Geodesic
PARACONTROLLED DISTRIBUTIONS AND SINGULAR PDES
Forum of Mathematics, Pi, Tome 3 (2015)

Voir la notice de l'article provenant de la source Cambridge University Press

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. 
@article{10_1017_fmp_2015_2,
     author = {MASSIMILIANO GUBINELLI and PETER IMKELLER and NICOLAS PERKOWSKI},
     title = {PARACONTROLLED {DISTRIBUTIONS} {AND} {SINGULAR} {PDES}},
     journal = {Forum of Mathematics, Pi},
     publisher = {mathdoc},
     volume = {3},
     year = {2015},
     doi = {10.1017/fmp.2015.2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/fmp.2015.2/}
}
TY  - JOUR
AU  - MASSIMILIANO GUBINELLI
AU  - PETER IMKELLER
AU  - NICOLAS PERKOWSKI
TI  - PARACONTROLLED DISTRIBUTIONS AND SINGULAR PDES
JO  - Forum of Mathematics, Pi
PY  - 2015
VL  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.1017/fmp.2015.2/
DO  - 10.1017/fmp.2015.2
LA  - en
ID  - 10_1017_fmp_2015_2
ER  - 
%0 Journal Article
%A MASSIMILIANO GUBINELLI
%A PETER IMKELLER
%A NICOLAS PERKOWSKI
%T PARACONTROLLED DISTRIBUTIONS AND SINGULAR PDES
%J Forum of Mathematics, Pi
%D 2015
%V 3
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.1017/fmp.2015.2/
%R 10.1017/fmp.2015.2
%G en
%F 10_1017_fmp_2015_2
MASSIMILIANO GUBINELLI; PETER IMKELLER; NICOLAS PERKOWSKI. PARACONTROLLED DISTRIBUTIONS AND SINGULAR PDES. Forum of Mathematics, Pi, Tome 3 (2015). doi: 10.1017/fmp.2015.2

[BCD11] Bahouri, H., Chemin, J.-Y. and Danchin, R., Fourier Analysis and Nonlinear Partial Differential Equations (Springer, 2011).Google Scholar

[BGR05] Bessaih, H., Gubinelli, M. and Russo, F., ‘The evolution of a random vortex filament’, Ann. Probab. 33(5) (2005), 1825–1855.Google Scholar

[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), 209–246.Google Scholar

[BGN13] Brzeźniak, Z., Gubinelli, M. and Neklyudov, M., ‘Global evolution of random vortex filament equation’, Nonlinearity 26(9) (2013), 2499.Google Scholar

[CM94] Carmona, R. A. and Molchanov, S. A., Parabolic Anderson Problem and Intermittency (American Mathematical Society, 1994).Google Scholar | DOI

[CF09] Caruana, M. and Friz, P., ‘Partial differential equations driven by rough paths’, J. Differential Equations 247(1) (2009), 140–173.Google Scholar

[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), 27–46.Google Scholar

[CC13] Catellier, R. and Chouk, K., ‘Paracontrolled distributions and the 3-dimensional stochastic quantization equation’. Preprint, 2013, arXiv:1310.6869.Google Scholar

[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), 679–698.Google Scholar

[CGP15] 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

[CG14] Chouk, K. and Gubinelli, M., ‘Rough sheets’, Preprint, 2014, arXiv:1406.7748.Google Scholar

[DGT12] Deya, A., Gubinelli, M. and Tindel, S., ‘Non-linear rough heat equations’, Probab. Theory Related Fields 153(1–2) (2012), 97–147.Google Scholar

[DF12] Diehl, J. and Friz, P., ‘Backward stochastic differential equations with rough drivers’, Ann. Probab. 40(4) (2012), 1715–1758.Google Scholar | DOI

[FGGR12] Friz, P. K., Gess, B., Gulisashvili, A. and Riedel, S., ‘Spatial rough path lifts of stochastic convolutions’, Preprint, 2012, arXiv:1211.0046.Google Scholar

[FO11] 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

[FV10] Friz, P. and Victoir, N., Multidimensional Stochastic Processes as Rough Paths. Theory and Applications (Cambridge University Press, 2010).Google Scholar | DOI

[GIP14] Gubinelli, M., Imkeller, P. and Perkowski, N., ‘A Fourier approach to pathwise stochastic integration’, Preprint, 2014, arXiv:1410.4006.Google Scholar

[GLT06] Gubinelli, M., Lejay, A. and Tindel, S., ‘Young integrals and SPDEs’, Potential Anal. 25(4) (2006), 307–326.Google Scholar

[GP15] Gubinelli, M. and Perkowski, N., ‘KPZ reloaded’ (2015), in preparation.Google Scholar

[GP15a] Gubinelli, M. and Perkowski, N., ‘Lectures on singular stochastic PDEs’, Preprint, 2015, arXiv:1502.00157.Google Scholar

[Gub04] Gubinelli, M., ‘Controlling rough paths’, J. Funct. Anal. 216(1) (2004), 86–140.Google Scholar

[Gub12] Gubinelli, M., ‘Rough solutions for the periodic Korteweg–de Vries equation’, Commun. Pure Appl. Anal. 11(2) (2012), 709–733.Google Scholar

[Hai11] Hairer, M., ‘Rough stochastic PDEs’, Commun. Pure Appl. Math. 64(11) (2011), 1547–1585.Google Scholar

[Hai13] Hairer, M., ‘Solving the KPZ equation’, Ann. of Math. (2) 178(2) (2013), 559–664.Google Scholar | DOI

[Hai14] Hairer, M., ‘A theory of regularity structures’, Invent. Math. 198(2) (2014), 269–504.Google Scholar | DOI

[HMW14] Hairer, M., Maas, J. and Weber, H., ‘Approximating rough stochastic PDEs’, Commun. Pure Appl. Math. 67(5) (2014), 776–870.Google Scholar

[HW13] Hairer, M. and Weber, H., ‘Rough Burgers-like equations with multiplicative noise’, Probab. Theory Related Fields 155(1–2) (2013), 71–126.Google Scholar

[Hu2002] Hu, Y., ‘Chaos expansion of heat equations with white noise potentials’, Potential Anal. 16(1) (2002), 45–66.Google Scholar

[Jan97] Janson, S., Gaussian Hilbert Spaces, Cambridge Tracts in Mathematics 129 (Cambridge University Press, Cambridge, 1997).Google Scholar

[KPZ86] Kardar, M., Parisi, G. and Zhang, Y.-C., ‘Dynamic scaling of growing interfaces’, Phys. Rev. Lett. 56(9) (1986), 889–892.Google Scholar

[Kön15] König, W., ‘The parabolic Anderson model’. Available at http://www.wias-berlin.de/people/koenig/www/PAMsurveyBook.pdf (2015), in preparation.Google Scholar

[Lyo98] Lyons, T. J., ‘Differential equations driven by rough signals’, Rev. Mat. Iberoam. 14(2) (1998), 215–310.Google Scholar

[LCL07] Lyons, T. J., Caruana, M. and Lévy, T., Differential Equations Driven by Rough Paths, Lecture Notes in Mathematics 1908 (Springer, Berlin, 2007).Google Scholar

[LQ02] Lyons, T. and Qian, Z., System Control and Rough Paths (Oxford University Press, 2002).Google Scholar

[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), 1061–1096.Google Scholar

[Per14] Perkowski, N., ‘Studies of Robustness in Stochastic Analysis and Mathematical Finance’, PhD Thesis, Humboldt-Universität zu Berlin, 2014.Google Scholar

[ST87] 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

[Tei11] Teichmann, J., ‘Another approach to some rough and stochastic partial differential equations’, Stoch. Dyn. 11(2–3) (2011), 535–550.Google Scholar

[Tri06] Triebel, H., Theory of Function Spaces III, Monographs in Mathematics 100 (Birkhäuser Verlag, Basel, 2006).Google Scholar

[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), 1444–1472.Google Scholar

[Unt10b] Unterberger, J., ‘Hölder—continuous rough paths by fourier normal ordering’, Commun. Math. Phys. 298(1) (2010), 1–36.Google Scholar

Cité par Sources :