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An extension of the sewing lemma to hyper-cubes and hyperbolic equations driven by multi-parameter Young fields


This article extends the celebrated Sewing lemma, known from the theory of rough paths, to multi-parameter fields on hyper-cubes. We use this to construct Young integrals for multi-parameter Hölder fields on general domains \(\left[ 0,T\right] ^{k}\) with \(k\ge 1\) taking values in \({\mathbb {R}}^{d}\). Moreover, we show existence, uniqueness and stability of some particular types of hyperbolic SPDEs driven by space-time Hölder noise in a Young regime.

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  1. Lyons, T.: Differential Equations driven by rough signals. Rev. Mat. Iberoam. 14, 215–310 (1998)

    Article  MathSciNet  Google Scholar 

  2. Young, L.C.: An inequality of the Hölder type, connected with Stieltjes integration. Acta Mathematica 67, 251 (1936)

    Article  MathSciNet  Google Scholar 

  3. Victoir, N., Friz, P.: Multidimensional Stochastic Processes as Rough Paths. Cambridge Studies in Advanced Mathematics (2009)

  4. Caruana, M., Lyons, T., Lévy, T.: Differential Equations driven by Rough Paths. Springer Lecture Series (2004)

  5. Hairer, M., Friz, P.: A Course on Rough Paths with an Introduction to Regularity Structures. Springer, Berlin (2014)

    MATH  Google Scholar 

  6. Cairoli, R., Walsh, J.: Stochastic integral in the plane. Acta Math. 134, 111 (1975)

    Article  MathSciNet  Google Scholar 

  7. Barndorff-Nielsen, O., Benth, F.E., Veraart, A.: Ambit Stochastics. Springer, Berlin (2018)

    Book  Google Scholar 

  8. Barndorff-Nielsen, O., Benth, F.E., Pedersen, J., Veraart, A.: On stochastic integration for volatility modulated Levy driven Volterra processes. arXiv:1205.3275v1 (2012)

  9. Quer-Sardanyons, L., Tindel, S.: The 1-d stochastic wave equation driven by fractional Brownian sheet. Stoch. Process. Theory Appl. 117, 1448–1472 (2007)

    Article  MathSciNet  Google Scholar 

  10. Towghi, N.: Multidimensional extension of LC Young’s inequality. J. Inequal. Pure Appl. Math. 3(2), 22 (2001)

    MathSciNet  MATH  Google Scholar 

  11. Gulisashvili, A., Friz, P., Gess, B., Riedel, S.: The Jain Monrad criterion for rough paths and applications to random Fourier series and non Markovian Hormander theory. Ann. Probab. 44(1), 684–738 (2016)

    MathSciNet  MATH  Google Scholar 

  12. Gubinelli, M., Chouk, K.: Rough sheets. arXiv:1406.7748v1 (2014)

  13. Gubinelli, M.: Controlling rough paths. J. Funct. Anal. 216, 86–140 (2004)

    Article  MathSciNet  Google Scholar 

  14. Gubinelli, M., Tindel, S.: Rough evolution equations. Ann. Probab. 38(1), 1–75 (2010)

    Article  MathSciNet  Google Scholar 

  15. Khoshnevisan, D.: Multi-parameter Processes—An Introduction to Random Fields. Springer, Berlin (2002)

    MATH  Google Scholar 

  16. Hu, Y., Le, K.: A multi-parameter Garsia Rodemich Rumsey Inequality and some applications. Stoch. Process. Appl. 123(9), 3359–3377 (2013)

    Article  Google Scholar 

  17. Victoir, N., Friz, P.: A note on higher dimensional p-variation. Electron. J. Probab. 16, 1880–1899 (2011)

    MathSciNet  MATH  Google Scholar 

  18. Hardy, M.: Combinatorics of partial derivatives. Electron. J. Combin. 13, 1 (2006)

    Article  MathSciNet  Google Scholar 

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The author is grateful for all the assistance, comments and advice received from Professor Frank N. Proske during the work on this project. Furthermore, we thank Professor Fred E. Benth and Professor Samy Tindel, for great discussions and comments on this project. At last we would like to thank the anonymous referee, whose detailed report with comments and suggestions is highly appreciated, and has helped improve the article to its current form.

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Correspondence to Fabian A. Harang.

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Harang, F.A. An extension of the sewing lemma to hyper-cubes and hyperbolic equations driven by multi-parameter Young fields. Stoch PDE: Anal Comp 9, 746–788 (2021).

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