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 or -dimensional parabolic Anderson model equation (PAM)
the equation of stochastic quantization
or the one dimensional KPZ equation
to name but a few examples. In each of these equations, the letter stands for a space or time/space white noise who is so irregular that we do not expect any solution of the equation to be regular enough for the nonlinear terms, or the product , 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.
@article{JEDP_2016____A1_0, author = {Bailleul, Isma\"el}, title = {Paracontrolled calculus}, journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles}, note = {talk:1}, pages = {1--11}, publisher = {Groupement de recherche 2434 du CNRS}, year = {2016}, doi = {10.5802/jedp.642}, language = {en}, url = {http://www.numdam.org./articles/10.5802/jedp.642/} }
TY - JOUR AU - Bailleul, Ismaël TI - Paracontrolled calculus JO - Journées équations aux dérivées partielles N1 - talk:1 PY - 2016 SP - 1 EP - 11 PB - Groupement de recherche 2434 du CNRS UR - http://www.numdam.org./articles/10.5802/jedp.642/ DO - 10.5802/jedp.642 LA - en ID - JEDP_2016____A1_0 ER -
Bailleul, Ismaël. Paracontrolled calculus. Journées équations aux dérivées partielles (2016), Exposé no. 1, 11 p. doi : 10.5802/jedp.642. http://www.numdam.org./articles/10.5802/jedp.642/
[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).
[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 model, arXiv:1508.05613.
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