An Invariance Principle for some Reaction-Diffusion Equations with a Multiplicative Random Source
Davar Khoshnevisan, Kunwoo Kim, Carl Mueller
TL;DR
This work establishes a stochastic invariance principle for a broad class of dissipative reaction-diffusion SPDEs with multiplicative noise, showing that large-time behavior of the nonlinear equation is universal with respect to the parabolic Anderson model (PAM) having parameters $\mu=f'(0+)$ and $\sigma=g'(0+)$. The authors develop a robust coupling framework and a sequence of technical results (moment bounds, dissipation, short-time stability, local tightness, tracking) to compare the nonlinear SPDE with PAM, yielding high-probability convergence of $\log w(t)$ to $\log u(t)$ as $t\to\infty$. A key contribution is a meticulous multi-stage coupling argument that handles nonlocality and intermittency, providing quantitative error controls and paving the way to PPP-type limit theorems (LLN and CLT) for PAM-driven growth rates. The work also connects to KPZ-type asymptotics via the Hopf-Cole transform, indicating that large-time statistics of the nonlinear system inherit universal PAM behavior in the high-noise regime, with implications for universality and potential extensions to real-line spatial domains.
Abstract
We establish a notion of universality for the parabolic Anderson model via an invariance principle for a wide family of parabolic stochastic partial differential equations. We then use this invariance principle in order to provide an asymptotic theory for a wide class of non-linear SPDEs. A novel ingredient of this invariance principle is the dissipativity of the underlying stochastic PDE.