A Sobolev rough path extension theorem via regularity structures

Abstract. We show that every R-valued Sobolev path with regularity α and integrability p can be lifted to a Sobolev rough path provided α < 1/p < 1/3. The novelty of our approach is its use of ideas underlying Hairer’s reconstruction theorem generalized to a framework allowing for Sobolev models and Sobolev modelled distributions. Moreover, we show that the corresponding lifting map is locally Lipschitz continuous with respect to the inhomogeneous Sobolev metric.

[1]  P. Bahr,et al.  Sampling: Theory and Applications , 2020, Applied and Numerical Harmonic Analysis.

[2]  S. Hensel,et al.  Modelled distributions of Triebel–Lizorkin type , 2017, Studia Mathematica.

[3]  L. Zambotti,et al.  The geometry of the space of branched rough paths , 2018, Proceedings of the London Mathematical Society.

[4]  Jacques Simon,et al.  Sobolev, Besov and Nikolskii fractional spaces: Imbeddings and comparisons for vector valued spaces on an interval , 1990 .

[5]  David J. Promel,et al.  On Sobolev rough paths , 2020, 2006.03322.

[6]  Y. Meyer.  Wavelets and Operators , 1993 .

[7]  Terry Lyons,et al.  An extension theorem to rough paths , 2007 .

[8]  A variation embedding theorem and applications , 2005, math/0511520.

[9]  Martin Hairer,et al.  A Course on Rough Paths , 2020, Universitext.

[10]  I. Daubechies.  Orthonormal bases of compactly supported wavelets , 1988 .

[11]  David J. Prömel,et al.  Optimal Extension to Sobolev Rough Paths , 2018, Potential Analysis.

[12]  J. Unterberger,et al.  Hölder-Continuous Rough Paths by Fourier Normal Ordering , 2009, 0903.2716.

[13]  H. Weber,et al.  Stochastic PDEs, Regularity Structures, and Interacting Particle Systems , 2015, 1508.03616.

[14]  École d'été de probabilités de Saint-Flour,et al.  Differential equations driven by rough paths , 2007 .

[15]  Martin Hairer.  Introduction to regularity structures , 2014, Universitext.

[16]  Martin Hairer,et al.  The reconstruction theorem in Besov spaces , 2016, 1609.04543.

[17]  DaubechiesIngrid.  Orthonormal bases of compactly supported wavelets II , 1993 .

[18]  Terry Lyons.  Di erential equations driven by rough signals , 1998 .