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H 1 The structure group revisited

Auteurs : Otto, Felix (Auteur de la Conférence)
CIRM (Editeur )

Résumé : Following the treatment of a class of quasi-linear SPDE with Sauer, Smith, and Weber, we approach Hairer's regularity structure $(\mathrm{A},\ \mathrm{T},\ \mathrm{G})$ from a different angle. In this approach, the model space $\mathrm{T}$ is a direct sum over an index set that corresponds to specific linear combination of (decorated) trees, and thus amounts to a more parsimonious parameterization of the solution manifold. Moreover, the same structure group $\mathrm{G}$ captures different classes of equations; depending on the class, different (sub)spaces $\mathrm{T}$ matter, which correspond to linear combinations of different types of trees.

In our approach to $\mathrm{G}$, we start from the space of tuples $(a,p)$ of (polynomial) nonlinearities $a$ and space-time polynomials $p$, which we think of parameterizing the entire manifold of solutions $u$ (satisfying the equation up to space-time polynomials) via re-centering. We consider the actions of a shift by a space-time vector $h\in \mathbb{R}^{d+1}$ and of tilt by space-time polynomial $q$ on $(a,p)$-space, where, crucially, the tilt by a constant is treated as a shift of the (one-dimensional) $u$-space. We consider the infinitesimal generators of these actions, and pull them back as derivations on the algebra of formal power series $\mathbb{R}[[\mathrm{z}_{k},\ \mathrm{z}_{\mathrm{n}}]]$ in the natural coordinates $\{\mathrm{z}_{k}\}_{k\in \mathbb{N}_{0}}$ and $\{\mathrm{z}_{\mathrm{n}}\}_{\mathrm{n}\in \mathbb{N}_{0}^{d+1}-\{\mathrm{O}\}}$ of $(a,\ p)$-space. This defines a Lie algebra $\mathrm{L}\subset \mathrm{D}\mathrm{e}\mathrm{r}(\mathbb{R}[[\mathrm{z}_{k},\ \mathrm{z}_{\mathrm{n}}]])$ . Loosely speaking, the corresponding Lie group coincides with $\mathrm{G}^{*}$, but we follow an algebraic path to construct G.

As a group, $\mathrm{G}\subset(\mathrm{T}^{+})^{*}$ arises in the standard way from the Hopf algebra $\mathrm{T}^{+}$ that is obtained from dualizing the universal enveloping algebra $\mathrm{U}(\mathrm{L})$ . Here, gradedness and finiteness properties are needed for the well-posedness of the co-product $\triangle^{+}:\mathrm{T}^{+}\rightarrow \mathrm{T}^{+}\otimes \mathrm{T}^{+}$ and the antipode. The passage from $\mathbb{R}[[\mathrm{z}_{k},\ \mathrm{z}_{\mathrm{n}}]]$ to a smaller (linear) subspace $\mathrm{T}^{*}$ is needed for dualizing the module defined through $\mathrm{L}\subset$ End(T$*$) to obtain the co-module $\triangle:\mathrm{T}\rightarrow \mathrm{T}^{+}\otimes \mathrm{T}$. This yields the representation $\mathrm{G}\subset$ End(T). Both $\triangle$ and $\triangle^{+}$ satisfy the postulates of regularity structures, in particular the properties that intertwine $\triangle, \triangle^{+}$, and the family of re-centering maps $\mathcal{J}_{\mathrm{n}}:\mathrm{T}\rightarrow \mathrm{T}^{+}$. The latter relies on choosing a natural basis of $\mathrm{U}(\mathrm{L})$ , different from the standard Poincar\'{e}-Birkhoff-Witt basis, for dualization.

This is joint work with P. Linares and M. Tempelmayr.

Keywords : regularity structures; structure group; universal enveloping algebra; Hopf algebra

 Informations sur la Vidéo Réalisateur : Hennenfent, Guillaume Langue : Anglais Date de publication : 26/03/2021 Date de captation : 09/03/2021 Collection : Research talks Durée : 00:43:19 Domaine : PDE Audience : Chercheurs ; Doctorants , Post - Doctorants Download : https://videos.cirm-math.fr/2021-03-09_Otto.mp4 Informations sur la Rencontre Virtuelle Nom de la rencontre : Pathwise Stochastic Analysis and Applications / Analyse stochastique trajectorielle et applicationsOrganisateurs de la rencontre : Coutin, Laure ; Gassiat, Paul ; Lejay, Antoine ; Marie, Nicolas ; Tindel, SamyDates : 08/03/2021 - 12/03/2021 Année de la rencontre : 2021 URL Congrès : https://conferences.cirm-math.fr/2322.html Citation Data DOI : 10.24350/CIRM.V.19728803 Cite this video as: Otto, Felix (2021). The structure group revisited.CIRM . Audiovisual resource. doi:10.24350/CIRM.V.19728803 URI : http://dx.doi.org/10.24350/CIRM.V.19728803

### Voir aussi

Bibliographie

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5. HAIRER, Martin. A theory of regularity structures. Inventiones mathematicae, 2014, vol. 198, no 2, p. 269-504. - https://doi.org/10.1007/s00222-014-0505-4

6. LINARES, Pablo, OTTO, Felix et TEMPELMAYR, Markus. The structure group for quasi-linear equations via universal enveloping algebras. arXiv preprint arXiv:2103.04187, 2021. - https://arxiv.org/abs/2103.04187

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