# zbMATH — the first resource for mathematics

Stochastic PDEs, regularity structures, and interacting particle systems. (English. French summary) Zbl 1421.81001
These lecture notes deals with stochastic partial differential equations (SPDEs), regularity structures and interacting particle systems, and aims to explain some aspects of the theory of “Regularity structure” developed by Hairer. In Lecture 1 the authors outline the scope of the theory of regularity structures. It is discussed the Kardar-Parisi-Zhang equation which is given by $$\partial_{t}h(t,x)=\partial^2_{x}h(t,x)+(\partial_{x}h(t,x))^2/2+\xi(t,x)$$, where the term $$\xi(t,x)$$ denotes a spacetime white noise. The authors focus more on the second example, namely the dynamic $$\Phi_{d}^4$$ model given by $$\partial_{t}\phi(t,x)=\Delta\phi(t,x)-\phi^3(t,x)-m^2\phi(t,x)+\xi(t,x)$$. They consider the scaling behaviour for SPDEs and subcriticality. Then they discuss the need for renormalisation and consider the approximation of renormalised SPDEs by interacting particle systems. Lecture 2 starts by describing how we will keep track of the regularity of spacetime functions and distributions. Then the authors give a review of classical solution techniques for semilinear SPDEs. It is explained how a lack of regularity causes problems for these theories, using $$(\Phi_{d}^4)$$ in $$d=2,3$$ as an examples. Then they describe a perturbative approach to these equations. Lecture 3 starts by discussing the Da Prato-Debussche argument and then sketch why it fails for $$\Phi_{3}^4$$. The authors introduce some of basic objects of the theory of regularity structures: regularity structures, models, and modelled distributions. In the last Lecture 4 the authors discuss regularity structures and models associated with controlled rough paths, $$\Phi_{2}^4$$, and $$\Phi_{3}^4$$. The construction of canonical models is presented, the convergence of the models and renormalization are discussed.

##### MSC:
 81-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to quantum theory 35R60 PDEs with randomness, stochastic partial differential equations 81T15 Perturbative methods of renormalization applied to problems in quantum field theory 82C22 Interacting particle systems in time-dependent statistical mechanics
Full Text:
##### References:
  Michael Aizenman , “ Geometric analysis of $$\phi^4$$ fields and Ising models. I, II ”, Commun. Math. Phys. 86 (1982) no. 1, p. 1-48 · Zbl 0533.58034  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. 64 (2011) no. 4, p. 466-537 · Zbl 1222.82070  Hajer Bahouri , Jean-Yves Chemin & Raphaël Danchin , Fourier analysis and nonlinear partial differential equations , Grundlehren der Mathematischen Wissenschaften 343, Springer, 2011 · Zbl 1227.35004  Lorenzo Bertini & Giambattista Giacomin , “ Stochastic Burgers and KPZ equations from particle systems ”, Commun. Math. Phys. 183 (1997) no. 3, p. 571-607 · Zbl 0874.60059  Lorenzo Bertini , Errico Presutti , Barbara Rüdiger & Ellen Saada , “ Dynamical fluctuations at the critical point: convergence to a nonlinear stochastic PDE ”, Teor. Veroyatnost. i Primenen. 38 (1993) no. 4, p. 689-741 · Zbl 0819.60070  Jean Bourgain , “ Invariant measures for the 2D-defocusing nonlinear Schrödinger equation ”, Commun. Math. Phys. 176 (1996) no. 2, p. 421-445 · Zbl 0852.35131  David C Brydges , Jürg Fröhlich & Alan D Sokal , “ A new proof of the existence and nontriviality of the continuum $$\varphi_2^4$$ and $$\varphi_3^4$$ quantum field theories ”, Commun. Math. Phys. 91 (1983) no. 2, p. 141-186  Rémi Catellier & Khalil Chouk , “ Paracontrolled distributions and the 3-dimensional stochastic quantization equation ”, , 2013  Ivan Corwin & Jeremy Quastel , “ Renormalization fixed point of the KPZ universality class ”, J. Stat. Phys. 160 (2015) no. 4, p. 815-834 · Zbl 1327.82064  Laure Coutin & Zhongmin Qian , “ Stochastic analysis, rough path analysis and fractional Brownian motions ”, Probab. Theory Relat. Fields 122 (2002) no. 1, p. 108-140 · Zbl 1047.60029  Giuseppe Da Prato & Arnaud Debussche , “ Strong solutions to the stochastic quantization equations ”, Ann. Probab. 31 (2003) no. 4, p. 1900-1916 · Zbl 1071.81070  Giuseppe Da Prato & Jerzy Zabczyk , Stochastic equations in infinite dimensions , Encyclopedia of Mathematics and its Applications 44, Cambridge University Press, Cambridge, 1992 · Zbl 1140.60034  Joel Feldman , “ The $$\lambda$$$$\varphi_3^4$$ field theory in a finite volume ”, Commun. Math. Phys. 37 (1974) no. 2, p. 93-120  Jochen Fritz & Bernd Rüdiger , “ Time dependent critical fluctuations of a one-dimensional local mean field model ”, Probab. Theory Relat. Fields 103 (1995) no. 3, p. 381-407 · Zbl 0833.60095  Peter K. Friz & Martin Hairer , A course on rough paths , Universitext, Springer, 2014  Jurg Fröhlich , “ On the triviality of $$\Lambda \Phi^4$$ theories and the approach to the critical-point in D$$\geq 4$$ -dimensions ”, Nuclear Physics B 200 (1982) no. 2, p. 281-296  Tadahisa Funaki , “ Random motion of strings and related stochastic evolution equations ”, Nagoya Math. J. 89 (1983), p. 129-193 · Zbl 0531.60095  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 64, American Mathematical Society, 1999, p. 107-152 · Zbl 0927.60060  James Glimm & Arthur Jaffe , Quantum physics. A functional integral point of view , Springer, 1981 · Zbl 0461.46051  James Glimm , Arthur Jaffe & Thomas Spencer , “ The Wightman axioms and particle structure in the P$$(\phi)_2$$ quantum field model ”, Ann. Math. 100 (1974) no. 3, p. 585-632  James Glimm , Arthur Jaffe & Thomas Spencer , “ Phase transitions for $$\phi_2^4$$ quantum fields ”, Commun. Math. Phys. 45 (1975) no. 3, p. 203-216 · Zbl 0956.82501  Massimiliano Gubinelli , “ Controlling rough paths ”, J. Funct. Anal. 216 (2004) no. 1, p. 86-140 · Zbl 1058.60037  Massimiliano Gubinelli , Peter Imkeller & Nicolas Perkowski , “ Paracontrolled distributions and singular PDEs ”, Forum Math. Pi 3 (2015), Article ID e6, 75 p. · Zbl 1333.60149  Martin Hairer , “ An Introduction to Stochastic PDEs ”, , 2009  Martin Hairer , “ Solving the KPZ equation ”, Ann. Math. 178 (2013) no. 2, p. 559-664 · Zbl 1281.60060  Martin Hairer , “ Singular Stochastic PDES ”, , 2014 · Zbl 1373.60110  Martin Hairer , “ A theory of regularity structures ”, Invent. Math. 198 (2014) no. 2, p. 269-504 · Zbl 1332.60093  Martin Hairer , “ Introduction to regularity structures ”, Braz. J. Probab. Stat. 29 (2015) no. 2, p. 175-210 · Zbl 1316.81061  Martin Hairer & Cyril Labbé , “ Multiplicative stochastic heat equations on the whole space ”, , to appear in $$J. Eur. Math. Soc.$$ , 2015 · Zbl 1332.60094  Martin Hairer & Étienne Pardoux , “ A Wong-Zakai theorem for Stochastic PDEs ”, J. Math. Soc. Japan 67 (2015) no. 4, p. 1551-1604 · Zbl 1341.60062  Martin Hairer , Marc D. Ryser & Hendrik Weber , “ Triviality of the 2D stochastic Allen-Cahn equation ”, Electron. J. Probab 17 (2012) no. 39, p. 1-14 · Zbl 1245.60063  Mehran Kardar , Giorgio Parisi & Yi-Cheng Zhang , “ Dynamic Scaling of Growing Interfaces ”, Phys. Rev. Lett. 56 (1986) no. 9, p. 889-892 · Zbl 1101.82329  Peter E. Kloeden & Eckhard Platen , Numerical solution of stochastic differential equations , Applications of Mathematics 23, Springer, 1992 · Zbl 0752.60043  Nicolaĭ Vladimirovich Krylov , Lectures on Elliptic and Parabolic Equations in Hölder Spaces , Graduate Studies in Mathematics 12, American Mathematical Society, 1996  Antti Kupiainen , “ Renormalization Group and Stochastic PDEs ”, Ann. Henri Poincaré 17 (2015) no. 3, p. 1-39 · Zbl 1347.81063  Terry J. Lyons , “ Differential equations driven by rough signals ”, Rev. Mat. Iberoam. 14 (1998) no. 2, p. 215-310 · Zbl 0923.34056  Jean-Christophe Mourrat & Hendrik Weber , “ Convergence of the two-dimensional dynamic Ising-Kac model to $$\Phi_2^4$$ ”, Commun. Pure Appl. Math. 70 (2017) no. 4, p. 717-812 · Zbl 1364.82013  Jean-Christophe Mourrat & Hendrik Weber , “ Global well-posedness of the dynamic $$\Phi^4$$ model in the plane ”, Ann. Probab. 45 (2017) no. 4, p. 2398-2476 · Zbl 1381.60098  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) 17 (2015) no. 7, p. 1687-1759 · Zbl 1326.35353  David Nualart , The Malliavin calculus and related topics , Probability and Its Applications, Springer, 2006 · Zbl 1099.60003  Rudolf Peierls , “ On Ising’s model of ferromagnetism ”, Proc. Camb. Philos. Soc. 32 (1936), p. 477-481 · Zbl 0014.33604  Claudia Prévôt & Michael Röckner , A concise course on stochastic partial differential equations , Lecture Notes in Mathematics 1905, Springer, Berlin, 2007  Jeremy Quastel & Herbert Spohn , “ The one-dimensional KPZ equation and its universality class ”, J. Stat. Phys. 160 (2015) no. 4, p. 965-984 · Zbl 1327.82069  Tomohiro Sasamoto & Herbert Spohn , “ One-dimensional Kardar-Parisi-Zhang equation: an exact solution and its universality ”, Phys. Rev. Lett. 104 (2010) no. 23, Article ID 230602 · Zbl 1204.35137  Kazumasa A. Takeuchi & Masaki Sano , “ Universal Fluctuations of Growing Interfaces: Evidence in Turbulent Liquid Crystals ”, Phys. Rev. Lett. 104 (2010) no. 23, Article ID 230601 · Zbl 1246.82109  Eugene Wong & Moshe Zakai , “ On the convergence of ordinary integrals to stochastic integrals ”, Ann. Math. Stat. 36 (1965), p. 1560-1564 · Zbl 0138.11201  Eugene Wong & Moshe Zakai , “ On the relation between ordinary and stochastic differential equations ”, Int. J. Eng. Sci. 3 (1965), p. 213-229 Search for an article Search within the site · Zbl 0131.16401
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.