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 stochastic partial differential equations of the form ∂t u = L u+ F(u, ξ), where the differential operator fails to be elliptic. This is achieved by interpreting the base space Rd as a non-trivial homogeneous Lie group G such that the differential operator ∂t - L becomes a translation invariant hypoelliptic operator on G. Prime examples are the kinetic Fokker-Planck operator ∂t - Δv -v · ∇x and heat-type operators associated with sub-Laplacians. As an application of the developed framework, we solve a class of parabolic Anderson type equations ∂tu = σ Xi2 + u (ξ-c) on the compact quotient of an arbitrary Carnot group.