Abstract
We show that every \(\mathbb {R}^{d}\)-valued Sobolev path with regularity α and integrability p can be lifted to a weakly geometric rough path in the sense of T. Lyons with exactly the same regularity and integrability, provided α > 1/p > 0. Moreover, we prove the existence of unique rough path lifts which are optimal w.r.t. strictly convex functionals among all possible rough path lifts given a Sobolev path. This paves a way towards classifying rough path lifts as solutions of optimization problems. As examples, we consider the rough path lift with minimal Sobolev norm and characterize the Stratonovich rough path lift of a Brownian motion as optimal lift w.r.t. a suitable convex functional. Generalizations of the results to Besov spaces are briefly discussed.
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References
Brault, A.: Solving Rough Differential Equations with the Theory of Regularity Structures, pp. 127–164. Springer International Publishing, Cham (2019)
Coutin, L., Lejay, A.: Semi-martingales and rough paths theory. Electron. J. Probab. 10, 761–785 (2005)
Cass, T., Lyons, T.: Evolving communities with individual preferences. Proc. Lond. Math. Soc. (3) 110(1), 83–107 (2015)
Coutin, L., Qian, Z.: Stochastic analysis, rough path analysis and fractional brownian motions. Probab. Theory Relat. Fields 122(1), 108–140 (2002)
Diehl, J., Oberhauser, H., Riedel, S.: A Lévy area between brownian motion and rough paths with applications to robust nonlinear filtering and rough partial differential equations. Stochastic Process. Appl. 125(1), 161–181 (2015)
Friz, P.K., Hairer, M.: A Course on Rough Paths. Universitext, Springer, Cham (2014). With an introduction to regularity structures
Friz, P., Seeger, B.: Besov rough path analysis. arXiv:2105.05978 (2021)
Friz, P., Victoir, N.: A variation embedding theorem and applications. J. Funct. Anal. 239(2), 631–637 (2006)
Friz, P., Victoir, N.: Multidimensional Stochastic Processes as Rough Paths. Theory and Applications. Cambridge University Press, Cambridge (2010)
Gromov, M.: Carnot-Carathéodory Spaces Seen from Within, Sub-riemannian Geometry. Progr. Math., vol. 144, pp. 79–323. Birkhäuser, Basel (1996)
Lyons, T.J., Caruana, M., Lévy, T.: Differential Equations Driven by Rough Paths. Lecture Notes in Mathematics, vol. 1908. Springer, Berlin (2007)
Lejay, A.: Yet Another Introduction to Rough Paths. Séminaire de Probabilités XLII. Lecture Notes in Math, vol. 1979, pp. 1–101. Springer, Berlin (2009)
Liu, C., Prömel, D.J., Teichmann, J.: Characterization of nonlinear besov spaces. Trans. Amer. Math. Soc. 373(1), 529–550 (2020)
Liu, C., Prömel, D.J., Teichmann, J.: On Sobolev rough paths. J. Math. Anal. Appl. 497(1), 124876 (2021)
Liu, C., Prömel, D.J., Teichmann, J.: A Sobolev rough path extension theorem via regularity structures. arXiv:2104.06158 (2021)
Ledoux, T., Lyons, M., Qian, Z.: Lévy area of wiener processes in banach spaces. Ann. Probab. 30(2), 546–578 (2002)
Lyons, T., Victoir, N.: An extension theorem to rough paths. Ann. Inst. H. Poincaré Anal. Non Linéaire 24(5), 835–847 (2007)
Lyons, T.: On the nonexistence of path integrals. Proc. Roy. Soc. London Ser. A 432(1885), 281–290 (1991)
Lyons, T.J.: Differential equations driven by rough signals. Rev. Mat. Iberoam. 14(2), 215–310 (1998)
Qian, Z., Tudor, J.: Differential structure and flow equations on rough path space. Bull. Sci. Math. 135(6-7), 695–732 (2011)
Rosenbaum, M.: First order p-variations and Besov spaces. Statist. Probab. Lett. 79(1), 55–62 (2009)
Tapia, N., Zambotti, L.: The geometry of the space of branched rough paths. Proc. Lond. Math. Soc. (3) 121(2), 220–251 (2020)
Unterberger, J.: Hölder-continuous rough paths by Fourier normal ordering. Comm. Math. Phys. 298(1), 1–36 (2010)
Yang, D.: Notes on area operator, geometric 2-rough path and Young integral when p− 1 + q− 1 = 1. Int. J. Math. Anal. (Ruse) 6(33–36), 1717–1746 (2012)
Young, L.C.: An inequality of the Hölder type, connected with stieltjes integration. Acta. Math. 67(1), 251–282 (1936)
Zălinescu, C.: Convex Analysis in General Vector Spaces. World Scientific Publishing Co., Inc., River Edge (2002)
Acknowledgements
C. Liu and J. Teichmann gratefully acknowledge support by the ETH foundation. C. Liu was employed at ETH Zurich when this project was commenced. D.J. Prömel is grateful to Martin Huesmann for inspiring discussions about the problem of lifting a path in a “optimal” manner. D.J. Prömel and J. Teichmann gratefully acknowledge support by the SNF Project 163014.
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Liu, C., Prömel, D.J. & Teichmann, J. Optimal Extension to Sobolev Rough Paths. Potential Anal 59, 1399–1424 (2023). https://doi.org/10.1007/s11118-022-10017-w
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DOI: https://doi.org/10.1007/s11118-022-10017-w
Keywords
- Besov space
- Brownian motion
- Convex optimization
- Lyons–Victoir extension theorem
- Sobolev space
- Stratonovich integration
- Rough path