Implications of Kunita-It\^o-Wentzell formula for $k$-forms in stochastic fluid dynamics
Probability
2020-03-18 v1
Abstract
We extend the It\^o-Wentzell formula for the evolution of a time-dependent stochastic field along a semimartingale to -form-valued stochastic processes. The result is the Kunita-It\^o-Wentzell (KIW) formula for -forms. We also establish a correspondence between the KIW formula for -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
Cite
@article{arxiv.1903.07201,
title = {Implications of Kunita-It\^o-Wentzell formula for $k$-forms in stochastic fluid dynamics},
author = {Aythami Bethencourt de Léon and Darryl Holm and Erwin Luesink and So Takao},
journal= {arXiv preprint arXiv:1903.07201},
year = {2020}
}