# Implications of Kunita-It\^o-Wentzell formula for $k$-forms in stochastic fluid dynamics

## TL;DR

This paper generalizes the Itô-Wentzell formula to $k$-forms in stochastic fluid dynamics, linking stochastic calculus with geometric fluid models and preserving key structures like Kelvin's theorem.

## Contribution

It introduces the Kunita-Itô-Wentzell formula for $k$-forms and connects it to stochastic fluid models that maintain geometric properties of deterministic fluids.

## Key findings

- Derived the KIW formula for $k$-forms.
- Established correspondence with geometric stochastic fluid models.
- Preserved structures like Kelvin's circulation in stochastic setting.

## Abstract

We extend the It\^o-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-It\^o-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.