On A Priori Estimates for Rough PDEs

Conference paper
Part of the Progress in Probability book series (PRPR, volume 72)


In this note, we present a new and simple method which allows to get a priori bounds on rough partial differential equations. The technique is based on a weak formulation of the equation and a rough version of Gronwall’s lemma. The method is presented on a simple linear example, but might be generalized to a wide number of situations.


A priori estimate Rough Gonwall lemma Rough paths Stochastic PDEs 


  1. 1.
    I. Bailleul, M. Gubinelli, Unbounded rough drivers (2015). arXiv:1501.02074Google Scholar
  2. 2.
    M. Caruana, P. Friz, H. Oberhauser, A (rough) pathwise approach to a class of non-linear stochastic partial differential equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 28(1), 27–46 (2011)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    L. Chen, Y. Hu, K. Kalbasi, D. Nualart, Intermittency for the stochastic heat equation driven by a rough time fractional Gaussian noise (2016). arXiv:1602.05617Google Scholar
  4. 4.
    A. Deya, A discrete approach to rough parabolic equations. Electron. J. Probab. 16(54), 1489–1518 (2011)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    A. Deya, M. Gubinelli, S. Tindel, Non-linear rough heat equations. Probab. Theory Relat. Fields 153, 97–147 (2012)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    A. Deya, M. Gubinelli, M. Hofmanova, S. Tindel, General a priori estimates for rough PDEs with applications to rough conservation laws (2016). arXiv:1604.00437Google Scholar
  7. 7.
    P. Friz, M. Hairer, A Course on Rough Paths (Springer, Berlin, 2014)CrossRefMATHGoogle Scholar
  8. 8.
    P. Friz, N. Victoir, Multidimensional Dimensional Processes Seen as Rough Paths (Cambridge University Press, Cambridge, 2010)CrossRefMATHGoogle Scholar
  9. 9.
    M. Gubinelli, S. Tindel, Rough evolution equations. Ann. Probab. 38(1), 1–75 (2010)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    M. Gubinelli, P. Imkeller, N. Perkowski, Paracontrolled distributions and singular PDEs. Forum Math. Pi 3(e6), 75 (2015)Google Scholar
  11. 11.
    M. Gubinelli, S. Tindel, I. Torrecilla, Controlled viscosity solutions of fully nonlinear rough PDEs (2014). arXiv:1403.2832Google Scholar
  12. 12.
    M. Hairer, Solving the KPZ equation. Ann. Math. 178(2), 559–664 (2013)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    M. Hairer, A theory of regularity structures. Invent. Math. 198(2), 269–504 (2014)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    S. Mallat, A Wavelet Tour of Signal Processing (Academic, London, 1998)MATHGoogle Scholar
  15. 15.
    D. Nualart, P.-A. Vuillermot, Variational solutions for partial differential equations driven by a fractional noise. J. Funct. Anal. 232, 390–454 (2006)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of MathematicsPurdue UniversityWest LafayetteUSA

Personalised recommendations