Skip to main content
×
Home
    • Aa
    • Aa

PARACONTROLLED DISTRIBUTIONS AND SINGULAR PDES

  • MASSIMILIANO GUBINELLI (a1), PETER IMKELLER (a2) and NICOLAS PERKOWSKI (a3)
Abstract

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 no-reply@cambridge.org 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 @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ 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.

      PARACONTROLLED DISTRIBUTIONS AND SINGULAR PDES
      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.

      PARACONTROLLED DISTRIBUTIONS AND SINGULAR PDES
      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.

      PARACONTROLLED DISTRIBUTIONS AND SINGULAR PDES
      Available formats
      ×
Copyright
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/3.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

H. Bahouri , J.-Y. Chemin  and R. Danchin , Fourier Analysis and Nonlinear Partial Differential Equations (Springer, 2011).

H. Bessaih , M. Gubinelli  and F. Russo , ‘The evolution of a random vortex filament’, Ann. Probab. 33(5) (2005), 18251855.

Z. Brzeźniak , M. Gubinelli  and M. Neklyudov , ‘Global evolution of random vortex filament equation’, Nonlinearity 26(9) (2013), 2499.

M. Caruana  and P. Friz , ‘Partial differential equations driven by rough paths’, J. Differential Equations 247(1) (2009), 140173.

M. Caruana , P. K. Friz  and 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) (2011), 2746.

A. Deya , M. Gubinelli  and S. Tindel , ‘Non-linear rough heat equations’, Probab. Theory Related Fields 153(1–2) (2012), 97147.

J. Diehl  and P. Friz , ‘Backward stochastic differential equations with rough drivers’, Ann. Probab. 40(4) (2012), 17151758.

P. Friz  and H. Oberhauser , ‘On the splitting-up method for rough (partial) differential equations’, J. Differential Equations 251(2) (2011), 316338.

M. Gubinelli , A. Lejay  and S. Tindel , ‘Young integrals and SPDEs’, Potential Anal. 25(4) (2006), 307326.

M. Gubinelli , ‘Controlling rough paths’, J. Funct. Anal. 216(1) (2004), 86140.

M. Gubinelli , ‘Rough solutions for the periodic Korteweg–de Vries equation’, Commun. Pure Appl. Anal. 11(2) (2012), 709733.

M. Hairer , ‘Solving the KPZ equation’, Ann. of Math. (2) 178(2) (2013), 559664.

M. Hairer , ‘A theory of regularity structures’, Invent. Math. 198(2) (2014), 269504.

M. Hairer , J. Maas  and H. Weber , ‘Approximating rough stochastic PDEs’, Commun. Pure Appl. Math. 67(5) (2014), 776870.

M. Hairer  and H. Weber , ‘Rough Burgers-like equations with multiplicative noise’, Probab. Theory Related Fields 155(1–2) (2013), 71126.

Y. Hu , ‘Chaos expansion of heat equations with white noise potentials’, Potential Anal. 16(1) (2002), 4566.

M. Kardar , G. Parisi  and Y.-C. Zhang , ‘Dynamic scaling of growing interfaces’, Phys. Rev. Lett. 56(9) (1986), 889892.

T. Lyons  and Z. Qian , System Control and Rough Paths (Oxford University Press, 2002).

D. Nualart  and S. Tindel , ‘A construction of the rough path above fractional Brownian motion using Volterra’s representation’, Ann. Probab. 39(3) (2011), 10611096.

J. Teichmann , ‘Another approach to some rough and stochastic partial differential equations’, Stoch. Dyn. 11(2–3) (2011), 535550.

J. Unterberger , ‘A rough path over multidimensional fractional Brownian motion with arbitrary Hurst index by Fourier normal ordering’, Stochastic Process. Appl. 120(8) (2010), 14441472.

J. Unterberger , ‘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? *
×
MathJax

Keywords:

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 328 *
Loading metrics...

Abstract views

Total abstract views: 407 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 18th August 2017. This data will be updated every 24 hours.