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Stochastic PDEs, Regularity structures, and interacting particle systems
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 26 (2017) no. 4, pp. 847-909.

These lecture notes grew out of a series of lectures given by the second named author in short courses in Toulouse, Matsumoto, and Darmstadt. The main aim is to explain some aspects of the theory of “Regularity structures” developed recently by Hairer in [27]. This theory gives a way to study well-posedness for a class of stochastic PDEs that could not be treated previously. Prominent examples include the KPZ equation as well as the dynamic Φ 3 4 model.

Such equations can be expanded into formal perturbative expansions. Roughly speaking the theory of regularity structures provides a way to truncate this expansion after finitely many terms and to solve a fixed point problem for the “remainder”. The key ingredient is a new notion of “regularity” which is based on the terms of this expansion.

Ces notes sont basées sur trois cours que le deuxième auteur a donnés à Toulouse, Matsumoto et Darmstadt. L’objectif principal est d’expliquer certains aspects de la théorie des « structures de régularité » développée récemment par Hairer [27]. Cette théorie permet de montrer que certaines EDP stochastiques, qui ne pouvaient pas être traitées auparavant, sont bien posées. Parmi les exemples se trouvent l’équation KPZ et le modèle Φ 3 4 dynamique.

Telles équations peuvent être développées en séries perturbatives formelles. La théorie des structures de régularité permet de tronquer ce développement aprés un nombre fini de termes, et de résoudre un problème de point fixe pour le reste. L’idée principale est une nouvelle notion de régularité des distributions, qui dépend des termes de ce développement.

Published online:
DOI: 10.5802/afst.1555
Ajay Chandra 1; Hendrik Weber 1

1 University of Warwick
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Ajay Chandra; Hendrik Weber. Stochastic PDEs, Regularity structures, and interacting particle systems. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 26 (2017) no. 4, pp. 847-909. doi : 10.5802/afst.1555. https://afst.centre-mersenne.org/articles/10.5802/afst.1555/

[1] Michael Aizenman Geometric analysis of ϕ 4 fields and Ising models. I, II, Commun. Math. Phys., Volume 86 (1982) no. 1, pp. 1-48 http://0-projecteuclid.org.pugwash.lib.warwick.ac.uk/getRecord?id=euclid.cmp/1103921614 | Article | MR: 678000 (84f:81078) | Zbl: 0533.58034

[2] Gideon Amir; Ivan Corwin; Jeremy Quastel Probability distribution of the free energy of the continuum directed random polymer in 1+1 dimensions, Commun. Pure Appl. Math., Volume 64 (2011) no. 4, pp. 466-537 | Article | MR: 2796514 (2012b:60304) | Zbl: 1222.82070

[3] Hajer Bahouri; Jean-Yves Chemin; Raphaël Danchin Fourier analysis and nonlinear partial differential equations, Grundlehren der Mathematischen Wissenschaften, Volume 343, Springer, 2011, xvi+523 pages | Zbl: 1227.35004

[4] Lorenzo Bertini; Giambattista Giacomin Stochastic Burgers and KPZ equations from particle systems, Commun. Math. Phys., Volume 183 (1997) no. 3, pp. 571-607 | Article | MR: 1462228 (99e:60212) | Zbl: 0874.60059

[5] Lorenzo Bertini; Errico Presutti; Barbara Rüdiger; Ellen Saada Dynamical fluctuations at the critical point: convergence to a nonlinear stochastic PDE, Teor. Veroyatnost. i Primenen., Volume 38 (1993) no. 4, pp. 689-741 | Article | MR: 1317994 (96m:60235) | Zbl: 0819.60070

[6] Jean Bourgain Invariant measures for the 2D-defocusing nonlinear Schrödinger equation, Commun. Math. Phys., Volume 176 (1996) no. 2, pp. 421-445 | Article | Zbl: 0852.35131

[7] David C Brydges; Jürg Fröhlich; Alan D Sokal A new proof of the existence and nontriviality of the continuum φ 2 4 and φ 3 4 quantum field theories, Commun. Math. Phys., Volume 91 (1983) no. 2, pp. 141-186 | Article

[8] Rémi Catellier; Khalil Chouk Paracontrolled distributions and the 3-dimensional stochastic quantization equation (2013) (https://arxiv.org/abs/1310.6869v1)

[9] Ivan Corwin; Jeremy Quastel Renormalization fixed point of the KPZ universality class, J. Stat. Phys., Volume 160 (2015) no. 4, pp. 815-834 | Article | Zbl: 1327.82064

[10] Laure Coutin; Zhongmin Qian Stochastic analysis, rough path analysis and fractional Brownian motions, Probab. Theory Relat. Fields, Volume 122 (2002) no. 1, pp. 108-140 | Article | Zbl: 1047.60029

[11] Giuseppe Da Prato; Arnaud Debussche Strong solutions to the stochastic quantization equations, Ann. Probab., Volume 31 (2003) no. 4, pp. 1900-1916 | Article | MR: 2016604 (2005e:81117) | Zbl: 1071.81070

[12] Giuseppe Da Prato; Jerzy Zabczyk Stochastic equations in infinite dimensions, Encyclopedia of Mathematics and its Applications, Volume 44, Cambridge University Press, Cambridge, 1992, xviii+454 pages | Article | MR: 1207136 (95g:60073) | Zbl: 076.60052

[13] Joel Feldman The λφ 3 4 field theory in a finite volume, Commun. Math. Phys., Volume 37 (1974) no. 2, pp. 93-120 | Article

[14] Jochen Fritz; Bernd Rüdiger Time dependent critical fluctuations of a one-dimensional local mean field model, Probab. Theory Relat. Fields, Volume 103 (1995) no. 3, pp. 381-407 | Article | MR: 1358083 (97a:60142) | Zbl: 0833.60095

[15] Peter K. Friz; Martin Hairer A course on rough paths, Universitext, Springer, 2014, xiv+251 pages | Zbl: 1327.60013

[16] Jurg Fröhlich On the triviality of ΛΦ 4 theories and the approach to the critical-point in D4-dimensions, Nuclear Physics B, Volume 200 (1982) no. 2, pp. 281-296 | Article

[17] Tadahisa Funaki Random motion of strings and related stochastic evolution equations, Nagoya Math. J., Volume 89 (1983), pp. 129-193 | Article | Zbl: 0531.60095

[18] Giambattista Giacomin; Joel L. Lebowitz; Errico Presutti Deterministic and stochastic hydrodynamic equations arising from simple microscopic model systems, Stochastic partial differential equations: six perspectives (Mathematical Surveys and Monographs) Volume 64, American Mathematical Society, 1999, pp. 107-152 | Article | MR: 1661764 (2000f:60151) | Zbl: 0927.60060

[19] James Glimm; Arthur Jaffe Quantum physics. A functional integral point of view, Springer, 1981, xx+417 pages | Zbl: 0461.46051

[20] James Glimm; Arthur Jaffe; Thomas Spencer The Wightman axioms and particle structure in the P(ϕ) 2 quantum field model, Ann. Math., Volume 100 (1974) no. 3, pp. 585-632 | Article

[21] James Glimm; Arthur Jaffe; Thomas Spencer Phase transitions for ϕ 2 4 quantum fields, Commun. Math. Phys., Volume 45 (1975) no. 3, pp. 203-216 | Article | Zbl: 0956.82501

[22] Massimiliano Gubinelli Controlling rough paths, J. Funct. Anal., Volume 216 (2004) no. 1, pp. 86-140 | Article | MR: MR2091358 (2005k:60169) | Zbl: 1058.60037

[23] Massimiliano Gubinelli; Peter Imkeller; Nicolas Perkowski Paracontrolled distributions and singular PDEs, Forum Math. Pi, Volume 3 (2015) (Article ID e6, 75 p.) | Article | Zbl: 1333.60149

[24] Martin Hairer An Introduction to Stochastic PDEs (2009) (https://arxiv.org/abs/0907.4178)

[25] Martin Hairer Solving the KPZ equation, Ann. Math., Volume 178 (2013) no. 2, pp. 559-664 | Article | Zbl: 1281.60060

[26] Martin Hairer Singular Stochastic PDES (2014) (https://arxiv.org/abs/1403.6353)

[27] Martin Hairer A theory of regularity structures, Invent. Math., Volume 198 (2014) no. 2, pp. 269-504 | Article | Zbl: 1332.60093

[28] Martin Hairer Introduction to regularity structures, Braz. J. Probab. Stat., Volume 29 (2015) no. 2, pp. 175-210 | Article | Zbl: 1316.81061

[29] Martin Hairer; Cyril Labbé Multiplicative stochastic heat equations on the whole space (2015) (https://arxiv.org/abs/1504.07162, to appear in J. Eur. Math. Soc.)

[30] Martin Hairer; Étienne Pardoux A Wong-Zakai theorem for Stochastic PDEs, J. Math. Soc. Japan, Volume 67 (2015) no. 4, pp. 1551-1604 | Article | Zbl: 1341.60062

[31] Martin Hairer; Marc D. Ryser; Hendrik Weber Triviality of the 2D stochastic Allen-Cahn equation, Electron. J. Probab, Volume 17 (2012) no. 39, pp. 1-14 | Zbl: 1245.60063

[32] Mehran Kardar; Giorgio Parisi; Yi-Cheng Zhang Dynamic Scaling of Growing Interfaces, Phys. Rev. Lett., Volume 56 (1986) no. 9, pp. 889-892 | Article | Zbl: 1101.82329

[33] Peter E. Kloeden; Eckhard Platen Numerical solution of stochastic differential equations, Applications of Mathematics, Volume 23, Springer, 1992, xxxvi+632 pages | Article | MR: 1214374 (94b:60069) | Zbl: 0752.60043

[34] Nicolaĭ Vladimirovich Krylov Lectures on Elliptic and Parabolic Equations in Hölder Spaces, Graduate Studies in Mathematics, Volume 12, American Mathematical Society, 1996, xii+164 pages | Zbl: 0865.35001

[35] Antti Kupiainen Renormalization Group and Stochastic PDEs, Ann. Henri Poincaré, Volume 17 (2015) no. 3, pp. 1-39 | Article | Zbl: 1347.81063

[36] Terry J. Lyons Differential equations driven by rough signals, Rev. Mat. Iberoam., Volume 14 (1998) no. 2, pp. 215-310 | Article | MR: 1654527 (2000c:60089) | Zbl: 0923.34056

[37] Jean-Christophe Mourrat; Hendrik Weber Convergence of the two-dimensional dynamic Ising-Kac model to Φ 2 4 , Commun. Pure Appl. Math., Volume 70 (2017) no. 4, pp. 717-812 | Article | Zbl: 1364.82013

[38] Jean-Christophe Mourrat; Hendrik Weber Global well-posedness of the dynamic Φ 4 model in the plane, Ann. Probab., Volume 45 (2017) no. 4, pp. 2398-2476 | Article | Zbl: 06786085

[39] Andrea R. Nahmod; Gigliola Staffilani Almost sure well-posedness for the periodic 3D quintic nonlinear Schrödinger equation below the energy space, J. Eur. Math. Soc. (JEMS), Volume 17 (2015) no. 7, pp. 1687-1759 | Article | MR: 3361727 | Zbl: 1326.35353

[40] David Nualart The Malliavin calculus and related topics, Probability and Its Applications, Springer, 2006, xiv+382 pages | Zbl: 1099.60003

[41] Rudolf Peierls On Ising’s model of ferromagnetism, Proc. Camb. Philos. Soc., Volume 32 (1936), pp. 477-481 | Article | Zbl: 0014.33604

[42] Claudia Prévôt; Michael Röckner A concise course on stochastic partial differential equations, Lecture Notes in Mathematics, Volume 1905, Springer, Berlin, 2007, vi+144 pages | MR: 2329435 (2009a:60069) | Zbl: 1123.60001

[43] Jeremy Quastel; Herbert Spohn The one-dimensional KPZ equation and its universality class, J. Stat. Phys., Volume 160 (2015) no. 4, pp. 965-984 | Article | Zbl: 1327.82069

[44] Tomohiro Sasamoto; Herbert Spohn One-dimensional Kardar-Parisi-Zhang equation: an exact solution and its universality, Phys. Rev. Lett., Volume 104 (2010) no. 23 (Article ID 230602) | Article

[45] Kazumasa A. Takeuchi; Masaki Sano Universal Fluctuations of Growing Interfaces: Evidence in Turbulent Liquid Crystals, Phys. Rev. Lett., Volume 104 (2010) no. 23 http://link.aps.org/doi/10.1103/PhysRevLett.104.230601 (Article ID 230601) | Article

[46] Eugene Wong; Moshe Zakai On the convergence of ordinary integrals to stochastic integrals, Ann. Math. Stat., Volume 36 (1965), pp. 1560-1564 | Article | MR: 0195142 (33 #3345) | Zbl: 0138.11201

[47] Eugene Wong; Moshe Zakai On the relation between ordinary and stochastic differential equations, Int. J. Eng. Sci., Volume 3 (1965), pp. 213-229 | Article | MR: 0183023 (32 #505) | Zbl: 0131.16401

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