of deterministic ideal fluid dynamics. This geometric structure includes Eulerian and Lagrangian variational principles, Lie-Poisson Hamiltonian formulations and natural analogues of the Kelvin circulation theorem, all derived in the stochastic setting.Keywords Stochastic geometric mechanics · Lie derivatives with respect to stochastic vector fields · Pull-back by smooth maps with stochastic time parameterization Communicated byPurpose of this paper. This paper aims to derive stochastic partial differential equations (SPDEs) for continuum dynamics with stochastic advective Lie transport (SALT). These derivations require stochastic counterparts of the deterministic approaches for deriving fluid equations (PDEs). The approach we follow is the stochastic counterpart of the Euler-Poincaré variational principle as in Holm et al. (1998) which reveals the geometric structure of deterministic ideal fluid dynamics. Our goal is to create stochastic counterparts which preserve this geometric structure. Such variational formulations of stochastic fluid PDEs will also possess auxiliary SPDEs for stochastic advection of material properties which will correspond to various differential k-forms. The auxiliary SPDE for advective transport of a given material property by a stochastic fluid flow corresponds to a type of stochastic "chain rule" in which a k-form-valued semimartingale K (t, x) is evaluated (via pullback) along a stochastic flow φ t . The stochastic aspect of the flow map φ t represents uncertainty in the Lagrange-to-Euler map for the fluid. Examples of the k-form-valued process K (t, x) advected by the pullback of the stochastic flow map φ t include mass density, regarded as a volume form, and magnetic field, interpreted as a two-form for ideal magnetohydrodynamics (MHD) (Holm et al. 1998). We denote pullback of a k-form K by φ * t K , which for scalar functions f takes the simple form φ * t f := f • φ t . We will refer to this "chain rule" that gives us the auxiliary SPDE for SALT satisfied by the k-form-valued process φ * t K as the Kunita-Itô-Wentzell (KIW) formula for k-forms.In Kunita (1981Kunita ( , 1997, Kunita derived an extension of Itô's formula showing that if K is a sufficiently regular (l, m)-tensor and φ t is the flow of the SDE(1.1) with sufficiently regular coefficients, then an analogue of Itô's formula holds for tensor fields, namelyRemark 1.1 In applications, we will sometimes express (1.6) using the differential notation

Stochastic effects of waves on currents in the ocean mixed layer

“…Computational simulations of the equations resulting from the SFLT and SFLP modelling approaches introduced here, as well as simulations of the EA SFLT equations in section 4 have all been left for future work. Formulas (A.1) and (A.2) are compact forms of equations in [19] which are written in integral notation to make the stochastic processes more explicit.…”

“…The class of SFLP equations in (3. 19) can be obtained via a phase-space variational principle, as we discuss next.…”

“…Here is a simplified statement of the theorem for applying the Kunita Itô-Wentzell change of variables formula to stochastic advection of differential k-forms proved in [19], based on [43,44]. See also [48].…”

Semi-martingale driven variational principles

“…Suppose at:Dfalse→D is invariant under the flow, meaning that it is an advected quantity. Then a0false(x0false)=atfalse(xtfalse)=false(at∘gtfalse)x0=false(gt∗atfalse)x0, and hence, by an application of the stochastic Kunita–Itô–Wentzell formula [42], right left right left right left right left right left right left3pt0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em0=da0(x0)=d(at∘gt)x0=d(gt∗at)x0=gt∗(dat(x0)+Luat(x0) dt+false∑iLξiat(x0)∘…”

Semi-martingale driven variational principles

“…Note that we intrinsically assume here that L satisfies similar integrability conditions as specified for G i,j 8. The methodology presented here can be easily extended to spaces of non-finite measures (e.g.…”