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An invariance principle for the two-dimensional parabolic Anderson model with small potential. (English) Zbl 1391.60154

The discrete parabolic Anderson model (PAM) is the infinite-dimensional random ODE \[ \begin{aligned} \partial _t v (t, i) = \Delta v (t, i) + v (t, i) \eta (i),\quad (t,i)\in [0,+\infty )\times \mathbb {Z}^d, \end{aligned} \] where \(\Delta\) is the discrete Laplacian and \((\eta (i) : i \in \mathbb {Z}^d)\) is an i.i.d. family of random variables with sufficiently many moments. An invariance principle for this model with small potential is proved. The proof is based on paracontrolled distributions and a certain operator technique.
As a corollary a Donsker-type invariance principle for a certain random polymer measure is shown, as well as a universality result for the spectrum of discrete random Schrödinger operators on large boxes with small potentials.

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
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[1] Alberts, T., Khanin, K., Quastel, J.: The intermediate disorder regime for directed polymers in dimension \[1+ 11+1\]. Ann. Probab. 42(3), 1212-1256 (2014) · Zbl 1292.82014
[2] Allez, R., Chouk, K.: The continuous Anderson Hamiltonian in dimension two (2015), preprint arXiv:1511.02718 · Zbl 1416.35303
[3] Bahouri, H., Chemin, J.-Y.: Danchin, Raphael, Fourier analysis and nonlinear partial differential equations. Springer, Berlin (2011) · Zbl 1227.35004
[4] Bailleul, I., Bernicot, F.: Heat semigroup and singular PDEs. J. Funct. Anal. 270(9), 3344-3452 (2016) · Zbl 1416.35303
[5] Bailleul, Ismaël, Bernicot, F., Frey, D.: Higher order paracontrolled calculus and 3d-PAM equation, (2015), arXiv preprint arXiv:1506.08773 · Zbl 1430.60053
[6] Biskup, M., Fukushima, R., König, W.: Eigenvalue fluctuations for lattice Anderson Hamiltonians. SIAM J. Math. Anal. 48(4), 2674-2700 (2016) · Zbl 1345.60058
[7] 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) · Zbl 0495.35024
[8] Brown, B.M.: Martingale central limit theorems. Ann. Math. Statist. 42(1), 59-66 (1971) · Zbl 0218.60048
[9] Bruned, Y.: Singular KPZ type equations, Ph.D. Thesis (2015) · Zbl 1336.60120
[10] Cannizzaro, G., Chouk, K.: Multidimensional SDEs with singular drift and universal construction of the polymer measure with white noise potential (2015), arXiv preprint arXiv:1501.04751 · Zbl 1407.60109
[11] Caravenna, F., Sun, R., Zygouras, N.: Polynomial chaos and scaling limits of disordered systems (2013), arXiv preprint arXiv:1312.3357 · Zbl 1364.82026
[12] Carmona, R., Molchanov, S.A.: Parabolic Anderson problem and intermittency, vol. 518. American Mathematical Soc, Providence (1994) · Zbl 0925.35074
[13] Catellier, R., Chouk, K.: Paracontrolled distributions and the 3-dimensional stochastic quantization equation (2013), arXiv preprint arXiv:1310.6869 · Zbl 1433.60048
[14] Chandra, A., Shen, H.: Glauber dynamics of 2D Kac-Blume-Capel model and their stochastic PDE limits (2016), arXiv preprint arXiv:1608.06556
[15] Chandra, A., Shen, H.: Moment bounds for SPDEs with non-Gaussian fields and application to the Wong-Zakai problem (2016), arXiv preprint arXiv:1605.05683 · Zbl 1379.60064
[16] Ethier, S.N., Kurtz, T.G.: Markov Processes: Characterization and Convergence. Wiley, Hoboken (2005) · Zbl 1089.60005
[17] Friz, P.K., Hairer, Martin: A Course on Rough Paths: With an Introduction to Regularity Structures. Springer, Berlin (2014) · Zbl 1327.60013
[18] Gubinelli, M.: Controlling rough paths. J. Funct. Anal. 216(1), 86-140 (2004) · Zbl 1058.60037
[19] Gubinelli, M., Imkeller, P., Perkowski, N.: Paracontrolled Distributions and Singular PDEs, p. e6. Cambridge University Press, Cambridge (2015) · Zbl 1333.60149
[20] Gubinelli, M., Perkowski, N.: KPZ reloaded (2015), arXiv preprint arXiv:1508.03877 · Zbl 1388.60110
[21] Gubinelli, M., Perkowski, N.: Lectures on singular stochastic PDEs, Ensaois Mat. 29 (2015) · Zbl 1337.60005
[22] Hairer, M.: Rough stochastic PDEs. Comm. Pure Appl. Math. 64(11), 1547-1585 (2011) · Zbl 1229.60079
[23] Hairer, M.: Solving the KPZ equation. Ann. Math. 178(2), 559-664 (2013) · Zbl 1281.60060
[24] Hairer, M.: A theory of regularity structures. Invent. Math. (2014). doi:10.1007/s00222-014-0505-4 · Zbl 1332.60093
[25] Hairer, M.: The motion of a random string (2016), arXiv preprint arXiv:1605.02192 · Zbl 1037.82022
[26] Hairer, M., Labbé, C.: Multiplicative stochastic heat equations on the whole space (2015), arXiv preprint arXiv:1504.07162 · Zbl 1447.60102
[27] Hairer, M., Maas, J.: A spatial version of the Itô-Stratonovich correction. Ann. Probab. 40(4), 1675-1714 (2012) · Zbl 1262.60060
[28] Hairer, M., Maas, J., Weber, H.: Approximating rough stochastic PDEs. Comm. Pure Appl. Math. 67(5), 776-870 (2014) · Zbl 1302.60095
[29] Hairer, M., Matetski, K.: Discretisations of rough stochastic PDEs (2015), arXiv preprint arXiv:1511.06937 · Zbl 1357.60069
[30] Hairer, M., Pardoux, É.: A Wong-Zakai theorem for stochastic PDEs. J. Math. Soc. Jpn. 67(4), 1551-1604 (2015) · Zbl 1341.60062
[31] Hairer, M., Quastel, J.: A class of growth models rescaling to KPZ (2015), arXiv preprint arXiv:1512.07845 · Zbl 1429.60057
[32] Hairer, M., Shen, H.: A central limit theorem for the KPZ equation (2015), arXiv preprint arXiv:1507.01237 · Zbl 1388.60111
[33] Hairer, M., Shen, H.: The dynamical sine-Gordon model. Comm. Math. Phys. 341(3), 933-989 (2016) · Zbl 1336.60120
[34] Hairer, M., Xu, W.: Large scale behaviour of 3D phase coexistence models (2016), arXiv preprint arXiv:1601.05138
[35] Hoshino, M.: KPZ equation with fractional derivatives of white noise (2016), arXiv preprint arXiv:1602.04570 · Zbl 1380.35116
[36] Hoshino, M.: Paracontrolled calculus and Funaki-Quastel approximation for the KPZ equation (2016), arXiv preprint arXiv:1605.02624 · Zbl 1384.35149
[37] Janson, Svante: Gaussian Hilbert spaces, Cambridge Tracts in Mathematics, vol. 129. Cambridge University Press, Cambridge (1997) · Zbl 0887.60009
[38] König, W.: The Parabolic Anderson Model. Random Walk in Random Potential. Birkhäuser, Basel (2016) · Zbl 1378.60007
[39] König, W., Schmidt, S.: The parabolic Anderson model with acceleration and deceleration. In: Deuschel, J.D., Gentz, B., König, W., von Renesse, M., Scheutzow, M., Schmock, U. (eds.) Probability in Complex Physical Systems. Springer Proceedings in Mathematics, vol. 11. Springer, Berlin, Heidelberg (2012)
[40] Kupiainen, A.: Renormalization group and stochastic PDEs. Annales Henri Poincaré 17(3), 497-535 (2016) · Zbl 1347.81063
[41] Kupiainen, A., Marcozzi, M.: Renormalization of generalized KPZ equation (2016), arXiv preprint arXiv:1604.08712 · Zbl 1369.82011
[42] Merkl, F., Wüthrich, M.V.: Annealed survival asymptotics for Brownian motion in a scaled Poissonian potential. Stochastic processes and their applications 96(2), 191-211 (2001) · Zbl 1062.82022
[43] Merkl, F., Wüthrich, M.V.: Phase transition of the principal Dirichlet eigenvalue in a scaled Poissonian potential. Probab. Theory Relat. Fields 119(4), 475-507 (2001) · Zbl 1037.82022
[44] Merkl, F., Wüthrich, M.V.: Infinite volume asymptotics of the ground state energy in a scaled Poissonian potential. Annales de l’IHP Probabilités et statistiques 38, 253-284 (2002) · Zbl 0996.82036
[45] Mossel, E., O’Donnell, R., Oleszkiewicz, K.: Noise stability of functions with low influences: invariance and optimality. Ann. Math. 171, 295-341 (2010) · Zbl 1201.60031
[46] Mourrat, J.-C., Weber, H.: Convergence of the two-dimensional dynamic Ising-Kac model to \[\phi^4_2\] ϕ24 (2014), arXiv preprint arXiv:1410.1179 · Zbl 1037.82022
[47] Prömel, D.J., Trabs, M.: Rough differential equations driven by signals in Besov spaces. J. Differ. Eq. 260(6), 5202-5249 (2016) · Zbl 1339.34065
[48] Schmidt, S.: Das parabolische Anderson-Modell mit Be- und Entschleunigung. Universität Leipzig, Leipzig (2010)
[49] Shen, H., Xu, W.: Weak universality of dynamical \[\phi^4_3\] ϕ34: non-Gaussian noise, (2016), arXiv preprint arXiv:1601.05724 · Zbl 1201.60031
[50] Zhu, R., Zhu, X.: Approximating three-dimensional Navier-Stokes equations driven by space-time white noise (2014), arXiv preprint arXiv:1409.4864
[51] Zhu, R., Zhu, X.: Lattice approximation to the dynamical \[\phi_3^4\] ϕ34 model (2015) arXiv preprint arXiv:1508.05613
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