Boundary renormalisation of SPDEs

TL;DR

This paper studies boundary renormalisation issues in SPDEs, specifically the parabolic Anderson model and $\

Contribution

It introduces a boundary renormalisation framework for SPDEs with Neumann and Robin conditions, and proves boundary triviality for $\

Findings

Neumann/Robin boundary conditions cause divergent boundary renormalisation.

Boundary triviality result shows Neumann and Dirichlet limits coincide under bulk renormalisation.

Provides a rigorous treatment of boundary effects in SPDE renormalisation.

Abstract

We consider the continuum parabolic Anderson model (PAM) and the dynamical $\Phi^4$ equation on the $3$-dimensional cube with boundary conditions. While the Dirichlet solution theories are relatively standard, the case of Neumann/Robin boundary conditions gives rise to a divergent boundary renormalisation. Furthermore for $\Phi^4_3$ a `boundary triviality' result is obtained: if one approximates the equation with Neumann boundary conditions and the usual bulk renormalisation, then the limiting process coincides with the one obtained using Dirichlet boundary conditions.