Regularity Structures for Interacting Photon Fields

We adapt Mr. Martin Hairer’s well-written theory of regularity structures to a class of stochastic Maxwell-type PDEs modeling interacting photon fields with nonlinear coupling and additive space-time noise. We construct a vector-valued regularity structure encoding the gauge/vector-symbols needed for electromagnetic fields, identify the renormalisation group, and compute the explicit counterterms required to obtain finite models. Our main result is a local (in time) well-posedness theorem for the renormalised stochastic Maxwell system in a subcritical regime (spatial dimension d≤3) and under suitable gauge fixing/damping. We provide a proof sketch following model construction, renormalisation (BPHZ-type), and reconstruction, and give a toy numerical scheme for computing renormalisation constants on a lattice via mollifier-dependent diagram evaluation. The approach gives a principled renormalisation prescription for interacting photon SPDEs and a framework for numerical experiments.