Abstract
We analyze the one-dimensional periodic Kardar–Parisi–Zhang equation in the language of paracontrolled distributions, giving an alternative viewpoint on the seminal results of Hairer. Apart from deriving a basic existence and uniqueness result for paracontrolled solutions to the KPZ equation we perform a thorough study of some related problems. We rigorously prove the links between the KPZ equation, stochastic Burgers equation, and (linear) stochastic heat equation and also the existence of solutions starting from quite irregular initial conditions. We also show that there is a natural approximation scheme for the nonlinearity in the stochastic Burgers equation. Interpreting the KPZ equation as the value function of an optimal control problem, we give a pathwise proof for the global existence of solutions and thus for the strict positivity of solutions to the stochastic heat equation. Moreover, we study Sasamoto–Spohn type discretizations of the stochastic Burgers equation and show that their limit solves the continuous Burgers equation possibly with an additional linear transport term. As an application, we give a proof of the invariance of the white noise for the stochastic Burgers equation that does not rely on the Cole–Hopf transform.
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Allez, R., Chouk, K.: The continuous Anderson hamiltonian in dimension two. arXiv:1511.02718 (2015)
Amir G., Corwin I., Quastel J.: Probability distribution of the free energy of the continuum directed random polymer in 1 + 1 dimensions. Commun. Pure Appl. Math. 64(4), 466–537 (2011)
Alberts T., Khanin K., Quastel J.: The intermediate disorder regime for directed polymers in dimension 1 + 1. Ann. Probab. 42(3), 1212–1256 (2014)
Assing S.: A pregenerator for Burgers equation forced by conservative noise. Commun. Math. Phys. 225(3), 611–632 (2002)
Assing S.: A rigorous equation for the Cole–Hopf solution of the conservative KPZ equation. Stoch. Partial Differ. Equ. Anal. Comput. 1(2), 365–388 (2013)
Bailleul I., Bernicot F.: Heat semigroup and singular PDEs. J. Funct. Anal. 270(9), 3344–3452 (2016)
Bailleul, I., Bernicot, F., Frey, D.: Higher order paracontrolled calculus and 3d-PAM equation. arXiv:1506.08773 (2015)
Bahouri H., Chemin J.-Y., Danchin R.: Fourier Analysis and Nonlinear Partial Differential Equations. Springer, New York (2011)
Bakhtin Y., Cator E., Khanin K.: Space–time stationary solutions for the Burgers equation. J. Am. Math. Soc. 27(1), 193–238 (2014)
Boué M., Dupuis P.: A variational representation for certain functionals of Brownian motion. Ann. Probab. 26(4), 1641–1659 (1998)
Bertini L., Giacomin G.: Stochastic Burgers and KPZ equations from particle systems. Commun. Math. Phys. 183(3), 571–607 (1997)
Bony J.-M.: Calcul symbolique et propagation des singularites pour les équations aux dérivées partielles non linéaires, Ann. Sci. Éc. Norm. Supér. (4) 14, 209–246 (1981)
Catellier, R., Chouk, K.: Paracontrolled distributions and the 3-dimensional stochastic quantization equation. arXiv:1310.6869 (2013)
Cannizzaro, G., Chouk, K.: Multidimensional SDEs with singular drift and universal construction of the polymer measure with white noise potential. arXiv:1501.04751 (2015)
Chouk, K., Friz, P.: Support theorem for a singular semilinear stochastic partial differential equation. arXiv:1409.4250 (2014)
Cannizzaro, G., Friz, P., Gassiat, P.: Malliavin calculus for regularity structures: the case of gPAM. arXiv:1511.08888 (2015)
Chen, L., Kim, K.: On comparison principle and strict positivity of solutions to the nonlinear stochastic fractional heat equations. arXiv:1410.0604 (2014)
Catuogno P., Olivera C.: Strong solution of the stochastic Burgers equation. Appl. Anal. 93(3), 646–652 (2014)
Corwin I.: The Kardar–Parisi–Zhang equation and universality class. Random Matrices Theory Appl. 1(1), 1130001, 76 (2012)
Corwin, I., Tsai, L.-C.: KPZ equation limit of higher-spin exclusion processes. arXiv:1505.04158 (2015)
Delarue F., Diel R.: Rough paths and 1D SDE with a time dependent distributional drift: application to polymers. Probab. Theory Relat. Fields 165(1), 1–63 (2016)
Da Prato G., Zabczyk J.: Stochastic Equations in Infinite Dimensions, pp. 152. Cambridge University Press, Cambridge (2014)
Dembo A., Tsai L.-C.: Weakly asymmetric non-simple exclusion process and the Kardar–Parisi–Zhang equation. Commun. Math. Phys. 341(1), 219–261 (2013)
Echeverría P.: A criterion for invariant measures of Markov processes. Z. Wahrsch. Verw. Gebiete. 61(1), 1–16 (1982)
Friz, P.K., Hairer, M.: A course on rough paths. Universitext, Springer, Cham (2014) (with an introduction to regularity structures)
Forster D., Nelson David R., Stephen Michael J.: Large-distance and long-time properties of a randomly stirred fluid. Phys. Rev. A. 16(2), 732–749 (1977)
Funaki T., Quastel J.: KPZ equation, its renormalization and invariant measures. Stoch. Partial Differ. Equ. Anal. Comput. 3(2), 159–220 (2014)
Friz P., Shekhar A.: Doob–Meyer for rough paths. Bull. Inst. Math. Acad. Sin. (N.S.). 8(1), 73–84 (2013)
Furlan, M.: Stochastic Navier–Stokes equation in 3 dimensions. Master’s thesis (2014) (supervised by Massimiliano Gubinelli)
Gärtner J.: Convergence towards Burgers’ equation and propagation of chaos for weakly asymmetric exclusion processes. Stoch. Process. Appl. 27(2), 233–260 (1988)
Gubinelli M.: Controlling rough paths. J. Funct. Anal. 1, 86–140 (2004)
Gubinelli, M., Imkeller, P., Perkowski, N.: A Fourier analytic approach to pathwise stochastic integration. Electron. J. Probab. 21(2) (2016)
Gubinelli, M., Imkeller, P., Perkowski, N.: Paracontrolled distributions and singular PDEs. Forum Math. Pi. 3(6) (2015)
Gubinelli M., Jara M.: Regularization by noise and stochastic Burgers equations. Stoch. Partial Differ. Equ. Anal. Comput. 1(2), 325–350 (2013)
Gonçalves P., Jara M.: Nonlinear fluctuations of weakly asymmetric interacting particle systems. Arch. Ration. Mech. Anal. 212(2), 597–644 (2014)
Gubinelli M., Lejay A., Tindel S.: Young integrals and SPDEs. Potential Anal. 25(4), 307–326 (2006)
Gubinelli M., Perkowski N.: Lectures on singular stochastic PDEs. Ensaios Math. 29, 1–89 (2015)
Gubinelli, M., Perkowski, N.: Energy solutions of KPZ are unique. arXiv:1508.07764 (2015)
Hairer M.: Rough stochastic PDEs. Commun. Pure Appl. Math. 64(11), 1547–1585 (2011)
Hairer M.: Solving the KPZ equation. Ann. Math. 178(2), 559–664 (2013)
Hairer M.: A theory of regularity structures. Invent. Math. 198(2), 269–504 (2014)
Hairer, M., Labbé, C.: Multiplicative stochastic heat equations on the whole space. preprint arXiv:1504.07162 (2015)
Hairer, M., Matetski, K.: Discretisations of rough stochastic PDEs. arXiv:1511.06937 (2015)
Hairer M., Maas J.: A spatial version of the Itô–Stratonovich correction. Ann. Probab. 40(4), 1675–1714 (2012)
Hairer M., Maas J., Weber H.: Approximating rough stochastic PDEs. Commun. Pure Appl. Math. 67(5), 776–870 (2014)
Hairer M., Pillai Natesh S.: Regularity of laws and ergodicity of hypoelliptic SDEs driven by rough paths. Ann. Probab. 41(4), 2544–2598 (2013)
Hairer, M., Quastel, J.: A class of growth models rescaling to KPZ. arXiv:1512.07845 (2015)
Hairer, M., Shen, H.: A central limit theorem for the KPZ equation. arXiv:1507.01237 (2015)
Kardar M., Parisi G., Zhang Y.-C.: Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56(9), 889–892 (1986)
Karatzas I., Shreve Steven E.: Brownian Motion and Stochastic Calculus. Springer, New York (1988)
Krug, J., Spohn, H.: Kinetic Roughening of Growing Surfaces, vol. 1(99), p. 1. C. Godreche, Cambridge University Press, Cambridge (1991)
Kupiainen, A.: Renormalization group and stochastic PDEs. Ann. Henri Poincaré (Springer), 1–39 (2014)
Lam C.-H., Shin Franklin G.: Improved discretization of the Kardar–Parisi–Zhang equation. Phys. Rev. E 58(5), 5592–5595 (1998)
Meyer, Y.: Remarques sur un théorème de J.-M. Bony. Rend. Circ. Mate. Palermo. Ser. II, 1–20 (1981)
Moreno F., Gregorio R.: On the (strict) positivity of solutions of the stochastic heat equation. Ann. Probab. 42(4), 1635–1643 (2014)
Mueller C.: On the support of solutions to the heat equation with noise. Stoch. Stoch. Rep. 37(4), 225–245 (1991)
Mourrat, J.-C., Weber, H.: Convergence of the two-dimensional dynamic Ising–Kac model to \({\phi^{4}_{2}}\). arXiv:1410.1179 (2014)
Perkowski, N.: Studies of robustness in stochastic analysis and mathematical finance. Ph.D. thesis (2014)
Prömel David J., Trabs M.: Rough differential equations driven by signals in Besov spaces. J. Differ. Equ. 260(6), 5202–5249 (2016)
Quastel J., Spohn H.: The one-dimensional KPZ equation and its universality class. J. Stat. Phys. 160(4), 965–984 (2015)
Quastel, J.: The Kardar–Parisi–Zhang equation and universality class. In: XVIIth International Congress on Mathematical Physics, pp. 113–133 (2014)
Sasamoto T., Spohn H.: Superdiffusivity of the 1D lattice Kardar–Parisi–Zhang equation. J. Stat. Phys. 137(5–6), 917–935 (2009)
Sasamoto T., Spohn H.: Exact height distributions for the KPZ equation with narrow wedge initial condition. Nuclear Phys. B. 834(3), 523–542 (2010)
Üstünel A.S.: Variational calculation of Laplace transforms via entropy on Wiener space and applications. J. Funct. Anal. 267(8), 3058–3083 (2014)
Weinan E., Khanin K., Mazel A., Sinai Ya.: Invariant measures for Burgers equation with stochastic forcing. Ann. Math. (2) 151(3), 877–960 (2000)
Zabusky Norman J., Kruskal Martin D.: Interaction of “solitons” in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett. 15(6), 240 (1965)
Zhu, R., Zhu, X.: Approximating three-dimensional Navier–Stokes equations driven by space–time white noise. arXiv:1409.4864 (2014)
Zhu R., Zhu X.: Three-dimensional Navier–Stokes equations driven by space–time white noise. J. Differ. Equ. 259(9), 4443–4508 (2015)
Zhu, R., Zhu, X.: Lattice approximation to the dynamical \({\Phi_{3}^{4}}\) model. arXiv:1508.05613 (2015)
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Gubinelli, M., Perkowski, N. KPZ Reloaded. Commun. Math. Phys. 349, 165–269 (2017). https://doi.org/10.1007/s00220-016-2788-3
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DOI: https://doi.org/10.1007/s00220-016-2788-3