Canonical solutions to non-translation invariant singular SPDEs

TL;DR

This paper constructs a canonical finite-dimensional solution family for certain singular SPDEs with non-translation invariant operators, including g-PAM, ^2, ^3, and KPZ, showing renormalization functions depend locally on the coefficients.

Contribution

It introduces a canonical solution framework for non-translation invariant singular SPDEs, extending existing methods to variable coefficient operators.

Findings

Constructed finite-dimensional solution families for specified SPDEs.

Demonstrated local dependence of renormalization functions on coefficients.

Proved continuity of solutions with respect to differential operator variations.

Abstract

We exhibit a canonical, finite dimensional solution family to certain singular SPDEs of the form \begin{equation} \left(\partial_t- \sum_{i,j=1}^d a_{i,j}(x,t) \partial_i \partial_j - \sum_{i=1}^d b_i(x,t) \partial_i - c(x,t)\right) u = F(u, \partial u, \xi) \ , \end{equation} where $a_{i,j}, b_i, c: \mathbb{T}^d\times \mathbb{R} \to \mathbb{R}$ and $A=\{a_{i,j}\}_{i,j=1}^d$ is uniformly elliptic. More specifically, we solve the non-translation invariant g-PAM, $\phi^4_2$, $\phi^4_3$ and KPZ-equation and show that the diverging renormalisation-functions are local functions of $A$. We also establish a continuity result for the solution map with respect to the differential operator for these equations.