Title: Stochastic Evolution of Augmented Born--Infeld Equations,
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.