Singular SPDEs on Homogeneous Lie Groups

The aim of this article is to extend the scope of the theory of regularity structures in order to deal with a large class of singular SPDEs of the form $${\mathrm{\partial }}_{t}u=\mathfrak{L}u+F(u,\xi ),$$ where the differential operator $\mathfrak{L}$ fails to be elliptic. This is achieved by interpreting the base space ${\mathbb{R}}^d$ as a non-trivial homogeneous Lie group $\mathbb{G}$ such that the differential operator ${\mathrm{\partial }}_t-\mathfrak{L}$ becomes a translation invariant hypoelliptic operator on $\mathbb{G}$. Prime examples are the kinetic Fokker-Planck operator ${\mathrm{\partial }}_t-{\mathrm{\Delta }}_v-v\cdot {\mathrm{\nabla }}_x$ and heat-type operators associated to sub-Laplacians. As an application of the developed framework, we solve a class of parabolic Anderson type equations $${\mathrm{\partial }}_{t}u=\sum _i{X}_i^{2}u+u(\xi -c)$$ on the compact quotient of an arbitrary Carnot group.