Hairer’s multilevel Schauder estimates without regularity structures

Broux, Lucas; Caravenna, Francesco; Zambotti, Lorenzo

doi:10.1090/tran/9245

Hairer’s multilevel Schauder estimates without regularity structures
HTML articles powered by AMS MathViewer

by Lucas Broux, Francesco Caravenna and Lorenzo Zambotti;

Trans. Amer. Math. Soc. 377 (2024), 6981-7035

DOI: https://doi.org/10.1090/tran/9245

Published electronically: August 9, 2024

HTML | PDF | Request permission

Abstract:

We investigate the regularising properties of singular kernels at the level of germs, i.e. families of distributions indexed by points in $\mathbb {R}^d$. First we construct a suitable integration map which acts on general coherent germs. Then we focus on germs that can be decomposed along a basis (corresponding to the so-called modelled distributions in Regularity Structures) and we prove a version of Hairer’s multilevel Schauder estimates in this setting, with minimal assumptions.

References

Ismaël Bailleul and Masato Hoshino, A tourist’s guide to regularity structures and singular stochastic pdes, arXiv preprint (2020).
Nils Berglund, An introduction to singular stochastics PDEs—Allen-Cahn equations, metastability, and regularity structures, EMS Series of Lectures in Mathematics, EMS Press, Berlin, [2022] ©2022. MR 4458524, DOI 10.4171/ELM/34
Nils Berglund and Christian Kuehn, Model spaces of regularity structures for space-fractional SPDEs, J. Stat. Phys. 168 (2017), no. 2, 331–368. MR 3667364, DOI 10.1007/s10955-017-1801-3
Lucas Broux and David Lee, Besov reconstruction, Potential Anal. 59 (2023), no. 4, 1875–1912. MR 4684379, DOI 10.1007/s11118-022-10028-7
Y. Bruned, A. Chandra, I. Chevyrev, and M. Hairer, Renormalising SPDEs in regularity structures, J. Eur. Math. Soc. (JEMS) 23 (2021), no. 3, 869–947. MR 4210726, DOI 10.4171/jems/1025
Y. Bruned, M. Hairer, and L. Zambotti, Algebraic renormalisation of regularity structures, Invent. Math. 215 (2019), no. 3, 1039–1156. MR 3935036, DOI 10.1007/s00222-018-0841-x
Francesco Caravenna and Lorenzo Zambotti, Hairer’s reconstruction theorem without regularity structures, EMS Surv. Math. Sci. 7 (2020), no. 2, 207–251. MR 4261666, DOI 10.4171/emss/39
Ajay Chandra and Martin Hairer, An analytic bphz theorem for regularity structures, arXiv preprint (2016).
Ingrid Daubechies, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math. 41 (1988), no. 7, 909–996. MR 951745, DOI 10.1002/cpa.3160410705
Ingrid Daubechies, Ten lectures on wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 61, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. MR 1162107, DOI 10.1137/1.9781611970104
A. M. Davie, Differential equations driven by rough paths: an approach via discrete approximation, Appl. Math. Res. Express. AMRX (2008), Art. ID abm009, 40. [Issue information previously given as no. 2 (2007)]. MR 2387018
Denis Feyel and Arnaud de La Pradelle, Curvilinear integrals along enriched paths, Electron. J. Probab. 11 (2006), no. 34, 860–892. MR 2261056, DOI 10.1214/EJP.v11-356
Peter K. Friz and Martin Hairer, A course on rough paths, Universitext, Springer, Cham, [2020] ©2020. With an introduction to regularity structures; Second edition of [ 3289027]. MR 4174393, DOI 10.1007/978-3-030-41556-3
Máté Gerencsér and Martin Hairer, Boundary renormalisation of SPDEs, Comm. Partial Differential Equations 47 (2022), no. 10, 2070–2123. MR 4492924, DOI 10.1080/03605302.2022.2109173
M. Gubinelli, Controlling rough paths, J. Funct. Anal. 216 (2004), no. 1, 86–140. MR 2091358, DOI 10.1016/j.jfa.2004.01.002
Massimiliano Gubinelli, A panorama of singular SPDEs, Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. III. Invited lectures, World Sci. Publ., Hackensack, NJ, 2018, pp. 2311–2338. MR 3966852
M. Hairer, A theory of regularity structures, Invent. Math. 198 (2014), no. 2, 269–504. MR 3274562, DOI 10.1007/s00222-014-0505-4
Martin Hairer and Cyril Labbé, The reconstruction theorem in Besov spaces, J. Funct. Anal. 273 (2017), no. 8, 2578–2618. MR 3684891, DOI 10.1016/j.jfa.2017.07.002
Martin Hairer and Étienne Pardoux, Fluctuations around a homogenised semilinear random PDE, Arch. Ration. Mech. Anal. 239 (2021), no. 1, 151–217. MR 4198718, DOI 10.1007/s00205-020-01574-8
Martin Hairer and Harprit Singh, Periodic space-time homogenisation of the $\phi ^4_2$ equation, arXiv preprint (2023).
Martin Hairer and Rhys Steele, The BPHZ theorem for regularity structures via the spectral gap inequality, Arch. Ration. Mech. Anal. 248 (2024), no. 1, Paper No. 9, 81. MR 4693281, DOI 10.1007/s00205-023-01946-w
Hannes Kern, A stochastic reconstruction theorem, arXiv preprint (2021).
Cyril Labbé, The continuous Anderson Hamiltonian in $d\leq 3$, J. Funct. Anal. 277 (2019), no. 9, 3187–3235. MR 3997634, DOI 10.1016/j.jfa.2019.05.027
Terry J. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana 14 (1998), no. 2, 215–310. MR 1654527, DOI 10.4171/RMI/240
Yves Meyer, Wavelets and operators, Cambridge Studies in Advanced Mathematics, vol. 37, Cambridge University Press, Cambridge, 1992. Translated from the 1990 French original by D. H. Salinger. MR 1228209
Augustin Moinat and Hendrik Weber, Space-time localisation for the dynamic $\Phi ^4_3$ model, Comm. Pure Appl. Math. 73 (2020), no. 12, 2519–2555. MR 4164267, DOI 10.1002/cpa.21925
Felix Otto, Jonas Sauer, Scott Smith, and Hendrik Weber, Parabolic equations with rough coefficients and singular forcing, arXiv preprint (2018).
Felix Otto, Jonas Sauer, Scott Smith, and Hendrik Weber, A priori bounds for quasi-linear spdes in the full sub-critical regime, arXiv preprint (2021).
Felix Otto and Hendrik Weber, Quasilinear SPDEs via rough paths, Arch. Ration. Mech. Anal. 232 (2019), no. 2, 873–950. MR 3925533, DOI 10.1007/s00205-018-01335-8
Paolo Rinaldi and Federico Sclavi, Reconstruction theorem for germs of distributions on smooth manifolds, J. Math. Anal. Appl. 501 (2021), no. 2, Paper No. 125215, 14. MR 4242052, DOI 10.1016/j.jmaa.2021.125215
Pavel Zorin-Kranich, The reconstruction theorem in quasinormed spaces, Rev. Mat. Iberoam. 39 (2023), no. 4, 1233–1246. MR 4615888, DOI 10.4171/rmi/1355