Centre de diffusion de revues académiques mathématiques

 
 
 
 

Journées équations aux dérivées partielles

Table des matières de ce volume | Article suivant
Ismaël Bailleul
Paracontrolled calculus
Journées équations aux dérivées partielles (2016), Exp. No. 1, 11 p., doi: 10.5802/jedp.642
Article PDF
Class. Math.: 60H15, 35R60, 35R01

Résumé - Abstract

At the same time that Hairer introduced his theory of regularity structures, Gubinelli, Imkeller and Perkowski developed paracontrolled calculus as an alternative playground where to study a number of singular, classically ill-posed, stochastic partial differential equations, such as the $2$ or $3$-dimensional parabolic Anderson model equation (PAM)

$$ \partial _t u = \Delta u + u\zeta , $$

the $\Phi ^4_3$ equation of stochastic quantization

$$ \partial _t u = \Delta u - u^3 + \zeta , $$

or the one dimensional KPZ equation

$$ \partial _t u = \Delta u + (\partial _x u)^2 + \zeta , $$

to name but a few examples. In each of these equations, the letter $\zeta $ stands for a space or time/space white noise who is so irregular that we do not expect any solution $u$ of the equation to be regular enough for the nonlinear terms, or the product $u\zeta $, in the equations to make sense on the sole basis of the regularizing properties of the heat semigroup. Like Hairer’s theory of regularity structures, paracontrolled calculus provides a setting where one can make sense of such a priori ill-defined products, and finally give some meaning and solve some singular partial differential equations. We present here an overview of paracontrolled calculus, from its initial form to its recent extensions.

Bibliographie

[1] I. Bailleul, Flows driven by rough paths. Rev. Mat. Iberoamericana, 31(3), (2015), 901–934.
[2] I. Bailleul and F. Bernicot, Heat semigroup and singular PDEs. J. Funct. Anal., 270, (2016), 3344–3452.
[3] I. Bailleul and F. Bernicot and D. Frey, Space-time paraproducts for paracontrolled calculus, 3d-PAM and multiplicative Burgers equations. arXiv:1506.08773, (2015).
[4] I. Bailleul and F. Bernicot, Paracontrolled calculus. (2016).
[5] J.-M. Bony, Calcul symbolique et propagation des singulariés pour les équations aux dérivées partielles non linéaires, Ann. Sci. Eco. Norm. Sup. 114 (1981), 209–246.
[6] R. Catellier and K. Chouk, Paracontrolled Distributions and the 3-dimensional Stochastic Quantization Equation. To appear in Ann. Probab., (2016+).
[7] A. Chandra and H. Weber, Stochastic PDEs, regularity structures, and interacting particle systems. arXiv:1508.03616 (2015).
[8] K. Chouk and R. Allez. The continuous Anderson hamiltonian in dimension two. arXiv:1511.02718, (2015).
[9] K. Chouk and G. Cannizzaro. Multidimensional SDEs with singular drift and universal construction of the polymer measure with white noise potential. arXiv:1501.04751, (2015).
[10] M. Gubinelli, P. Imkeller and N. Perkowski, Paracontrolled distributions and singular PDEs. Forum Math. Pi, 3, (2015).
[11] M. Gubinelli and N. Perkowski, Lectures on singular stochastic PDEs. Ensaios Math., (2015).
[12] M. Gubinelli and N. Perkowski, KPZ reloaded. arXiv:1508.03877, (2015).
[13] M. Hairer, A theory of regularity structures. Invent. Math., 198 (2), (2014), 269–504.
[14] M. Hairer, Introduction to regularity structures. Braz. J. Probab. Stat., 29 (2), (2015), 175–210.
[15] T. Lyons, Differential equations driven by rough signals. Rev. Mat. Iberoamericana, 14 (2), (1998), 215–310.
[16] T.J. Lyons, On the nonexistence of path integrals. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci., 432, (1991), 281–290.
[17] M. Caruana and T. Lévy and T. Lyons, Differential equations driven by rough paths. Lect. Notes Math. 1908, (2006).  MR 2314753
[18] R. Zhu and X. Zhu, Three-dimensional Navier-Stokes equations driven by space-time white noise, arXiv:1406.0047, to appear in J. Diff. Eq., 2016.
[19] R. Zhu and X. Zhu, Approximating three-dimensional Navier-Stokes equations driven by space-time white noise, arXiv:1409.4864.
[20] R. Zhu and X. Zhu, Lattice approximation to the dynamical $\Phi ^4_3$ model, arXiv:1508.05613.
Copyright Cellule MathDoc 2018 | Crédit | Plan du site