Boundary renormalisation of SPDEs

Abstract

We consider the continuum parabolic Anderson model (PAM)(PAM) $$({\partial }_t-\Delta )u=u\xi$$(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 ${\Phi }_3^4$ 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.

Keywords:

• Boundary renormalisation
• singular SPDE
• regularity structures

MSC CLASSIFICATION:

• 60H15
• 60L30

Acknowledgements

The authors thank the referees for several valuable suggestions. MG thanks the support of the Austrian Science Fund (FWF) through the Lise Meitner programme M2250-N3 during a significant part of the project. MH gratefully acknowledges support from the Royal Society through a research professorship. Thanks also to Etienne Pardoux for numerous discussions on the topic of this article.

Notes

1 Note that ${\mathit{G}}_{\beta }^{\partial }$ is not actually a vector space! Scalar multiplication however is well-defined and our ‘norm’ is positive and one-homogeneous.