Canonical solutions to non-translation invariant singular SPDEs
Harprit Singh
TL;DR
The paper addresses singular parabolic SPDEs with non‑translation invariant operators and proves that a canonical, finite‑dimensional family of renormalised equations suffices to describe solutions for g‑PAM, φ^4_2, φ^4_3, and KPZ. It develops a covariant regularisation framework within Regularity Structures, identifying local, A‑dependent counterterms that render the solutions well‑defined and regulator‑independent. The authors also establish continuity of the solution map with respect to the differential operator, and show that the renormalisation constants depend only on local geometric data via A, hence the theory is robust under operator perturbations. These results provide a geometric, canonical approach to renormalisation in non‑translation invariant SPDEs, enabling explicit, regulator‑insensitive limits and paving the way for automated, geometry‑aware analyses of such equations.
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, ξ) \ , \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, $φ^4_2$, $φ^4_3$ 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.