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PARACONTROLLED DISTRIBUTIONS AND SINGULAR PDES

Part of: Miscellaneous topics - Partial differential equations Stochastic analysis

Published online by Cambridge University Press:  14 August 2015

MASSIMILIANO GUBINELLI ,
PETER IMKELLER  and
NICOLAS PERKOWSKI
MASSIMILIANO GUBINELLI
Affiliation:
CEREMADE & CNRS UMR 7534, Université Paris-Dauphine and Institut Universitaire de France, France; gubinelli@ceremade.dauphine.fr
PETER IMKELLER
Affiliation:
Institut für Mathematik, Humboldt-Universität zu Berlin, Germany; imkeller@math.hu-berlin.de
NICOLAS PERKOWSKI
Affiliation:
CEREMADE & CNRS UMR 7534, Université Paris-Dauphine, France; perkowski@ceremade.dauphine.fr

Abstract

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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.

MSC classification

Primary: 60H15: Stochastic partial differential equations
Secondary: 35S50: Paradifferential operators
Type
Research Article
Information
Forum of Mathematics, Pi , Volume 3 , 2015 , e6
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Creative Commons
Creative Common License - CCCreative Common License - BY
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.
Copyright
© The Author(s) 2015

References

Bahouri, H., Chemin, J.-Y. and Danchin, R., Fourier Analysis and Nonlinear Partial Differential Equations (Springer, 2011).Google Scholar
Bessaih, H., Gubinelli, M. and Russo, F., ‘The evolution of a random vortex filament’, Ann. Probab. 33(5) (2005), 1825–1855.Google Scholar
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), 209–246.Google Scholar
Brzeźniak, Z., Gubinelli, M. and Neklyudov, M., ‘Global evolution of random vortex filament equation’, Nonlinearity 26(9) (2013), 2499.Google Scholar
Carmona, R. A. and Molchanov, S. A., Parabolic Anderson Problem and Intermittency (American Mathematical Society, 1994).CrossRefGoogle Scholar
Caruana, M. and Friz, P., ‘Partial differential equations driven by rough paths’, J. Differential Equations 247(1) (2009), 140–173.Google Scholar
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), 27–46.Google Scholar
Catellier, R. and Chouk, K., ‘Paracontrolled distributions and the 3-dimensional stochastic quantization equation’. Preprint, 2013, arXiv:1310.6869.Google Scholar
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), 679–698.Google Scholar
Chouk, K., Gairing, J. and Perkowski, N., ‘An invariance principle for the two-dimensional parabolic Anderson model with small potential’ (2015), in preparation.Google Scholar
Chouk, K. and Gubinelli, M., ‘Rough sheets’, Preprint, 2014, arXiv:1406.7748.Google Scholar
Deya, A., Gubinelli, M. and Tindel, S., ‘Non-linear rough heat equations’, Probab. Theory Related Fields 153(1–2) (2012), 97–147.Google Scholar
Diehl, J. and Friz, P., ‘Backward stochastic differential equations with rough drivers’, Ann. Probab. 40(4) (2012), 1715–1758.CrossRefGoogle Scholar
Friz, P. K., Gess, B., Gulisashvili, A. and Riedel, S., ‘Spatial rough path lifts of stochastic convolutions’, Preprint, 2012, arXiv:1211.0046.Google Scholar
Friz, P. and Oberhauser, H., ‘On the splitting-up method for rough (partial) differential equations’, J. Differential Equations 251(2) (2011), 316–338.Google Scholar
Friz, P. and Victoir, N., Multidimensional Stochastic Processes as Rough Paths. Theory and Applications (Cambridge University Press, 2010).CrossRefGoogle Scholar
Gubinelli, M., Imkeller, P. and Perkowski, N., ‘A Fourier approach to pathwise stochastic integration’, Preprint, 2014, arXiv:1410.4006.Google Scholar
Gubinelli, M., Lejay, A. and Tindel, S., ‘Young integrals and SPDEs’, Potential Anal. 25(4) (2006), 307–326.Google Scholar
Gubinelli, M. and Perkowski, N., ‘KPZ reloaded’ (2015), in preparation.Google Scholar
Gubinelli, M. and Perkowski, N., ‘Lectures on singular stochastic PDEs’, Preprint, 2015, arXiv:1502.00157.Google Scholar
Gubinelli, M., ‘Controlling rough paths’, J. Funct. Anal. 216(1) (2004), 86–140.Google Scholar
Gubinelli, M., ‘Rough solutions for the periodic Korteweg–de Vries equation’, Commun. Pure Appl. Anal. 11(2) (2012), 709–733.Google Scholar
Hairer, M., ‘Rough stochastic PDEs’, Commun. Pure Appl. Math. 64(11) (2011), 1547–1585.Google Scholar
Hairer, M., ‘Solving the KPZ equation’, Ann. of Math. (2) 178(2) (2013), 559–664.CrossRefGoogle Scholar
Hairer, M., ‘A theory of regularity structures’, Invent. Math. 198(2) (2014), 269–504.CrossRefGoogle Scholar
Hairer, M., Maas, J. and Weber, H., ‘Approximating rough stochastic PDEs’, Commun. Pure Appl. Math. 67(5) (2014), 776–870.Google Scholar
Hairer, M. and Weber, H., ‘Rough Burgers-like equations with multiplicative noise’, Probab. Theory Related Fields 155(1–2) (2013), 71–126.Google Scholar
Hu, Y., ‘Chaos expansion of heat equations with white noise potentials’, Potential Anal. 16(1) (2002), 45–66.Google Scholar
Janson, S., Gaussian Hilbert Spaces, Cambridge Tracts in Mathematics 129 (Cambridge University Press, Cambridge, 1997).Google Scholar
Kardar, M., Parisi, G. and Zhang, Y.-C., ‘Dynamic scaling of growing interfaces’, Phys. Rev. Lett. 56(9) (1986), 889–892.Google Scholar
König, W., ‘The parabolic Anderson model’. Available at http://www.wias-berlin.de/people/koenig/www/PAMsurveyBook.pdf (2015), in preparation.Google Scholar
Lyons, T. J., ‘Differential equations driven by rough signals’, Rev. Mat. Iberoam. 14(2) (1998), 215–310.Google Scholar
Lyons, T. J., Caruana, M. and Lévy, T., Differential Equations Driven by Rough Paths, Lecture Notes in Mathematics 1908 (Springer, Berlin, 2007).Google Scholar
Lyons, T. and Qian, Z., System Control and Rough Paths (Oxford University Press, 2002).Google Scholar
Nualart, D. and Tindel, S., ‘A construction of the rough path above fractional Brownian motion using Volterra’s representation’, Ann. Probab. 39(3) (2011), 1061–1096.Google Scholar
Perkowski, N., ‘Studies of Robustness in Stochastic Analysis and Mathematical Finance’, PhD Thesis, Humboldt-Universität zu Berlin, 2014.Google Scholar
Schmeisser, H.-J. and Triebel, H., Topics in Fourier Analysis and Function Spaces, Vol. 42 (Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig, 1987).Google Scholar
Teichmann, J., ‘Another approach to some rough and stochastic partial differential equations’, Stoch. Dyn. 11(2–3) (2011), 535–550.Google Scholar
Triebel, H., Theory of Function Spaces III, Monographs in Mathematics 100 (Birkhäuser Verlag, Basel, 2006).Google Scholar
Unterberger, J., ‘A rough path over multidimensional fractional Brownian motion with arbitrary Hurst index by Fourier normal ordering’, Stochastic Process. Appl. 120(8) (2010), 1444–1472.Google Scholar
Unterberger, J., ‘Hölder—continuous rough paths by fourier normal ordering’, Commun. Math. Phys. 298(1) (2010), 1–36.Google Scholar