Some recent progress in singular stochastic partial differential equations

Corwin, Ivan; Shen, Hao

doi:10.1090/bull/1670

Some recent progress in singular stochastic partial differential equations

Authors: Ivan Corwin and Hao Shen
Journal: Bull. Amer. Math. Soc. 57 (2020), 409-454
DOI: https://doi.org/10.1090/bull/1670
Published electronically: September 26, 2019
MathSciNet review: 4108091
Full-text PDF Free Access
View in AMS MathViewer New

Abstract | References | Additional Information

Abstract: Stochastic partial differential equations are ubiquitous in mathematical modeling. Yet, many such equations are too singular to admit classical treatment. In this article we review some recent progress in defining, approximating, and studying the properties of a few examples of such equations. We focus mainly on the dynamical $\Phi ^4$ equation, the KPZ equation, and the parabolic Anderson model, as well as a few other equations which arise mainly in physics.

[AC15] R. Allez and K. Chouk,
The continuous Anderson Hamiltonian in dimension two,
arXiv:1511.02718, 2015.
[ACQ11] Gideon Amir, Ivan Corwin, and Jeremy Quastel, Probability distribution of the free energy of the continuum directed random polymer in 1+1 dimensions, Comm. Pure Appl. Math. 64 (2011), no. 4, 466–537. MR 2796514, https://doi.org/10.1002/cpa.20347
[Agg16] Amol Aggarwal, Current fluctuations of the stationary ASEP and six-vertex model, Duke Math. J. 167 (2018), no. 2, 269–384. MR 3754630, https://doi.org/10.1215/00127094-2017-0029
[AHR96] Sergio Albeverio, Zbigniew Haba, and Francesco Russo, Trivial solutions for a non-linear two-space-dimensional wave equation perturbed by space-time white noise, Stochastics Stochastics Rep. 56 (1996), no. 1-2, 127–160. MR 1396758, https://doi.org/10.1080/17442509608834039
[AK17] S. Albeverio and S. Kusuoka,
The invariant measure and the flow associated to the $\phi ^4_3$ -quantum field model,
arXiv:1711.07108, 2017.
[AR91] S. Albeverio and M. Röckner, Stochastic differential equations in infinite dimensions: solutions via Dirichlet forms, Probab. Theory Related Fields 89 (1991), no. 3, 347–386. MR 1113223, https://doi.org/10.1007/BF01198791
[Ass13] Sigurd Assing, A rigorous equation for the Cole-Hopf solution of the conservative KPZ equation, Stoch. Partial Differ. Equ. Anal. Comput. 1 (2013), no. 2, 365–388. MR 3327511, https://doi.org/10.1007/s40072-013-0013-3
[BB16] I. Bailleul and F. Bernicot, Heat semigroup and singular PDEs, J. Funct. Anal. 270 (2016), no. 9, 3344–3452. With an appendix by F. Bernicot and D. Frey. MR 3475460, https://doi.org/10.1016/j.jfa.2016.02.012
[BC95] Lorenzo Bertini and Nicoletta Cancrini, The stochastic heat equation: Feynman-Kac formula and intermittence, J. Statist. Phys. 78 (1995), no. 5-6, 1377–1401. MR 1316109, https://doi.org/10.1007/BF02180136
[BCCH17] Y. Bruned, A. Chandra, I. Chevyrev, and M. Hairer,
Renormalising SPDEs in regularity structures,
arXiv:1711.10239, 2017.
[BCD11] Hajer Bahouri, Jean-Yves Chemin, and Raphaël Danchin, Fourier analysis and nonlinear partial differential equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 343, Springer, Heidelberg, 2011. MR 2768550
[BDH19] I. Bailleul, A. Debussche, and M. Hofmanová, Quasilinear generalized parabolic Anderson model equation, Stoch. Partial Differ. Equ. Anal. Comput. 7 (2019), no. 1, 40–63. MR 3916262, https://doi.org/10.1007/s40072-018-0121-1
[Ber70] V. L. Berezinskiĭ, Destruction of long-range order in one-dimensional and two-dimensional systems having a continuous symmetry group. I. Classical systems, Ž. Èksper. Teoret. Fiz. 59 (1970), 907–920 (Russian, with English summary); English transl., Soviet Physics JETP 32 (1971), 493–500. MR 0314399
[BG97] Lorenzo Bertini and Giambattista Giacomin, Stochastic Burgers and KPZ equations from particle systems, Comm. Math. Phys. 183 (1997), no. 3, 571–607. MR 1462228, https://doi.org/10.1007/s002200050044
[BG18] A. Borodin and V. Gorin,
A stochastic telegraph equation from the six-vertex model,
arXiv:1803.09137, 2018.
[BGHZ19] Y. Bruned, F. Gabriel, M. Hairer, and L. Zambotti,
Geometric stochastic heat equations,
arXiv:1902.02884, 2019.
[BHZ16] Y. Bruned, M. Hairer, and L. Zambotti, Algebraic renormalisation of regularity structures, Invent. Math. 215 (2019), no. 3, 1039–1156. MR 3935036, https://doi.org/10.1007/s00222-018-0841-x
[BO17] Alexei Borodin and Grigori Olshanski, The ASEP and determinantal point processes, Comm. Math. Phys. 353 (2017), no. 2, 853–903. MR 3649488, https://doi.org/10.1007/s00220-017-2858-1
[Bou96] Jean Bourgain, Invariant measures for the 2D-defocusing nonlinear Schrödinger equation, Comm. Math. Phys. 176 (1996), no. 2, 421–445. MR 1374420
[BP15] Alexei Borodin and Leonid Petrov, Integrable probability: stochastic vertex models and symmetric functions, Stochastic processes and random matrices, Oxford Univ. Press, Oxford, 2017, pp. 26–131. MR 3728714
[Bru68] Stephen G. Brush, A history of random processes, Arch. History Exact Sci. 5 (1968), no. 1, 1–36. I. Brownian movement from Brown to Perrin. MR 1554112, https://doi.org/10.1007/BF00328110
[BS95] Dirk Jan Bukman and Joel D. Shore, The conical point in the ferroelectric six-vertex model, J. Statist. Phys. 78 (1995), no. 5-6, 1277–1309. MR 1316105, https://doi.org/10.1007/BF02180132
[CC18a] Giuseppe Cannizzaro and Khalil Chouk, Multidimensional SDEs with singular drift and universal construction of the polymer measure with white noise potential, Ann. Probab. 46 (2018), no. 3, 1710–1763. MR 3785598, https://doi.org/10.1214/17-AOP1213
[CC18b] Rémi Catellier and Khalil Chouk, Paracontrolled distributions and the 3-dimensional stochastic quantization equation, Ann. Probab. 46 (2018), no. 5, 2621–2679. MR 3846835, https://doi.org/10.1214/17-AOP1235
[CD18] S. Chatterjee and A. Dunlap.
Constructing a solution of the -dimensional KPZ equation,
arXiv:1809.00803, 2018.
[CGP17] Khalil Chouk, Jan Gairing, and Nicolas Perkowski, An invariance principle for the two-dimensional parabolic Anderson model with small potential, Stoch. Partial Differ. Equ. Anal. Comput. 5 (2017), no. 4, 520–558. MR 3736653, https://doi.org/10.1007/s40072-017-0096-3
[CGST18] I. Corwin, P. Ghosal, H. Shen, and L.-C. Tsai,
Stochastic PDE limit of the Six Vertex Model,
arXiv:1803.08120, 2018.
[CH16] A. Chandra and M. Hairer,
An analytic BPHZ theorem for regularity structures,
arXiv:1612.08138, 2016.
[CHI15] Dmitry Chelkak, Clément Hongler, and Konstantin Izyurov, Conformal invariance of spin correlations in the planar Ising model, Ann. of Math. (2) 181 (2015), no. 3, 1087–1138. MR 3296821, https://doi.org/10.4007/annals.2015.181.3.5
[CHS18] A. Chandra, M. Hairer, and H. Shen,
The dynamical sine-Gordon model in the full subcritical regime,
arXiv:1808.02594, 2018.
[CM94] René A. Carmona and S. A. Molchanov, Parabolic Anderson problem and intermittency, Mem. Amer. Math. Soc. 108 (1994), no. 518, viii+125. MR 1185878, https://doi.org/10.1090/memo/0518
[CM16] G. Cannizzaro and K. Matetski, Space-time discrete KPZ equation, Comm. Math. Phys. 358 (2018), no. 2, 521–588. MR 3774431, https://doi.org/10.1007/s00220-018-3089-9
[Con12] C. Conti,
Solitonization of the Anderson localization,
Physical Review A, 86 (2012), no. 6, 061801.
[Cor14] Ivan Corwin, Macdonald processes, quantum integrable systems and the Kardar-Parisi-Zhang universality class, Proceedings of the International Congress of Mathematicians—Seoul 2014. Vol. III, Kyung Moon Sa, Seoul, 2014, pp. 1007–1034. MR 3729062
[CP16] Ivan Corwin and Leonid Petrov, Stochastic higher spin vertex models on the line, Comm. Math. Phys. 343 (2016), no. 2, 651–700. MR 3477349, https://doi.org/10.1007/s00220-015-2479-5
[CR99] Rene A. Carmona and Boris Rozovskii (eds.), Stochastic partial differential equations: six perspectives, Mathematical Surveys and Monographs, vol. 64, American Mathematical Society, Providence, RI, 1999. MR 1661761
[CS18] Ivan Corwin and Hao Shen, Open ASEP in the weakly asymmetric regime, Comm. Pure Appl. Math. 71 (2018), no. 10, 2065–2128. MR 3861074, https://doi.org/10.1002/cpa.21744
[CST18] Ivan Corwin, Hao Shen, and Li-Cheng Tsai, 𝐴𝑆𝐸𝑃(𝑞,𝑗) converges to the KPZ equation, Ann. Inst. Henri Poincaré Probab. Stat. 54 (2018), no. 2, 995–1012 (English, with English and French summaries). MR 3795074, https://doi.org/10.1214/17-AIHP829
[CSZ17a] Francesco Caravenna, Rongfeng Sun, and Nikos Zygouras, Polynomial chaos and scaling limits of disordered systems, J. Eur. Math. Soc. (JEMS) 19 (2017), no. 1, 1–65. MR 3584558, https://doi.org/10.4171/JEMS/660
[CSZ17b] Francesco Caravenna, Rongfeng Sun, and Nikos Zygouras, Universality in marginally relevant disordered systems, Ann. Appl. Probab. 27 (2017), no. 5, 3050–3112. MR 3719953, https://doi.org/10.1214/17-AAP1276
[CSZ18] F. Caravenna, R. Sun, and N. Zygouras,
The two-dimensional KPZ equation in the entire subcritical regime,
arXiv:1812.03911, 2018.
[CT17] Ivan Corwin and Li-Cheng Tsai, KPZ equation limit of higher-spin exclusion processes, Ann. Probab. 45 (2017), no. 3, 1771–1798. MR 3650415, https://doi.org/10.1214/16-AOP1101
[CT18] I. Corwin and L.-C. Tsai,
SPDE limit of weakly inhomogeneous ASEP,
arXiv:1806.09682, 2018.
[CW17] Ajay Chandra and Hendrik Weber, Stochastic PDEs, regularity structures, and interacting particle systems, Ann. Fac. Sci. Toulouse Math. (6) 26 (2017), no. 4, 847–909 (English, with English and French summaries). MR 3746645, https://doi.org/10.5802/afst.1555
[CY92] Chih Chung Chang and Horng-Tzer Yau, Fluctuations of one-dimensional Ginzburg-Landau models in nonequilibrium, Comm. Math. Phys. 145 (1992), no. 2, 209–234. MR 1162798
[Daw72] D. A. Dawson, Stochastic evolution equations, Math. Biosci. 15 (1972), 287–316. MR 321178, https://doi.org/10.1016/0025-5564(72)90039-9
[DGP17] Joscha Diehl, Massimiliano Gubinelli, and Nicolas Perkowski, The Kardar-Parisi-Zhang equation as scaling limit of weakly asymmetric interacting Brownian motions, Comm. Math. Phys. 354 (2017), no. 2, 549–589. MR 3663617, https://doi.org/10.1007/s00220-017-2918-6
[DK90] S. K. Donaldson and P. B. Kronheimer, The geometry of four-manifolds, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1990. Oxford Science Publications. MR 1079726
[DKM$^+$09] Robert Dalang, Davar Khoshnevisan, Carl Mueller, David Nualart, and Yimin Xiao, A minicourse on stochastic partial differential equations, Lecture Notes in Mathematics, vol. 1962, Springer-Verlag, Berlin, 2009. Held at the University of Utah, Salt Lake City, UT, May 8–19, 2006; Edited by Khoshnevisan and Firas Rassoul-Agha. MR 1500166
[DM17] A. Debussche and J. Martin,
Solution to the stochastic Schrodinger equation on the full space,
arXiv:1707.06431, 2017.
[DPD02] Giuseppe Da Prato and Arnaud Debussche, Two-dimensional Navier-Stokes equations driven by a space-time white noise, J. Funct. Anal. 196 (2002), no. 1, 180–210. MR 1941997, https://doi.org/10.1006/jfan.2002.3919
[DPD03] Giuseppe Da Prato and Arnaud Debussche, Strong solutions to the stochastic quantization equations, Ann. Probab. 31 (2003), no. 4, 1900–1916. MR 2016604, https://doi.org/10.1214/aop/1068646370
[DPZ14] Giuseppe Da Prato and Jerzy Zabczyk, Stochastic equations in infinite dimensions, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 152, Cambridge University Press, Cambridge, 2014. MR 3236753
[DT16] Amir Dembo and Li-Cheng Tsai, Weakly asymmetric non-simple exclusion process and the Kardar-Parisi-Zhang equation, Comm. Math. Phys. 341 (2016), no. 1, 219–261. MR 3439226, https://doi.org/10.1007/s00220-015-2527-1
[DW18] Arnaud Debussche and Hendrik Weber, The Schrödinger equation with spatial white noise potential, Electron. J. Probab. 23 (2018), Paper No. 28, 16. MR 3785398, https://doi.org/10.1214/18-EJP143
[EH17] D. Erhard and M. Hairer,
Discretisation of regularity structures,
arXiv:1705.02836, 2017.
[Fee14] P. M. N. Feehan,
Global existence and convergence of smooth solutions to yang-mills gradient flow over compact four-manifolds,
arXiv:1409.1525, 2014.
[FG17] M. Furlan and M. Gubinelli, Weak universality for a class of 3d stochastic reaction-diffusion models, Probab. Theory Related Fields 173 (2019), no. 3-4, 1099–1164. MR 3936152, https://doi.org/10.1007/s00440-018-0849-6
[FG19] Marco Furlan and Massimiliano Gubinelli, Paracontrolled quasilinear SPDEs, Ann. Probab. 47 (2019), no. 2, 1096–1135. MR 3916943, https://doi.org/10.1214/18-AOP1280
[FGS16] Tertuliano Franco, Patrícia Gonçalves, and Marielle Simon, Crossover to the stochastic Burgers equation for the WASEP with a slow bond, Comm. Math. Phys. 346 (2016), no. 3, 801–838. MR 3537337, https://doi.org/10.1007/s00220-016-2607-x
[FH14] Peter K. Friz and Martin Hairer, A course on rough paths, Universitext, Springer, Cham, 2014. With an introduction to regularity structures. MR 3289027
[FH17] Tadahisa Funaki and Masato Hoshino, A coupled KPZ equation, its two types of approximations and existence of global solutions, J. Funct. Anal. 273 (2017), no. 3, 1165–1204. MR 3653951, https://doi.org/10.1016/j.jfa.2017.05.002
[Fla08] Franco Flandoli, An introduction to 3D stochastic fluid dynamics, SPDE in hydrodynamic: recent progress and prospects, Lecture Notes in Math., vol. 1942, Springer, Berlin, 2008, pp. 51–150. MR 2459085, https://doi.org/10.1007/978-3-540-78493-7_2
[Fle75] Wendell H. Fleming, Distributed parameter stochastic systems in population biology, Control theory, numerical methods and computer systems modelling (Internat. Sympos., IRIA LABORIA, Rocquencourt, 1974) Springer, Berlin, 1975, pp. 179–191. Lecture Notes in Econom. and Math. Systems, Vol. 107. MR 0378124
[FOT11] Masatoshi Fukushima, Yoichi Oshima, and Masayoshi Takeda, Dirichlet forms and symmetric Markov processes, Second revised and extended edition, De Gruyter Studies in Mathematics, vol. 19, Walter de Gruyter & Co., Berlin, 2011. MR 2778606
[FR95] J. Fritz and B. Rüdiger, Time dependent critical fluctuations of a one-dimensional local mean field model, Probab. Theory Related Fields 103 (1995), no. 3, 381–407. MR 1358083, https://doi.org/10.1007/BF01195480
[Fri87] J. Fritz, On the hydrodynamic limit of a one-dimensional Ginzburg-Landau lattice model. The a priori bounds, J. Statist. Phys. 47 (1987), no. 3-4, 551–572. MR 894407, https://doi.org/10.1007/BF01007526
[FS81] Jürg Fröhlich and Thomas Spencer, The Kosterlitz-Thouless transition in two-dimensional abelian spin systems and the Coulomb gas, Comm. Math. Phys. 81 (1981), no. 4, 527–602. MR 634447
[Fun83] Tadahisa Funaki, Random motion of strings and related stochastic evolution equations, Nagoya Math. J. 89 (1983), 129–193. MR 692348, https://doi.org/10.1017/S0027763000020298
[Gär88] Jürgen Gärtner, Convergence towards Burgers’ equation and propagation of chaos for weakly asymmetric exclusion processes, Stochastic Process. Appl. 27 (1988), no. 2, 233–260. MR 931030, https://doi.org/10.1016/0304-4149(87)90040-8
[GGF$^+$12] N. Ghofraniha, S. Gentilini, V. Folli, E. DelRe, and C. Conti,
Shock waves in disordered media,
Physical Review Letters, 109 (2012), no. 24, 243902.
[GH17a] Máté Gerencsér and Martin Hairer, Singular SPDEs in domains with boundaries, Probab. Theory Related Fields 173 (2019), no. 3-4, 697–758. MR 3936145, https://doi.org/10.1007/s00440-018-0841-1
[GH17b] M. Gerencsér and M. Hairer.
A solution theory for quasilinear singular SPDEs,
Comm. Pure Appl. Math., 2017.
[GH18a] M. Gubinelli and M. Hofmanová,
A PDE construction of the Euclidean $\Phi ^4_3$ quantum field theory,
arXiv:1810.01700, 2018.
[GH18b] M. Gubinelli and M. Hofmanová,
Global solutions to elliptic and parabolic $\phi ^4$ models in Euclidean space,
arXiv:1804.11253, 2018.
[Gho17] P. Ghosal,
Hall-Littlewood-PushTASEP and its KPZ limit,
arXiv:1701.07308, 2017.
[GIP15] Massimiliano Gubinelli, Peter Imkeller, and Nicolas Perkowski, Paracontrolled distributions and singular PDEs, Forum Math. Pi 3 (2015), e6, 75. MR 3406823, https://doi.org/10.1017/fmp.2015.2
[GJ10] P. Goncalves and M. Jara,
Universality of KPZ equation.
arXiv:1003.4478, 2010.
[GJ13] M. Gubinelli and M. Jara, Regularization by noise and stochastic Burgers equations, Stoch. Partial Differ. Equ. Anal. Comput. 1 (2013), no. 2, 325–350. MR 3327509, https://doi.org/10.1007/s40072-013-0011-5
[GJ14] Patrícia Gonçalves and Milton Jara, Nonlinear fluctuations of weakly asymmetric interacting particle systems, Arch. Ration. Mech. Anal. 212 (2014), no. 2, 597–644. MR 3176353, https://doi.org/10.1007/s00205-013-0693-x
[GJ17] Patrícia Gonçalves and Milton Jara, Stochastic Burgers equation from long range exclusion interactions, Stochastic Process. Appl. 127 (2017), no. 12, 4029–4052. MR 3718105, https://doi.org/10.1016/j.spa.2017.03.022
[GJS15] Patrícia Gonçalves, Milton Jara, and Sunder Sethuraman, A stochastic Burgers equation from a class of microscopic interactions, Ann. Probab. 43 (2015), no. 1, 286–338. MR 3298474, https://doi.org/10.1214/13-AOP878
[GKO18a] M. Gubinelli, H. Koch, and T. Oh,
Paracontrolled approach to the three-dimensional stochastic nonlinear wave equation with quadratic nonlinearity,
arXiv:1811.07808, 2018.
[GKO18b] Massimiliano Gubinelli, Herbert Koch, and Tadahiro Oh, Renormalization of the two-dimensional stochastic nonlinear wave equations, Trans. Amer. Math. Soc. 370 (2018), no. 10, 7335–7359. MR 3841850, https://doi.org/10.1090/tran/7452
[GKR18] Yu Gu, Tomasz Komorowski, and Lenya Ryzhik, The Schrödinger equation with spatial white noise: the average wave function, J. Funct. Anal. 274 (2018), no. 7, 2113–2138. MR 3762097, https://doi.org/10.1016/j.jfa.2018.01.015
[Gla63] Roy J. Glauber, Time-dependent statistics of the Ising model, J. Mathematical Phys. 4 (1963), 294–307. MR 148410, https://doi.org/10.1063/1.1703954
[GP16] M. Gubinelli and N. Perkowski,
The Hairer-Quastel universality result in equilibrium,
arXiv:1602.02428, 2016.
[GP17a] Massimiliano Gubinelli and Nicolas Perkowski, Energy solutions of KPZ are unique, J. Amer. Math. Soc. 31 (2018), no. 2, 427–471. MR 3758149, https://doi.org/10.1090/jams/889
[GP17b] Massimiliano Gubinelli and Nicolas Perkowski, KPZ reloaded, Comm. Math. Phys. 349 (2017), no. 1, 165–269. MR 3592748, https://doi.org/10.1007/s00220-016-2788-3
[GP18a] M. Gubinelli and N. Perkowski,
The infinitesimal generator of the stochastic Burgers equation,
arXiv:1810.12014, 2018.
[GP18b] Massimiliano Gubinelli and Nicolas Perkowski, An introduction to singular SPDEs, Stochastic partial differential equations and related fields, Springer Proc. Math. Stat., vol. 229, Springer, Cham, 2018, pp. 69–99. MR 3828162, https://doi.org/10.1007/978-3-319-74929-7_4
[GP18c] Massimiliano Gubinelli and Nicolas Perkowski, Probabilistic approach to the stochastic Burgers equation, Stochastic partial differential equations and related fields, Springer Proc. Math. Stat., vol. 229, Springer, Cham, 2018, pp. 515–527. MR 3828193, https://doi.org/10.1007/978-3-319-74929-7_4
[GPS17] P. Goncalves, N. Perkowski, and M. Simon,
Derivation of the stochastic Burgers equation with Dirichlet boundary conditions from the WASEP,
arXiv:1710.11011, 2017.
[GPV88] M. Z. Guo, G. C. Papanicolaou, and S. R. S. Varadhan, Nonlinear diffusion limit for a system with nearest neighbor interactions, Comm. Math. Phys. 118 (1988), no. 1, 31–59. MR 954674
[GS92] Leh-Hun Gwa and Herbert Spohn, Six-vertex model, roughened surfaces, and an asymmetric spin Hamiltonian, Phys. Rev. Lett. 68 (1992), no. 6, 725–728. MR 1147356, https://doi.org/10.1103/PhysRevLett.68.725
[Gu18] Y. Gu.
Gaussian fluctuations of the 2D KPZ equation,
arXiv:1812.07467, 2018.
[Gub04] M. Gubinelli, Controlling rough paths, J. Funct. Anal. 216 (2004), no. 1, 86–140. MR 2091358, https://doi.org/10.1016/j.jfa.2004.01.002
[Gub18] M. Gubinelli,
A panorama of singular SPDEs,
in Proc. Int. Cong. of Math., volume 2, pp. 2277-2304, 2018.
[GUZ18] M. Gubinelli, B. E. Ugurcan, and I. Zachhuber,
Semilinear evolution equations for the Anderson Hamiltonian in two and three dimensions,
arXiv:1807.06825, 2018.
[Hai13] Martin Hairer, Solving the KPZ equation, Ann. of Math. (2) 178 (2013), no. 2, 559–664. MR 3071506, https://doi.org/10.4007/annals.2013.178.2.4
[Hai14a] Martin Hairer, Singular stochastic PDEs, Proceedings of the International Congress of Mathematicians—Seoul 2014. Vol. 1, Kyung Moon Sa, Seoul, 2014, pp. 685–709. MR 3728488
[Hai14b] M. Hairer, A theory of regularity structures, Invent. Math. 198 (2014), no. 2, 269–504. MR 3274562, https://doi.org/10.1007/s00222-014-0505-4
[Hai15a] Martin Hairer, Introduction to regularity structures, Braz. J. Probab. Stat. 29 (2015), no. 2, 175–210. MR 3336866, https://doi.org/10.1214/14-BJPS241
[Hai15b] M. Hairer,
Regularity structures and the dynamical $\phi ^4_3$ model,
arXiv:1508.05261, 2015.
[Hai16] M. Hairer,
The motion of a random string,
arXiv:1605.02192, 2016.
[HL15] Martin Hairer and Cyril Labbé, A simple construction of the continuum parabolic Anderson model on 𝑅², Electron. Commun. Probab. 20 (2015), no. 43, 11. MR 3358965, https://doi.org/10.1214/ECP.v20-4038
[HL18] Martin Hairer and Cyril Labbé, Multiplicative stochastic heat equations on the whole space, J. Eur. Math. Soc. (JEMS) 20 (2018), no. 4, 1005–1054. MR 3779690, https://doi.org/10.4171/JEMS/781
[HM18] M. Hairer and K. Matetski, Discretisations of rough stochastic PDEs, Ann. Probab. 46 (2018), no. 3, 1651–1709. MR 3785597, https://doi.org/10.1214/17-AOP1212
[HP15] Martin Hairer and Étienne Pardoux, A Wong-Zakai theorem for stochastic PDEs, J. Math. Soc. Japan 67 (2015), no. 4, 1551–1604. MR 3417505, https://doi.org/10.2969/jmsj/06741551
[HQ18] Martin Hairer and Jeremy Quastel, A class of growth models rescaling to KPZ, Forum Math. Pi 6 (2018), e3, 112. MR 3877863, https://doi.org/10.1017/fmp.2018.2
[HS16] Martin Hairer and Hao Shen, The dynamical sine-Gordon model, Comm. Math. Phys. 341 (2016), no. 3, 933–989. MR 3452276, https://doi.org/10.1007/s00220-015-2525-3
[HS17] Martin Hairer and Hao Shen, A central limit theorem for the KPZ equation, Ann. Probab. 45 (2017), no. 6B, 4167–4221. MR 3737909, https://doi.org/10.1214/16-AOP1162
[HX18a] Martin Hairer and Weijun Xu, Large-scale behavior of three-dimensional continuous phase coexistence models, Comm. Pure Appl. Math. 71 (2018), no. 4, 688–746. MR 3772400, https://doi.org/10.1002/cpa.21738
[HX18b] M. Hairer and W. Xu,
Large-scale limit of interface fluctuation models,
arXiv:1802.08192, 2018.
[Hos18] Masato Hoshino, Paracontrolled calculus and Funaki-Quastel approximation for the KPZ equation, Stochastic Process. Appl. 128 (2018), no. 4, 1238–1293. MR 3769661, https://doi.org/10.1016/j.spa.2017.07.001
[Jaf00] Arthur Jaffe, Constructive quantum field theory, Mathematical physics 2000, Imp. Coll. Press, London, 2000, pp. 111–127. MR 1773042, https://doi.org/10.1142/9781848160224_0007
[Jan97] Svante Janson, Gaussian Hilbert spaces, Cambridge Tracts in Mathematics, vol. 129, Cambridge University Press, Cambridge, 1997. MR 1474726
[JLM85] G. Jona-Lasinio and P. K. Mitter, On the stochastic quantization of field theory, Comm. Math. Phys. 101 (1985), no. 3, 409–436. MR 815192
[Jos13] J. V. Jos,
40 years of Berezinskii-Kosterlitz-Thouless theory.
World Scientific, 2013.
[K16] Wolfgang König, The parabolic Anderson model, Pathways in Mathematics, Birkhäuser/Springer, [Cham], 2016. Random walk in random potential. MR 3526112
[KM17] Antti Kupiainen and Matteo Marcozzi, Renormalization of generalized KPZ equation, J. Stat. Phys. 166 (2017), no. 3-4, 876–902. MR 3607594, https://doi.org/10.1007/s10955-016-1636-3
[KPZ86] M. Kardar, G. Parisi, and Y.-C. Zhang,
Dynamic scaling of growing interfaces,
Phys. Rev. Lett., 56 (1986), 889-892.
[KR82] N. V. Krylov and B. L. Rozovskiĭ, Stochastic partial differential equations and diffusion processes, Uspekhi Mat. Nauk 37 (1982), no. 6(228), 75–95 (Russian). MR 683274
[KS91a] Ioannis Karatzas and Steven E. Shreve, Brownian motion and stochastic calculus, 2nd ed., Graduate Texts in Mathematics, vol. 113, Springer-Verlag, New York, 1991. MR 1121940
[KS91b] J. Krug and H. Spohn, Kinetic roughening of growing surfaces, in Solids Far From Equilibrium: Growth, Morphology and Defects (C. Godreche, ed.), Cambridge University Press, Cambridge, 1991.
[KS12] Sergei Kuksin and Armen Shirikyan, Mathematics of two-dimensional turbulence, Cambridge Tracts in Mathematics, vol. 194, Cambridge University Press, Cambridge, 2012. MR 3443633
[KT73] J. M. Kosterlitz and D. J. Thouless,
Ordering, metastability and phase transitions in two-dimensional systems,
J. Physics C: Solid State Physics, 6 (1973), no. 7, 1181.
[Kup10] A. Kupiainen,
Ergodicity of two dimensional turbulence,
arXiv:1005.0587, 2010.
[Kup16] Antti Kupiainen, Renormalization group and stochastic PDEs, Ann. Henri Poincaré 17 (2016), no. 3, 497–535. MR 3459120, https://doi.org/10.1007/s00023-015-0408-y
[Lab17] Cyril Labbé, Weakly asymmetric bridges and the KPZ equation, Comm. Math. Phys. 353 (2017), no. 3, 1261–1298. MR 3652491, https://doi.org/10.1007/s00220-017-2875-0
[LR15] Wei Liu and Michael Röckner, Stochastic partial differential equations: an introduction, Universitext, Springer, Cham, 2015. MR 3410409
[Lyo98] Terry J. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana 14 (1998), no. 2, 215–310. MR 1654527, https://doi.org/10.4171/RMI/240
[Mat03] Jonathan C. Mattingly, On recent progress for the stochastic Navier Stokes equations, Journées “Équations aux Dérivées Partielles”, Univ. Nantes, Nantes, 2003, pp. Exp. No. XI, 52. MR 2050597
[Mat18] K. Matetski,
Martingale-driven approximations of singular stochastic PDEs,
arXiv:1808.09429, 2018.
[McK95] H. P. McKean, Statistical mechanics of nonlinear wave equations. IV. Cubic Schrödinger, Comm. Math. Phys. 168 (1995), no. 3, 479–491. MR 1328250
[MP17] J. Martin and N. Perkowski,
Paracontrolled distributions on Bravais lattices and weak universality of the 2d parabolic anderson model,
arXiv:1704.08653, 2017.
[MS77] Oliver A. McBryan and Thomas Spencer, On the decay of correlations in 𝑆𝑂(𝑛)-symmetric ferromagnets, Comm. Math. Phys. 53 (1977), no. 3, 299–302. MR 441179
[MU18] Jacques Magnen and Jérémie Unterberger, The scaling limit of the KPZ equation in space dimension 3 and higher, J. Stat. Phys. 171 (2018), no. 4, 543–598. MR 3790153, https://doi.org/10.1007/s10955-018-2014-0
[Mue91] Carl Mueller, On the support of solutions to the heat equation with noise, Stochastics Stochastics Rep. 37 (1991), no. 4, 225–245. MR 1149348, https://doi.org/10.1080/17442509108833738
[MW17a] Jean-Christophe Mourrat and Hendrik Weber, Convergence of the two-dimensional dynamic Ising-Kac model to Φ⁴₂, Comm. Pure Appl. Math. 70 (2017), no. 4, 717–812. MR 3628883, https://doi.org/10.1002/cpa.21655
[MW17b] Jean-Christophe Mourrat and Hendrik Weber, The dynamic Φ⁴₃ model comes down from infinity, Comm. Math. Phys. 356 (2017), no. 3, 673–753. MR 3719541, https://doi.org/10.1007/s00220-017-2997-4
[MW18] A. Moinat and H. Weber,
Space-time localisation for the dynamic $\phi ^4_3$ model,
arXiv:1811.05764, 2018.
[OR98] M. Oberguggenberger and F. Russo, Nonlinear stochastic wave equations, Integral Transform. Spec. Funct. 6 (1998), no. 1-4, 71–83. Generalized functions—linear and nonlinear problems (Novi Sad, 1996). MR 1640497, https://doi.org/10.1080/10652469808819152
[OW18] Felix Otto and Hendrik Weber, Quasilinear SPDEs via rough paths, Arch. Ration. Mech. Anal. 232 (2019), no. 2, 873–950. MR 3925533, https://doi.org/10.1007/s00205-018-01335-8
[Par17] Shalin Parekh, The KPZ limit of ASEP with boundary, Comm. Math. Phys. 365 (2019), no. 2, 569–649. MR 3907953, https://doi.org/10.1007/s00220-018-3258-x
[Pau35] L. Pauling,
The structure and entropy of ice and of other crystals with some randomness of atomic arrangement,
J. Amer. Chem. Soc., 57 (1935), no. 12, 2680-2684.
[PR07] Claudia Prévôt and Michael Röckner, A concise course on stochastic partial differential equations, Lecture Notes in Mathematics, vol. 1905, Springer, Berlin, 2007. MR 2329435
[PR18] N. Perkowski and T. C. Rosati,
The KPZ equation on the real line,
arXiv:1808.00354, 2018.
[PW81] G. Parisi and Yong Shi Wu, Perturbation theory without gauge fixing, Sci. Sinica 24 (1981), no. 4, 483–496. MR 626795
[RWZZ18] Michael Röckner, Bo Wu, Rongchan Zhu, and Xiangchan Zhu, Stochastic heat equations with values in a Riemannian manifold, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 29 (2018), no. 1, 205–213. MR 3787728, https://doi.org/10.4171/RLM/801
[She07] Scott Sheffield, Gaussian free fields for mathematicians, Probab. Theory Related Fields 139 (2007), no. 3-4, 521–541. MR 2322706, https://doi.org/10.1007/s00440-006-0050-1
[She18] H. Shen,
Stochastic quantization of an Abelian gauge theory,
arXiv:1801.04596, 2018.
[Spo86] Herbert Spohn, Equilibrium fluctuations for interacting Brownian particles, Comm. Math. Phys. 103 (1986), no. 1, 1–33. MR 826856
[Spo14] Herbert Spohn, Nonlinear fluctuating hydrodynamics for anharmonic chains, J. Stat. Phys. 154 (2014), no. 5, 1191–1227. MR 3176405, https://doi.org/10.1007/s10955-014-0933-y
[ST18] H. Shen and L.-C. Tsai,
Stochastic telegraph equation limit for the stochastic six vertex model,
arXiv:1807.04678, 2018.
[SW18] Hao Shen and Hendrik Weber, Glauber dynamics of 2D Kac-Blume-Capel model and their stochastic PDE limits, J. Funct. Anal. 275 (2018), no. 6, 1321–1367. MR 3820327, https://doi.org/10.1016/j.jfa.2017.12.014
[SX16] Hao Shen and Weijun Xu, Weak universality of dynamical Φ⁴₃: non-Gaussian noise, Stoch. Partial Differ. Equ. Anal. Comput. 6 (2018), no. 2, 211–254. MR 3818405, https://doi.org/10.1007/s40072-017-0107-4
[Wal86] John B. Walsh, An introduction to stochastic partial differential equations, École d’été de probabilités de Saint-Flour, XIV—1984, Lecture Notes in Math., vol. 1180, Springer, Berlin, 1986, pp. 265–439. MR 876085, https://doi.org/10.1007/BFb0074920
[Yan18] K. Yang,
The KPZ equation, non-equilibrium solutions, and weak universality for long-range interactions,
arXiv:1810.02836, 2018.
[Zhu90] Ming Zhu, Equilibrium fluctuations for one-dimensional Ginzburg-Landau lattice model, Nagoya Math. J. 117 (1990), 63–92. MR 1044937, https://doi.org/10.1017/S0027763000001811
[ZZ15] Rongchan Zhu and Xiangchan Zhu, Three-dimensional Navier-Stokes equations driven by space-time white noise, J. Differential Equations 259 (2015), no. 9, 4443–4508. MR 3373412, https://doi.org/10.1016/j.jde.2015.06.002
[ZZ18] Rongchan Zhu and Xiangchan Zhu, Lattice approximation to the dynamical Φ₃⁴ model, Ann. Probab. 46 (2018), no. 1, 397–455. MR 3758734, https://doi.org/10.1214/17-AOP1188

Additional Information

Ivan Corwin
Affiliation: Department of Mathematics, Columbia University, 2990 Broadway, New York, New York 10027
Email: corwin@math.columbia.edu

Hao Shen
Affiliation: Department of Mathematics, University of Wisconsin–Madison, 480 Lincoln Drive, Madison, Wisconsin 53706
Email: pkushenhao@gmail.com

DOI: https://doi.org/10.1090/bull/1670
Received by editor(s): April 8, 2019
Published electronically: September 26, 2019
Additional Notes: The first author was partially supported by the Packard Fellowship for Science and Engineering, and by the NSF through DMS-1811143 and DMS-1664650
The second author was partially supported by the NSF through DMS-1712684 and DMS-1909525
Article copyright: © Copyright 2019 American Mathematical Society