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On global solutions of the random Hamilton-Jacobi equations and the KPZ problem. (English) Zbl 1387.37073

Summary: In this paper, we discuss possible qualitative approaches to the problem of KPZ universality. Throughout the paper, our point of view is based on the geometrical and dynamical properties of minimisers and shocks forming interlacing tree-like structures. We believe that the KPZ universality can be explained in terms of statistics of these structures evolving in time. The paper is focussed on the setting of the random Hamilton-Jacobi equations. We formulate several conjectures concerning global solutions and discuss how their properties are connected to the KPZ scalings in dimension \(1+1\). In the case of general viscous Hamilton-Jacobi equations with non-quadratic Hamiltonians, we define generalised directed polymers. We expect that their behaviour is similar to the behaviour of classical directed polymers, and present arguments in favour of this conjecture. We also define a new renormalisation transformation defined in purely geometrical terms and discuss conjectural properties of the corresponding fixed points. Most of our conjectures are widely open, and supported by only partial rigorous results for particular models.

MSC:

37L55 Infinite-dimensional random dynamical systems; stochastic equations
35R60 PDEs with randomness, stochastic partial differential equations
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60K37 Processes in random environments
82D60 Statistical mechanics of polymers
82C22 Interacting particle systems in time-dependent statistical mechanics
82C24 Interface problems; diffusion-limited aggregation in time-dependent statistical mechanics
82C41 Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics
82C44 Dynamics of disordered systems (random Ising systems, etc.) in time-dependent statistical mechanics
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References:

[1] Alberts T, Khanin K and Quastel J 2014 The intermediate disorder regime for directed polymers in dimension 1+1 {\it Ann. Probab.}42 1212-56 · Zbl 1292.82014
[2] Amir G, Corwin I and Quastel J 2011 Probability distribution of the free energy of the continuum directed random polymer in 1  +  1 dimensions {\it Commun. Pure Appl. Math.}64 466-537 · Zbl 1222.82070
[3] Arratia R A 1979 Coalescing Brownian motions on the line ProQuest LLC, Ann Arbor, MI {\it PhD Thesis} The University of Wisconsin
[4] Baik J, Deift P and Johansson K 1999 On the distribution of the length of the longest increasing subsequence of random permutations {\it J. Am. Math. Soc.}12 1119-78 · Zbl 0932.05001
[5] Bakhtin Y and Li L 2016 Thermodynamic limit for directed polymers and stationary solutions of the Burgers equation {\it Commun. Pure Appl. Math.} (arXiv: 1607.04864) accepted · Zbl 1455.60134
[6] Bakhtin Y and Li L 2017 The inviscid limit for space-time stationary solutions of the Burgers equation, and zero-temperature limit for directed polymers (arXiv: 1706.09950) · Zbl 1401.82061
[7] Bakhtin Y 2007 Burgers equation with random boundary conditions {\it Proc. Am. Math. Soc.}135 2257-62 · Zbl 1119.35126
[8] Bakhtin Y 2013 The Burgers equation with Poisson random forcing {\it Ann. Probab.}41 2961-89 · Zbl 1286.60099
[9] Bakhtin Y 2016 Inviscid Burgers equation with random kick forcing in noncompact setting {\it Electron. J. Probab.}21 50 · Zbl 1338.37117
[10] Bakhtin Y, Cator E and Khanin K 2014 Space-time stationary solutions for the Burgers equation {\it J. Am. Math. Soc.}27 193-238 · Zbl 1296.37051
[11] Balázs M, Cator E and Seppäläinen T 2006 Cube root fluctuations for the corner growth model associated to the exclusion process {\it Electron. J. Probab.}11 1094-132 · Zbl 1139.60046
[12] Bates E and Chatterjee S 2016 The endpoint distribution of directed polymers (arXiv: 1612.03443) · Zbl 1444.60087
[13] Bolthausen E 1989 A note on the diffusion of directed polymers in a random environment {\it Commun. Math. Phys.}123 529-34 · Zbl 0684.60013
[14] Boritchev A and Khanin K 2013 On the hyperbolicity of minimizers for 1D random Lagrangian systems {\it Nonlinearity}26 65-80 · Zbl 1263.35189
[15] Borodin A and Corwin I 2014 Macdonald processes {\it Probab. Theory Relat. Fields}158 225-400
[16] Borodin A, Corwin I and Ferrari P 2014 Free energy fluctuations for directed polymers in random media in 1+1 dimension {\it Commun. Pure Appl. Math.}67 1129-214 · Zbl 1295.82035
[17] Borodin A, Corwin I, Ferrari P and Vetõ B 2015 Height fluctuations for the stationary KPZ equation {\it Math. Phys. Anal. Geom.}18 95 · Zbl 1332.82068
[18] Borodin A and Ferrari P L 2008 Large time asymptotics of growth models on space-like paths. I. PushASEP {\it Electron. J. Probab.}13 1380-418 · Zbl 1187.82084
[19] Borodin A and Gorin V 2016 {\it Probability and Statistical Physics in St. Petersburg}{\it (Proc. of Symp. in Pure Mathematics vol 91)} (Providence, RI: American Mathematical Society) pp 155-214
[20] Borodin A, Ferrari P L and Sasamoto T 2008 Large time asymptotics of growth models on space-like paths. II. PNG and parallel TASEP {\it Commun. Math. Phys.}283 417-49 · Zbl 1201.82030
[21] Boué M and Dupuis P 1998 A variational representation for certain functionals of Brownian motion {\it Ann. Probab.}26 1641-59 · Zbl 0936.60059
[22] Calabrese P, Le Doussal P and Rosso A 2010 Free-energy distribution of the directed polymer at high temperature {\it Europhys. Lett.}90 20002
[23] Calabrese P and Le Doussal P 2011 Exact solution for the Kardar-Parisi-Zhang equation with flat initial conditions {\it Phys. Rev. Lett.}106 250603
[24] Carmona P and Hu Y 2004 Fluctuation exponents and large deviations for directed polymers in a random environment {\it Stoch. Process. Appl.}112 285-308 · Zbl 1072.60091
[25] Cator E and Groeneboom P 2006 Second class particles and cube root asymptotics for Hammersley’s process {\it Ann. Probab.}34 1273-95 · Zbl 1101.60076
[26] Cator E and Pimentel L P R 2012 Busemann functions and equilibrium measures in last passage percolation models {\it Probab. Theory Relat. Fields}154 89-125 · Zbl 1262.60094
[27] Chen Y, Georgiou T T and Pavon M 2016 On the relation between optimal transport and Schrödinger bridges: a stochastic control viewpoint {\it J. Optim. Theory Appl.}169 671-91 · Zbl 1344.49072
[28] Comets F, Shiga T and Yoshida N 2003 Directed polymers in a random environment: path localization and strong disorder {\it Bernoulli}9 705-23 · Zbl 1042.60069
[29] Corwin I 2012 The Kardar-Parisi-Zhang equation and universality class {\it Random Matrices Theory Appl.}1 1130001 · Zbl 1247.82040
[30] Corwin I, Quastel J and Remenik D 2015 Renormalization fixed point of the KPZ universality class {\it J. Stat. Phys.}160 815-34 · Zbl 1327.82064
[31] Dotsenko V 2010 Bethe ansatz derivation of the tracy-widom distribution for one-dimensional directed polymers {\it Europhys. Lett.}90 20003 · Zbl 1456.82249
[32] Weinan E, Khanin K, Mazel A and Sinai Y 2000 Invariant measures for Burgers equation with stochastic forcing {\it Ann. Math.}151 877-960 · Zbl 0972.35196
[33] Eyink G L and Drivas T D 2015 Spontaneous stochasticity and anomalous dissipation for Burgers equation {\it J. Stat. Phys.}158 386-432 · Zbl 1315.35163
[34] Fleming W H and Soner H M 2006 {\it Controlled Markov Processes and Viscosity Solutions}{\it (Stochastic Modelling and Applied Probability vol 25)} 2nd edn (New York: Springer) · Zbl 1105.60005
[35] Gomes D, Iturriaga R, Khanin K and Padilla P 2005 Viscosity limit of stationary distributions for the random forced Burgers equation {\it Mosc. Math. J.}5 613-31 · Zbl 1115.35081
[36] Gubinelli M and Perkowski N 2017 KPZ reloaded {\it Commun. Math. Phys.}349 165-269 · Zbl 1388.60110
[37] Hairer M 2013 Solving the KPZ equation {\it Ann. Math.}178 559-664 · Zbl 1281.60060
[38] Hairer M 2014 A theory of regularity structures {\it Inventory Math.}198 269-504 · Zbl 1332.60093
[39] Imbrie J Z and Spencer T 1988 Diffusion of directed polymers in a random environment {\it J. Stat. Phys.}52 609-26 · Zbl 1084.82595
[40] Iturriaga R and Khanin K 2003 Burgers turbulence and random Lagrangian systems {\it Commun. Math. Phys.}232 377-428 · Zbl 1029.76030
[41] Johansson K 2000 Shape fluctuations and random matrices {\it Commun. Math. Phys.}209 437-76 · Zbl 0969.15008
[42] Khanin K and Zhang K 2017 Hyperbolicity of minimizers and regularity of viscosity solutions for random Hamilton-Jacobi equations {\it Commun. Math. Phys.}355 803-37 · Zbl 1386.35503
[43] Kifer Y 1997 The Burgers equation with a random force and a general model for directed polymers in random environments {\it Probab. Theory Relat. Fields}108 29-65 · Zbl 0883.60059
[44] Le Doussal P and Calabrese P 2012 The KPZ equation with flat initial condition and the directed polymer with one free end {\it J. Stat. Mech.} P06001 · Zbl 1459.82352
[45] Ledrappier F and Young L S 1988 Entropy formula for random transformations {\it Probab. Theory Relat. Fields}80 217-40 · Zbl 0638.60054
[46] Moreno Flores G, Quastel J and Remenik D 2013 Endpoint distribution of directed polymers in 1+1 dimensions {\it Commun. Math. Phys.}317 363-80 · Zbl 1257.82117
[47] Munasinghe R, Rajesh R, Tribe R and Zaboronski O 2006 Multi-scaling of the n-point density function for coalescing Brownian motions {\it Commun. Math. Phys.}268 717-25 · Zbl 1122.60089
[48] Piterbarg L I and Piterbarg V V 1999 Intermittency of the tracer gradient {\it Commun. Math. Phys.}202 237-53
[49] Piterbarg V V 1998 Expansions and contractions of isotropic stochastic flows of homeomorphisms {\it Ann. Probab.}26 479-99 · Zbl 0936.60041
[50] Piterbarg V V 1997 Expansions and contractions of Stochastic flows {\it PhD Thesis} University of Southern California
[51] Prähofer M and Spohn H 2002 Scale invariance of the PNG droplet and the Airy process {\it J. Stat. Phys.}108 1071-106 · Zbl 1025.82010
[52] Quastel J and Spohn H 2015 The one-dimensional KPZ equation and its universality class {\it J. Stat. Phys.}160 965-84 · Zbl 1327.82069
[53] Sasamoto T and Spohn H 2010 Exact height distributions for the KPZ equation with narrow wedge initial condition {\it Nucl. Phys.} B 834 523-42 · Zbl 1204.35137
[54] Sasamoto T and Spohn H 2010 One-dimensional Kardar-Parisi-Zhang equation: an exact solution and its universality {\it Phys. Rev. Lett.}104 230602 · Zbl 1459.82195
[55] Seppäläinen T 2012 Scaling for a one-dimensional directed polymer with boundary conditions {\it Ann. Probab.}40 19-73 · Zbl 1254.60098
[56] Sinai Y G 1995 A remark concerning random walks with random potentials {\it Fundam. Math.}147 173-80 · Zbl 0835.60062
[57] Tracy C A and Widom H 1996 On orthogonal and symplectic matrix ensembles {\it Commun. Math. Phys.}177 727-54 · Zbl 0851.60101
[58] Tribe R and Zaboronski O 2011 Pfaffian formulae for one dimensional coalescing and annihilating systems {\it Electron. J. Probab.}16 2080-103 · Zbl 1244.60097
[59] Üstünel A S 2014 Variational calculation of Laplace transforms via entropy on Wiener space and applications {\it J. Funct. Anal.}267 3058-83 · Zbl 1306.60064
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