# Stochastic Evolution of Augmented Born--Infeld Equations

## TL;DR

This paper applies a stochastic uncertainty quantification method to the Born-Infeld electromagnetism model, revealing that stochastic perturbations induce similar Lie transport effects as in fluid dynamics, highlighting a deep structural analogy.

## Contribution

It demonstrates that a stochastic quantification method for fluid dynamics can be extended to nonlinear electromagnetism, preserving Hamiltonian structure and revealing fundamental similarities.

## Key findings

- Stochastic Lie transport appears in both theories with cylindrical noise.
- The stochastic equations retain their form with an additional divergence-free term.
- The Hamiltonian construction links spatial correlations with the momentum map.

## Abstract

This paper compares the results of applying a recently developed method of stochastic uncertainty quantification designed for fluid dynamics to the Born-Infeld model of nonlinear electromagnetism. The similarities in the results are striking. Namely, the introduction of Stratonovich cylindrical noise into each of their Hamiltonian formulations introduces stochastic Lie transport into their dynamics in the same form for both theories. Moreover, the resulting stochastic partial differential equations (SPDE) retain their unperturbed form, except for an additional term representing induced Lie transport by the set of divergence-free vector fields associated with the spatial correlations of the cylindrical noise. The explanation for this remarkable similarity lies in the method of construction of the Hamiltonian for the Stratonovich stochastic contribution to the motion in both cases; which is done via pairing spatial correlation eigenvectors for cylindrical noise with the momentum map for the deterministic motion. This momentum map is responsible for the well-known analogy between hydrodynamics and electromagnetism. The momentum map for the Maxwell and Born-Infeld theories of electromagnetism treated here is the 1-form density known as the Poynting vector. Two Appendices treat the Hamiltonian structures underlying these results.