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Ajay Chandra; Hendrik Weber
Stochastic PDEs, Regularity structures, and interacting particle systems
Annales de la faculté des sciences de Toulouse Sér. 6, 26 no. 4 (2017), p. 847-909, doi: 10.5802/afst.1555
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[1] Michael Aizenman, “Geometric analysis of $\varphi ^{4}$ fields and Ising models. I, II”, Commun. Math. Phys. 86 (1982) no. 1, p. 1-48 Article
[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. 64 (2011) no. 4, p. 466-537 Article
[3] Hajer Bahouri, Jean-Yves Chemin & Raphaël Danchin, Fourier analysis and nonlinear partial differential equations, Grundlehren der Mathematischen Wissenschaften 343, Springer, 2011
[4] Lorenzo Bertini & Giambattista Giacomin, “Stochastic Burgers and KPZ equations from particle systems”, Commun. Math. Phys. 183 (1997) no. 3, p. 571-607 Article
[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. 38 (1993) no. 4, p. 689-741 Article
[6] Jean Bourgain, “Invariant measures for the 2D-defocusing nonlinear Schrödinger equation”, Commun. Math. Phys. 176 (1996) no. 2, p. 421-445 Article
[7] David C Brydges, Jürg Fröhlich & Alan D Sokal, “A new proof of the existence and nontriviality of the continuum $\phi _2^4$ and $\phi _3^4$ quantum field theories”, Commun. Math. Phys. 91 (1983) no. 2, p. 141-186 Article
[8] Rémi Catellier & Khalil Chouk, “Paracontrolled distributions and the 3-dimensional stochastic quantization equation”, https://arxiv.org/abs/1310.6869v1, 2013
[9] Ivan Corwin & Jeremy Quastel, “Renormalization fixed point of the KPZ universality class”, J. Stat. Phys. 160 (2015) no. 4, p. 815-834 Article
[10] Laure Coutin & Zhongmin Qian, “Stochastic analysis, rough path analysis and fractional Brownian motions”, Probab. Theory Relat. Fields 122 (2002) no. 1, p. 108-140 Article
[11] Giuseppe Da Prato & Arnaud Debussche, “Strong solutions to the stochastic quantization equations”, Ann. Probab. 31 (2003) no. 4, p. 1900-1916 Article
[12] Giuseppe Da Prato & Jerzy Zabczyk, Stochastic equations in infinite dimensions, Encyclopedia of Mathematics and its Applications 44, Cambridge University Press, Cambridge, 1992 Article
[13] Joel Feldman, “The $\lambda $$\phi ^{4}_{3}$ field theory in a finite volume”, Commun. Math. Phys. 37 (1974) no. 2, p. 93-120 Article
[14] 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 Article
[15] Peter K. Friz & Martin Hairer, A course on rough paths, Universitext, Springer, 2014
[16] Jurg Fröhlich, “On the triviality of $\Lambda \Phi ^{4}$ theories and the approach to the critical-point in D$\ge 4$-dimensions”, Nuclear Physics B 200 (1982) no. 2, p. 281-296 Article
[17] Tadahisa Funaki, “Random motion of strings and related stochastic evolution equations”, Nagoya Math. J. 89 (1983), p. 129-193 Article
[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 64, American Mathematical Society, 1999, p. 107–152 Article
[19] James Glimm & Arthur Jaffe, Quantum physics. A functional integral point of view, Springer, 1981
[20] James Glimm, Arthur Jaffe & Thomas Spencer, “The Wightman axioms and particle structure in the P$(\varphi )_2$ quantum field model”, Ann. Math. 100 (1974) no. 3, p. 585-632 Article
[21] James Glimm, Arthur Jaffe & Thomas Spencer, “Phase transitions for $\varphi _2^4$ quantum fields”, Commun. Math. Phys. 45 (1975) no. 3, p. 203-216 Article
[22] Massimiliano Gubinelli, “Controlling rough paths”, J. Funct. Anal. 216 (2004) no. 1, p. 86-140 Article
[23] Massimiliano Gubinelli, Peter Imkeller & Nicolas Perkowski, “Paracontrolled distributions and singular PDEs”, Forum Math. Pi 3 (2015), Article ID e6, 75 p. Article
[24] Martin Hairer, “An Introduction to Stochastic PDEs”, https://arxiv.org/abs/0907.4178, 2009
[25] Martin Hairer, “Solving the KPZ equation”, Ann. Math. 178 (2013) no. 2, p. 559-664 Article
[26] Martin Hairer, “Singular Stochastic PDES”, https://arxiv.org/abs/1403.6353, 2014
[27] Martin Hairer, “A theory of regularity structures”, Invent. Math. 198 (2014) no. 2, p. 269-504 Article
[28] Martin Hairer, “Introduction to regularity structures”, Braz. J. Probab. Stat. 29 (2015) no. 2, p. 175-210 Article
[29] Martin Hairer & Cyril Labbé, “Multiplicative stochastic heat equations on the whole space”, https://arxiv.org/abs/1504.07162, to appear in J. Eur. Math. Soc., 2015
[30] Martin Hairer & Étienne Pardoux, “A Wong-Zakai theorem for Stochastic PDEs”, J. Math. Soc. Japan 67 (2015) no. 4, p. 1551-1604 Article
[31] 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
[32] Mehran Kardar, Giorgio Parisi & Yi-Cheng Zhang, “Dynamic Scaling of Growing Interfaces”, Phys. Rev. Lett. 56 (1986) no. 9, p. 889-892 Article
[33] Peter E. Kloeden & Eckhard Platen, Numerical solution of stochastic differential equations, Applications of Mathematics 23, Springer, 1992 Article
[34] Nicolaĭ Vladimirovich Krylov, Lectures on Elliptic and Parabolic Equations in Hölder Spaces, Graduate Studies in Mathematics 12, American Mathematical Society, 1996
[35] Antti Kupiainen, “Renormalization Group and Stochastic PDEs”, Ann. Henri Poincaré 17 (2015) no. 3, p. 1-39 Article
[36] Terry J. Lyons, “Differential equations driven by rough signals”, Rev. Mat. Iberoam. 14 (1998) no. 2, p. 215-310 Article
[37] Jean-Christophe Mourrat & Hendrik Weber, “Convergence of the two-dimensional dynamic Ising-Kac model to $\Phi ^4_2$”, Commun. Pure Appl. Math. 70 (2017) no. 4, p. 717-812 Article
[38] 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 Article
[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) 17 (2015) no. 7, p. 1687-1759 Article
[40] David Nualart, The Malliavin calculus and related topics, Probability and Its Applications, Springer, 2006
[41] Rudolf Peierls, “On Ising’s model of ferromagnetism”, Proc. Camb. Philos. Soc. 32 (1936), p. 477-481 Article
[42] Claudia Prévôt & Michael Röckner, A concise course on stochastic partial differential equations, Lecture Notes in Mathematics 1905, Springer, Berlin, 2007
[43] Jeremy Quastel & Herbert Spohn, “The one-dimensional KPZ equation and its universality class”, J. Stat. Phys. 160 (2015) no. 4, p. 965-984 Article
[44] 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 Article
[45] 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 Article
[46] Eugene Wong & Moshe Zakai, “On the convergence of ordinary integrals to stochastic integrals”, Ann. Math. Stat. 36 (1965), p. 1560-1564 Article
[47] Eugene Wong & Moshe Zakai, “On the relation between ordinary and stochastic differential equations”, Int. J. Eng. Sci. 3 (1965), p. 213-229 Article
Ajay Chandra; Hendrik Weber
Stochastic PDEs, Regularity structures, and interacting particle systems
Annales de la faculté des sciences de Toulouse Sér. 6, 26 no. 4 (2017), p. 847-909, doi: 10.5802/afst.1555
Article PDF
Résumé - Abstract
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 $\Phi ^4_3$ 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.
Bibliography
[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. 64 (2011) no. 4, p. 466-537 Article
[3] Hajer Bahouri, Jean-Yves Chemin & Raphaël Danchin, Fourier analysis and nonlinear partial differential equations, Grundlehren der Mathematischen Wissenschaften 343, Springer, 2011
[4] Lorenzo Bertini & Giambattista Giacomin, “Stochastic Burgers and KPZ equations from particle systems”, Commun. Math. Phys. 183 (1997) no. 3, p. 571-607 Article
[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. 38 (1993) no. 4, p. 689-741 Article
[6] Jean Bourgain, “Invariant measures for the 2D-defocusing nonlinear Schrödinger equation”, Commun. Math. Phys. 176 (1996) no. 2, p. 421-445 Article
[7] David C Brydges, Jürg Fröhlich & Alan D Sokal, “A new proof of the existence and nontriviality of the continuum $\phi _2^4$ and $\phi _3^4$ quantum field theories”, Commun. Math. Phys. 91 (1983) no. 2, p. 141-186 Article
[8] Rémi Catellier & Khalil Chouk, “Paracontrolled distributions and the 3-dimensional stochastic quantization equation”, https://arxiv.org/abs/1310.6869v1, 2013
[9] Ivan Corwin & Jeremy Quastel, “Renormalization fixed point of the KPZ universality class”, J. Stat. Phys. 160 (2015) no. 4, p. 815-834 Article
[10] Laure Coutin & Zhongmin Qian, “Stochastic analysis, rough path analysis and fractional Brownian motions”, Probab. Theory Relat. Fields 122 (2002) no. 1, p. 108-140 Article
[11] Giuseppe Da Prato & Arnaud Debussche, “Strong solutions to the stochastic quantization equations”, Ann. Probab. 31 (2003) no. 4, p. 1900-1916 Article
[12] Giuseppe Da Prato & Jerzy Zabczyk, Stochastic equations in infinite dimensions, Encyclopedia of Mathematics and its Applications 44, Cambridge University Press, Cambridge, 1992 Article
[13] Joel Feldman, “The $\lambda $$\phi ^{4}_{3}$ field theory in a finite volume”, Commun. Math. Phys. 37 (1974) no. 2, p. 93-120 Article
[14] 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 Article
[15] Peter K. Friz & Martin Hairer, A course on rough paths, Universitext, Springer, 2014
[16] Jurg Fröhlich, “On the triviality of $\Lambda \Phi ^{4}$ theories and the approach to the critical-point in D$\ge 4$-dimensions”, Nuclear Physics B 200 (1982) no. 2, p. 281-296 Article
[17] Tadahisa Funaki, “Random motion of strings and related stochastic evolution equations”, Nagoya Math. J. 89 (1983), p. 129-193 Article
[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 64, American Mathematical Society, 1999, p. 107–152 Article
[19] James Glimm & Arthur Jaffe, Quantum physics. A functional integral point of view, Springer, 1981
[20] James Glimm, Arthur Jaffe & Thomas Spencer, “The Wightman axioms and particle structure in the P$(\varphi )_2$ quantum field model”, Ann. Math. 100 (1974) no. 3, p. 585-632 Article
[21] James Glimm, Arthur Jaffe & Thomas Spencer, “Phase transitions for $\varphi _2^4$ quantum fields”, Commun. Math. Phys. 45 (1975) no. 3, p. 203-216 Article
[22] Massimiliano Gubinelli, “Controlling rough paths”, J. Funct. Anal. 216 (2004) no. 1, p. 86-140 Article
[23] Massimiliano Gubinelli, Peter Imkeller & Nicolas Perkowski, “Paracontrolled distributions and singular PDEs”, Forum Math. Pi 3 (2015), Article ID e6, 75 p. Article
[24] Martin Hairer, “An Introduction to Stochastic PDEs”, https://arxiv.org/abs/0907.4178, 2009
[25] Martin Hairer, “Solving the KPZ equation”, Ann. Math. 178 (2013) no. 2, p. 559-664 Article
[26] Martin Hairer, “Singular Stochastic PDES”, https://arxiv.org/abs/1403.6353, 2014
[27] Martin Hairer, “A theory of regularity structures”, Invent. Math. 198 (2014) no. 2, p. 269-504 Article
[28] Martin Hairer, “Introduction to regularity structures”, Braz. J. Probab. Stat. 29 (2015) no. 2, p. 175-210 Article
[29] Martin Hairer & Cyril Labbé, “Multiplicative stochastic heat equations on the whole space”, https://arxiv.org/abs/1504.07162, to appear in J. Eur. Math. Soc., 2015
[30] Martin Hairer & Étienne Pardoux, “A Wong-Zakai theorem for Stochastic PDEs”, J. Math. Soc. Japan 67 (2015) no. 4, p. 1551-1604 Article
[31] 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
[32] Mehran Kardar, Giorgio Parisi & Yi-Cheng Zhang, “Dynamic Scaling of Growing Interfaces”, Phys. Rev. Lett. 56 (1986) no. 9, p. 889-892 Article
[33] Peter E. Kloeden & Eckhard Platen, Numerical solution of stochastic differential equations, Applications of Mathematics 23, Springer, 1992 Article
[34] Nicolaĭ Vladimirovich Krylov, Lectures on Elliptic and Parabolic Equations in Hölder Spaces, Graduate Studies in Mathematics 12, American Mathematical Society, 1996
[35] Antti Kupiainen, “Renormalization Group and Stochastic PDEs”, Ann. Henri Poincaré 17 (2015) no. 3, p. 1-39 Article
[36] Terry J. Lyons, “Differential equations driven by rough signals”, Rev. Mat. Iberoam. 14 (1998) no. 2, p. 215-310 Article
[37] Jean-Christophe Mourrat & Hendrik Weber, “Convergence of the two-dimensional dynamic Ising-Kac model to $\Phi ^4_2$”, Commun. Pure Appl. Math. 70 (2017) no. 4, p. 717-812 Article
[38] 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 Article
[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) 17 (2015) no. 7, p. 1687-1759 Article
[40] David Nualart, The Malliavin calculus and related topics, Probability and Its Applications, Springer, 2006
[41] Rudolf Peierls, “On Ising’s model of ferromagnetism”, Proc. Camb. Philos. Soc. 32 (1936), p. 477-481 Article
[42] Claudia Prévôt & Michael Röckner, A concise course on stochastic partial differential equations, Lecture Notes in Mathematics 1905, Springer, Berlin, 2007
[43] Jeremy Quastel & Herbert Spohn, “The one-dimensional KPZ equation and its universality class”, J. Stat. Phys. 160 (2015) no. 4, p. 965-984 Article
[44] 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 Article
[45] 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 Article
[46] Eugene Wong & Moshe Zakai, “On the convergence of ordinary integrals to stochastic integrals”, Ann. Math. Stat. 36 (1965), p. 1560-1564 Article
[47] Eugene Wong & Moshe Zakai, “On the relation between ordinary and stochastic differential equations”, Int. J. Eng. Sci. 3 (1965), p. 213-229 Article
