Skip to main content



We introduce an approach to study certain singular partial differential equations (PDEs) which is based on techniques from paradifferential calculus and on ideas from the theory of controlled rough paths. We illustrate its applicability on some model problems such as differential equations driven by fractional Brownian motion, a fractional Burgers-type stochastic PDE (SPDE) driven by space-time white noise, and a nonlinear version of the parabolic Anderson model with a white noise potential.

    • Send article to Kindle

      To send this article to your Kindle, first ensure is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle.

      Note you can select to send to either the or variations. ‘’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Available formats
      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about sending content to Dropbox.

      Available formats
      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about sending content to Google Drive.

      Available formats
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (, which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Hide All
[BCD11]Bahouri H., Chemin J.-Y. and Danchin R., Fourier Analysis and Nonlinear Partial Differential Equations (Springer, 2011).
[BGR05]Bessaih H., Gubinelli M. and Russo F., ‘The evolution of a random vortex filament’, Ann. Probab. 33(5) (2005), 18251855.
[Bon81]Bony J.-M., ‘Calcul symbolique et propagation des singularites pour les équations aux dérivées partielles non linéaires’, Ann. Sci. Éc. Norm. Supér. (4) 14 (1981), 209246.
[BGN13]Brzeźniak Z., Gubinelli M. and Neklyudov M., ‘Global evolution of random vortex filament equation’, Nonlinearity 26(9) (2013), 2499.
[CM94]Carmona R. A. and Molchanov S. A., Parabolic Anderson Problem and Intermittency (American Mathematical Society, 1994).
[CF09]Caruana M. and Friz P., ‘Partial differential equations driven by rough paths’, J. Differential Equations 247(1) (2009), 140173.
[CFO11]Caruana M., Friz P. K. and Oberhauser H., ‘A (rough) pathwise approach to a class of non-linear stochastic partial differential equations’, Ann. Inst. H. Poincaré Anal. Non Linéaire 28(1) (2011), 2746.
[CC13]Catellier R. and Chouk K., ‘Paracontrolled distributions and the 3-dimensional stochastic quantization equation’. Preprint, 2013, arXiv:1310.6869.
[CG06]Chemin J.-Y. and Gallagher I., ‘On the global wellposedness of the 3-D Navier–Stokes equations with large initial data’, Ann. Sci. Éc. Norm. Supér. (4) 39(4) (2006), 679698.
[CGP15]Chouk K., Gairing J. and Perkowski N., ‘An invariance principle for the two-dimensional parabolic Anderson model with small potential’ (2015), in preparation.
[CG14]Chouk K. and Gubinelli M., ‘Rough sheets’, Preprint, 2014, arXiv:1406.7748.
[DGT12]Deya A., Gubinelli M. and Tindel S., ‘Non-linear rough heat equations’, Probab. Theory Related Fields 153(1–2) (2012), 97147.
[DF12]Diehl J. and Friz P., ‘Backward stochastic differential equations with rough drivers’, Ann. Probab. 40(4) (2012), 17151758.
[FGGR12]Friz P. K., Gess B., Gulisashvili A. and Riedel S., ‘Spatial rough path lifts of stochastic convolutions’, Preprint, 2012, arXiv:1211.0046.
[FO11]Friz P. and Oberhauser H., ‘On the splitting-up method for rough (partial) differential equations’, J. Differential Equations 251(2) (2011), 316338.
[FV10]Friz P. and Victoir N., Multidimensional Stochastic Processes as Rough Paths. Theory and Applications (Cambridge University Press, 2010).
[GIP14]Gubinelli M., Imkeller P. and Perkowski N., ‘A Fourier approach to pathwise stochastic integration’, Preprint, 2014, arXiv:1410.4006.
[GLT06]Gubinelli M., Lejay A. and Tindel S., ‘Young integrals and SPDEs’, Potential Anal. 25(4) (2006), 307326.
[GP15]Gubinelli M. and Perkowski N., ‘KPZ reloaded’ (2015), in preparation.
[GP15a]Gubinelli M. and Perkowski N., ‘Lectures on singular stochastic PDEs’, Preprint, 2015, arXiv:1502.00157.
[Gub04]Gubinelli M., ‘Controlling rough paths’, J. Funct. Anal. 216(1) (2004), 86140.
[Gub12]Gubinelli M., ‘Rough solutions for the periodic Korteweg–de Vries equation’, Commun. Pure Appl. Anal. 11(2) (2012), 709733.
[Hai11]Hairer M., ‘Rough stochastic PDEs’, Commun. Pure Appl. Math. 64(11) (2011), 15471585.
[Hai13]Hairer M., ‘Solving the KPZ equation’, Ann. of Math. (2) 178(2) (2013), 559664.
[Hai14]Hairer M., ‘A theory of regularity structures’, Invent. Math. 198(2) (2014), 269504.
[HMW14]Hairer M., Maas J. and Weber H., ‘Approximating rough stochastic PDEs’, Commun. Pure Appl. Math. 67(5) (2014), 776870.
[HW13]Hairer M. and Weber H., ‘Rough Burgers-like equations with multiplicative noise’, Probab. Theory Related Fields 155(1–2) (2013), 71126.
[Hu2002]Hu Y., ‘Chaos expansion of heat equations with white noise potentials’, Potential Anal. 16(1) (2002), 4566.
[Jan97]Janson S., Gaussian Hilbert Spaces, Cambridge Tracts in Mathematics 129 (Cambridge University Press, Cambridge, 1997).
[KPZ86]Kardar M., Parisi G. and Zhang Y.-C., ‘Dynamic scaling of growing interfaces’, Phys. Rev. Lett. 56(9) (1986), 889892.
[Kön15]König W., ‘The parabolic Anderson model’. Available at (2015), in preparation.
[Lyo98]Lyons T. J., ‘Differential equations driven by rough signals’, Rev. Mat. Iberoam. 14(2) (1998), 215310.
[LCL07]Lyons T. J., Caruana M. and Lévy T., Differential Equations Driven by Rough Paths, Lecture Notes in Mathematics 1908 (Springer, Berlin, 2007).
[LQ02]Lyons T. and Qian Z., System Control and Rough Paths (Oxford University Press, 2002).
[NT11]Nualart D. and Tindel S., ‘A construction of the rough path above fractional Brownian motion using Volterra’s representation’, Ann. Probab. 39(3) (2011), 10611096.
[Per14]Perkowski N., ‘Studies of Robustness in Stochastic Analysis and Mathematical Finance’, PhD Thesis, Humboldt-Universität zu Berlin, 2014.
[ST87]Schmeisser H.-J. and Triebel H., Topics in Fourier Analysis and Function Spaces, Vol. 42 (Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig, 1987).
[Tei11]Teichmann J., ‘Another approach to some rough and stochastic partial differential equations’, Stoch. Dyn. 11(2–3) (2011), 535550.
[Tri06]Triebel H., Theory of Function Spaces III, Monographs in Mathematics 100 (Birkhäuser Verlag, Basel, 2006).
[Unt10a]Unterberger J., ‘A rough path over multidimensional fractional Brownian motion with arbitrary Hurst index by Fourier normal ordering’, Stochastic Process. Appl. 120(8) (2010), 14441472.
[Unt10b]Unterberger J., ‘Hölder—continuous rough paths by fourier normal ordering’, Commun. Math. Phys. 298(1) (2010), 136.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Forum of Mathematics, Pi
  • ISSN: -
  • EISSN: 2050-5086
  • URL: /core/journals/forum-of-mathematics-pi
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 2
Total number of PDF views: 416 *
Loading metrics...

Abstract views

Total abstract views: 591 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 13th December 2017. This data will be updated every 24 hours.