Small-time Wasserstein action for KPZ and ${\Phi }_3^4$

Abstract

We determine the leading small-time Wasserstein action of the full spatial laws of centered KPZ and dynamical ${\Phi }_3^4$, both started from zero. The leading term is Gaussian, with an explicit heat-trace constant. A common comparison also determines the ratio of length to displacement. For KPZ, a first-chaos cancellation yields a quadratic coupling error. For ${\Phi }_3^4$, two weighted entropies and an adjoint estimate give comparison bounds uniform in the spectral cutoff. Integrating a Gaussian rotation gives a differentiable realization of the full law; the comparison holds at every positive time.