Abstract
We construct renormalised models of regularity structures by using a recursive formulation for the structure group and for the renormalisation group. This construction covers all the examples of singular SPDEs which have been treated so far with the theory of regularity structures and improves the renormalisation procedure based on Hopf algebras given in Bruned–Hairer–Zambotti (Algebraic renormalisation of regularity structures, 2016. arXiv:1610.08468).
Highlights
The theory of regularity structures introduced by Martin Hairer in [10] has proven to be an essential tool for solving singular SPDEs of the form: M
Since [16], the rough path approach is a way to study SDEs driven by non-smooth paths with an enhancement of the underlying path which allows to recover continuity of the solution map
Stoch PDE: Anal Comp (2018) 6:525–564 of SPDEs the enhancement is represented by a model (, ), to which is associated a space of local Taylor expansions of the solution with new monomials, coded by an abstract space T of decorated trees
Summary
In order to obtain a nice expression of the renormalised model in [3], the authors use a co-interaction property, described in the context of B-series in [4,6,7], between the Hopf algebra for the structure group and the one for the renormalisation group This co-interaction gives powerful results but one has to work with extended decorations see [3] for the definitions. The main idea is to separate the renormalisation happening at the root from the ones happening inside the tree following the steps of3 This representation allows us to derive a recursive definition of the coproduct extending the one giving in [10] and covering the two coproducts used for G+ and G−. We show that some of the coassiociativity proofs given in [3] can be recovered by using the recursive formula for the coproducts
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