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Singular SPDEs on Homogeneous Lie Groups

Avi Mayorcas, Harprit Singh

TL;DR

This work generalises the regularity-structures framework to singular SPDEs with hypoelliptic, translation-invariant operators on homogeneous Lie groups, reframing Euclidean problems as analysis on non-commutative spaces. It develops intrinsic derivatives, polynomial structures, modelled distributions, and reconstruction in the group setting, and provides Schauder-type estimates for singular kernels and local operations, enabling fixed-point theories for semilinear evolution equations. The authors then specialise to Anderson-type equations on (compact quotients of) Carnot groups, constructing a regularity structure and proving convergence and renormalisation results for mollified noises, with explicit treatment of Heisenberg and matrix-exponential group examples. The framework yields a pathwise solution theory for subelliptic PAM-type models and opens avenues toward a full BPHZ renormalisation in homogeneous Lie groups, with potential applications in kinetic equations and kinetic-Fokker–Planck-type dynamics on sub-Riemannian spaces.

Abstract

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 $$\partial_t u = \mathfrak{L} u+ F(u, ξ)\ ,$$ 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 $\partial_t -\mathfrak{L}$ becomes a translation invariant hypoelliptic operator on $\mathbb{G}$. Prime examples are the kinetic Fokker-Planck operator $\partial_t -Δ_v - v\cdot \nabla_x$ and heat-type operators associated to sub-Laplacians. As an application of the developed framework, we solve a class of parabolic Anderson type equations $$\partial_t u = \sum_{i} X^2_i u + u (ξ-c)$$ on the compact quotient of an arbitrary Carnot group.

Singular SPDEs on Homogeneous Lie Groups

TL;DR

This work generalises the regularity-structures framework to singular SPDEs with hypoelliptic, translation-invariant operators on homogeneous Lie groups, reframing Euclidean problems as analysis on non-commutative spaces. It develops intrinsic derivatives, polynomial structures, modelled distributions, and reconstruction in the group setting, and provides Schauder-type estimates for singular kernels and local operations, enabling fixed-point theories for semilinear evolution equations. The authors then specialise to Anderson-type equations on (compact quotients of) Carnot groups, constructing a regularity structure and proving convergence and renormalisation results for mollified noises, with explicit treatment of Heisenberg and matrix-exponential group examples. The framework yields a pathwise solution theory for subelliptic PAM-type models and opens avenues toward a full BPHZ renormalisation in homogeneous Lie groups, with potential applications in kinetic equations and kinetic-Fokker–Planck-type dynamics on sub-Riemannian spaces.

Abstract

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 where the differential operator fails to be elliptic. This is achieved by interpreting the base space as a non-trivial homogeneous Lie group such that the differential operator becomes a translation invariant hypoelliptic operator on . Prime examples are the kinetic Fokker-Planck operator and heat-type operators associated to sub-Laplacians. As an application of the developed framework, we solve a class of parabolic Anderson type equations on the compact quotient of an arbitrary Carnot group.
Paper Structure (40 sections, 39 theorems, 250 equations)

This paper contains 40 sections, 39 theorems, 250 equations.

Table of Contents

  1. Introduction
  2. Related Literature
  3. A Motivating Class of Examples
  4. Structure and Contributions of the Paper
  5. Open Problems and Wider Context
  6. Notation:
  7. Acknowledgements:
  8. Analysis on Homogeneous Lie Groups
  9. Derivatives and Polynomials
  10. Distributions and Convolution
  11. Discrete Subgroups
  12. Concrete Examples
  13. The Heisenberg group
  14. Matrix Exponential Groups
  15. Regularity Structures and Models
  16. ...and 25 more sections

Key Result

Theorem 1.1

Let $\mathbb{G}$ be a homogeneous Lie group of homogeneous dimension $|\mathfrak{s}|$ and $\mathfrak{L}$ be a left-translation invariant (with respect to $\mathbb{G}$), homogeneous differential operator of degree $\beta\in (0,|\mathfrak{s}|)$ on $\mathbb{G}$, such that $\mathfrak{L}$ and its adjoint where the convolution is with respect to the given Lie structure.

Theorems & Definitions (124)

  • Theorem 1.1: folland_75_subelliptic
  • Theorem 1.2: hormander_67
  • Remark 1.3
  • Proposition 2.1: folland_stein_82_hardy
  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Definition 2.2
  • Remark 2.4
  • Proposition 2.5
  • ...and 114 more