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.
Highlights
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
Employed the KIW formula in deriving an Euler–Poincaré variational principle and a Clebsch-constrained Hamilton’s principle which each introduce stochastic advection by Lie transport (SALT) into the semidirect-product continuum equations derived in Holm et al (1998) while preserving their Kelvin–Noether theorem and Lie–Poisson Hamiltonian structure
Applied the KIW formula to provide a rigorous derivation of stochastic advection by Lie transport (SALT) equations, continuity equations in fluid dynamics, and Kelvin’s circulation Theorem
Summary
In Krylov (2011), Krylov considered an approach using mollifiers to provide a general proof of the classical Itô–Wentzell formula (Kunita 1981; Bismut 1981; Kunita 1997) This classical formula states that for a sufficiently smooth scalar function-valued semimartingale f , represented as df (t, x) = g(t, x) dt + h(t, x) ◦ dWt ,. We derive the Kunita–Itô–Wentzell Theorem which establishes the formula for the evolution of a k-form-valued process φt∗ K This result generalises Kunita’s formula (1.2) and the Itô–Wentzell formula for a scalar function (1.4) by allowing K to be any smooth-in-space, stochastic-in-time k-form on Rn. Omitting the technical regularity assumptions provided in the more detailed statement of the theorem, we state a simplified version of our main theorem, as follows. No mention of a transfer principle occurs in Krylov’s detailed technical proof of (1.4) for the scalar case in Krylov (2011)
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