Canonical solutions to non-translation invariant singular SPDEs

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, ${\varphi }_2^4$, ${\varphi }_3^4$ 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.

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