Paracontrolled calculus and regularity structures I

Abstract

We start in this work the study of the relation between the theory of regularity structures and paracontrolled calculus. We give a paracontrolled representation of the reconstruction operator and provide a natural parametrization of the space of admissible models.

Paracontrolled calculus and regularity structures II

“…The BPHZ renormalization procedure for the model involves a real-valued map k acting on a side space T − , which also defines a homogeneity-preserving linear map k from T into itself. It follows from [5,Th. 21] that the bracket data associated with the renormalized model k M ε is simply given by the [[ k(τ )]], for τ of negative homogeneity.…”

“…It happens nonetheless to be possible to compare the two languages, independently of their applications to the study of singular stochastic PDEs. This task was initiated in Gubinelli, Imkeller, Perkowski' seminal work [13] and Martin and Perkowski's work [22], and in our previous work [5], where we proved that the set of admissible models M = (g, Π) over a concrete regularity structure T = (T + , ∆ + ), (T, ∆) equipped with an abstract integration map is parameterized by a paracontrolled representation of Π on the set of elements τ with non-positive homogeneity. (Admissible models play a crucial in the regularity structures approach to the study of singular stochastic PDEs.)…”

“…(Admissible models play a crucial in the regularity structures approach to the study of singular stochastic PDEs.) Theorem 21 in [5] To understand the practical relevance of this linear parameterization of the space of admissible models on T , assume T stands for the Bruned, Hairer, and Zambotti's regularity structure [7] associated with a singular stochastic PDE and M ε = (g ε , Π ε ) stands for the naive interpretation model associated with a smoothened noise in the equation, with regularization parameter ε. The BPHZ renormalization procedure for the model involves a real-valued map k acting on a side space T − , which also defines a homogeneity-preserving linear map k from T into itself.…”

“…Here is another illustration of the use of the parameterization result of admissible models proved in [5] that will be developed in Section 4.1 and Section 4.4. Consider the elementary setting of branched rough paths; they are admissible models on particular examples of regularity structures.…”

“…Consider the elementary setting of branched rough paths; they are admissible models on particular examples of regularity structures. Theorem 21 in [5] gives a direct proof of Lyons' extension theorem, saying that a branched Hölder p-rough path has a unique extension into a branched Hölder q-rough path, for any q > p. (Recall weak geometric rough paths are branched rough paths.) This result allows to define the signature of a branched rough path.…”