Implications of Kunita-Itô-Wentzell formula for $k$-forms in stochastic fluid dynamics

Aythami Bethencourt de Léon; Darryl Holm; Erwin Luesink; So Takao

doi:10.1007/s00332-020-09613-0

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arXiv · 1903.07201

Implications of Kunita-Itô-Wentzell formula for $k$-forms in stochastic fluid dynamics

Aythami Bethencourt de Léon·Darryl Holm·Erwin Luesink·So Takao

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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.

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Aythami Bethencourt de Léon, Darryl Holm, Erwin Luesink, So Takao. 2019-03-17. Implications of Kunita-Itô-Wentzell formula for $k$-forms in stochastic fluid dynamics. https://doi.org/10.1007/s00332-020-09613-0

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