Corpus ID: 232148084

The structure group for quasi-linear equations via universal enveloping algebras

  title={The structure group for quasi-linear equations via universal enveloping algebras},
  author={P. Linares and Felix Otto and Markus Tempelmayr},
We consider the approach of replacing trees by (fewer) multi-indices as an index set of the abstract model space T, which was introduced in [15] to tackle quasi-linear singular SPDE. We show that this approach is consistent with the postulates of regularity structures in [10] when it comes to the structure group G. In particular, G ⊂ Aut(T) arises from a Hopf algebra T and a comodule ∆: T → T ⊗ T. In fact, this approach, where the dual T of the abstract model space T naturally embeds into a… Expand
A priori bounds for quasi-linear SPDEs in the full sub-critical regime
This paper is concerned with quasi-linear parabolic equations driven by an additive forcing ξ ∈ C, in the full subcritical regime α ∈ (0, 1). We are inspired by Hairer’s regularity structures,Expand
Smooth rough paths, their geometry and algebraic renormalization
We introduce the class of “smooth rough paths” and study their main properties. Working in a smooth setting allows us to discard sewing arguments and focus on algebraic and geometric aspects.Expand
The Sewing lemma for $0<\gamma \leq 1$
We establish a Sewing lemma in the regime γ ∈ (0, 1], constructing a Sewing map which is neither unique nor canonical, but which is nonetheless continuous with respect to the standard norms. TwoExpand


On the Lie envelopping algebra of a pre-Lie algebra
We construct an associative product on the symmetric module S(L) of any pre-Lie algebra L. Then we proove that in the case of rooted trees our construction is dual to that of Connes and Kreimer. WeExpand
Renormalising SPDEs in regularity structures
The formalism recently introduced in arXiv:1610.08468 allows one to assign a regularity structure, as well as a corresponding "renormalisation group", to any subcritical system of semilinearExpand
A theory of regularity structures
We introduce a new notion of “regularity structure” that provides an algebraic framework allowing to describe functions and/or distributions via a kind of “jet” or local Taylor expansion around eachExpand
Trees, Renormalization and Differential Equations
The Butcher group and its underlying Hopf algebra of rooted trees were originally formulated to describe Runge–Kutta methods in numerical analysis. In the past few years, these concepts turned out toExpand
Regularity structures and the dynamical Φ 43 model
We give a concise overview of the theory of regularity structures as first exposed in [Hai14]. In order to allow to focus on the conceptual aspects of the theory, many proofs are omitted andExpand
A rough path perspective on renormalization
Abstract We develop the algebraic theory of rough path translation. Particular attention is given to the case of branched rough paths, whose underlying algebraic structure (Connes-Kreimer,Expand
Di erential equations driven by rough signals
This paper aims to provide a systematic approach to the treatment of differential equations of the type dyt = Si fi(yt) dxti where the driving signal xt is a rough path. Such equations are veryExpand
Hopf π- Algebras
This paper introduces five notions, including π-algebras, π-ideals, Hopf π-algebras, π-modules and Hopf π-modules, verifies the fundamental isomorphism theorem of π-algebras and studies someExpand
Ramification of rough paths
The stack of iterated integrals of a path is embedded in a larger algebraic structure where iterated integrals are indexed by decorated rooted trees and where an extended Chen's multiplicativeExpand
Controlling rough paths
Abstract We formulate indefinite integration with respect to an irregular function as an algebraic problem which has a unique solution under some analytic constraints. This allows us to define a goodExpand