Renormalization of the two-dimensional stochastic nonlinear wave equations

Gubinelli, Massimiliano; Koch, Herbert; Oh, Tadahiro

doi:10.1090/tran/7452

Renormalization of the two-dimensional stochastic nonlinear wave equations
HTML articles powered by AMS MathViewer

by Massimiliano Gubinelli, Herbert Koch and Tadahiro Oh PDF

Trans. Amer. Math. Soc. 370 (2018), 7335-7359 Request permission

Abstract:

We study the two-dimensional stochastic nonlinear wave equations (SNLW) with an additive space-time white noise forcing. In particular, we introduce a time-dependent renormalization and prove that SNLW is pathwise locally well-posed. As an application of the local well-posedness argument, we also establish a weak universality result for the renormalized SNLW.

References

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, DOI 10.1080/17442509608834039
Richard F. Bass, Stochastic processes, Cambridge Series in Statistical and Probabilistic Mathematics, vol. 33, Cambridge University Press, Cambridge, 2011. MR 2856623, DOI 10.1017/CBO9780511997044
Árpád Bényi and Tadahiro Oh, Modulation spaces, Wiener amalgam spaces, and Brownian motions, Adv. Math. 228 (2011), no. 5, 2943–2981. MR 2838066, DOI 10.1016/j.aim.2011.07.023
Jean Bourgain, Invariant measures for the $2$D-defocusing nonlinear Schrödinger equation, Comm. Math. Phys. 176 (1996), no. 2, 421–445. MR 1374420, DOI 10.1007/BF02099556
N. Burq, L. Thomann, and N. Tzvetkov, Remarks on the Gibbs measures for nonlinear dispersive equations, Ann. Fac. Sci. Toulouse Math., to appear.
Nicolas Burq and Nikolay Tzvetkov, Random data Cauchy theory for supercritical wave equations. I. Local theory, Invent. Math. 173 (2008), no. 3, 449–475. MR 2425133, DOI 10.1007/s00222-008-0124-z
R. R. Coifman and Yves Meyer, On commutators of singular integrals and bilinear singular integrals, Trans. Amer. Math. Soc. 212 (1975), 315–331. MR 380244, DOI 10.1090/S0002-9947-1975-0380244-8
James Colliander and Tadahiro Oh, Almost sure well-posedness of the cubic nonlinear Schrödinger equation below $L^2(\Bbb T)$, Duke Math. J. 161 (2012), no. 3, 367–414. MR 2881226, DOI 10.1215/00127094-1507400
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, DOI 10.1006/jfan.2002.3919
Dashan Fan and Shuichi Sato, Transference on certain multilinear multiplier operators, J. Aust. Math. Soc. 70 (2001), no. 1, 37–55. MR 1808390, DOI 10.1017/S1446788700002263
J. Ginibre and G. Velo, Generalized Strichartz inequalities for the wave equation, J. Funct. Anal. 133 (1995), no. 1, 50–68. MR 1351643, DOI 10.1006/jfan.1995.1119
Massimiliano Gubinelli and Nicolas Perkowski, KPZ reloaded, Comm. Math. Phys. 349 (2017), no. 1, 165–269. MR 3592748, DOI 10.1007/s00220-016-2788-3
Massimiliano Gubinelli and Nicolas Perkowski, The Hairer-Quastel universality result at stationarity, Stochastic analysis on large scale interacting systems, RIMS Kôkyûroku Bessatsu, B59, Res. Inst. Math. Sci. (RIMS), Kyoto, 2016, pp. 101–115. MR 3675927
M. Hairer and J. Quastel, A class of growth models rescaling to KPZ, arXiv:1512.07845 [math-ph].
Markus Keel and Terence Tao, Endpoint Strichartz estimates, Amer. J. Math. 120 (1998), no. 5, 955–980. MR 1646048, DOI 10.1353/ajm.1998.0039
Hui-Hsiung Kuo, Introduction to stochastic integration, Universitext, Springer, New York, 2006. MR 2180429
Hans Lindblad and Christopher D. Sogge, On existence and scattering with minimal regularity for semilinear wave equations, J. Funct. Anal. 130 (1995), no. 2, 357–426. MR 1335386, DOI 10.1006/jfan.1995.1075
H. P. McKean, Statistical mechanics of nonlinear wave equations. IV. Cubic Schrödinger, Comm. Math. Phys. 168 (1995), no. 3, 479–491. MR 1328250, DOI 10.1007/BF02101840
Camil Muscalu and Wilhelm Schlag, Classical and multilinear harmonic analysis. Vol. II, Cambridge Studies in Advanced Mathematics, vol. 138, Cambridge University Press, Cambridge, 2013. MR 3052499
Edward Nelson, A quartic interaction in two dimensions, Mathematical Theory of Elementary Particles (Proc. Conf., Dedham, Mass., 1965) M.I.T. Press, Cambridge, Mass., 1966, pp. 69–73. MR 0210416
David Nualart, The Malliavin calculus and related topics, 2nd ed., Probability and its Applications (New York), Springer-Verlag, Berlin, 2006. MR 2200233
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, DOI 10.1080/10652469808819152
Michael Oberguggenberger and Francesco Russo, Singular limiting behavior in nonlinear stochastic wave equations, Stochastic analysis and mathematical physics, Progr. Probab., vol. 50, Birkhäuser Boston, Boston, MA, 2001, pp. 87–99. MR 1886565
Tadahiro Oh, Periodic stochastic Korteweg-de Vries equation with additive space-time white noise, Anal. PDE 2 (2009), no. 3, 281–304. MR 2603800, DOI 10.2140/apde.2009.2.281
T. Oh, J. Quastel, and P. Sosoe, Global dynamics for the stochastic KdV equation with white noise as initial data, in preparation.
T. Oh and L. Thomann, A pedestrian approach to the invariant Gibbs measure for the 2-d defocusing nonlinear Schrödinger equations, Stoch. Partial Differ. Equ. Anal. Comput. (2018), DOI 10.1007/s40072-018-0112-2.
T. Oh and L. Thomann, Invariant Gibbs measure for the 2-$d$ defocusing nonlinear wave equations, Ann. Fac. Sci. Toulouse Math., to appear.
Francesco Russo, Colombeau generalized functions and stochastic analysis, Stochastic analysis and applications in physics (Funchal, 1993) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 449, Kluwer Acad. Publ., Dordrecht, 1994, pp. 329–349. MR 1337971
Barry Simon, The $P(\phi )_{2}$ Euclidean (quantum) field theory, Princeton Series in Physics, Princeton University Press, Princeton, N.J., 1974. MR 0489552
Terence Tao, Nonlinear dispersive equations, CBMS Regional Conference Series in Mathematics, vol. 106, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006. Local and global analysis. MR 2233925, DOI 10.1090/cbms/106
Laurent Thomann and Nikolay Tzvetkov, Gibbs measure for the periodic derivative nonlinear Schrödinger equation, Nonlinearity 23 (2010), no. 11, 2771–2791. MR 2727169, DOI 10.1088/0951-7715/23/11/003
N. Tzvetkov, Construction of a Gibbs measure associated to the periodic Benjamin-Ono equation, Probab. Theory Related Fields 146 (2010), no. 3-4, 481–514. MR 2574736, DOI 10.1007/s00440-008-0197-z