- AutorIn
- Pablo Linares Ballesteros
- Titel
- On a tree-free approach to regularity structures for quasi-linear stochastic partial differential equations
- Zitierfähige Url:
- https://nbn-resolving.org/urn:nbn:de:bsz:15-qucosa2-807548
- Datum der Einreichung
- 26.01.2022
- Datum der Verteidigung
- 08.09.2022
- Abstract (EN)
- We consider the approach to regularity structures introduced by Otto, Sauer, Smith and Weber to obtain a priori bounds for quasi-linear SPDEs. This approach replaces the index set of trees, used in the original constructions of Hairer et. al., by multi-indices describing products of derivatives of the corresponding nonlinearity. The two tasks of this thesis are: - Construction and estimates of the model. We first provide the construction of a model in the regular, deterministic setting, where negative renormalization can be avoided. We later extend these ideas to the singular case, incorporating BPHZ-renormalization under spectral gap assumptions as a convenient input for an automated proof of the stochastic estimates of the singular model in the full subcritical regime. - Characterization of the algebraic structures generated by the multi-index setting. We consider natural actions on functionals of the nonlinearity and build a (pre-)Lie algebra from them. We use this as the starting point of an algebraic path towards the structure group, which as in the regularity structures literature is based on a Hopf algebra. This approach further allows us to explore the relation between multi-indices and trees, which we express through pre-Lie and Hopf algebra morphisms, in certain semi-linear equations. All the results are based on a series of joint works with Otto, Tempelmayr and Tsatsoulis.
- Freie Schlagwörter (EN)
- Quasi-linear SPDEs, regularity structures
- Klassifikation (DDC)
- 500
- Den akademischen Grad verleihende / prüfende Institution
- Universität Leipzig, Leipzig
- Version / Begutachtungsstatus
- publizierte Version / Verlagsversion
- URN Qucosa
- urn:nbn:de:bsz:15-qucosa2-807548
- Veröffentlichungsdatum Qucosa
- 23.09.2022
- Dokumenttyp
- Dissertation
- Sprache des Dokumentes
- Englisch
- Lizenz / Rechtehinweis
- CC BY-NC-ND 4.0