Abstract
We prove an invariance principle for the two-dimensional lattice parabolic Anderson model with small potential. As applications we deduce a Donsker type convergence result for a discrete random polymer measure, as well as a universality result for the spectrum of discrete random Schrödinger operators on large boxes with small potentials. Our proof is based on paracontrolled distributions and some basic results for multiple stochastic integrals of discrete martingales.
This is a preview of subscription content, access via your institution.
Notes
- 1.
But not for \(d\ge 4\) because in that case the scaling factor \(\varepsilon ^{2 - d/2}\) vanishes or even blows up, so the intermittency effect will be very strong. This corresponds to the fact that in the language of Hairer [24] the continuous parabolic Anderson model is locally subcritical in dimensions 1, 2, 3, it is critical in dimension 4, and supercritical in dimensions \(d > 4\).
References
- 1.
Alberts, T., Khanin, K., Quastel, J.: The intermediate disorder regime for directed polymers in dimension \(1+ 1\). Ann. Probab. 42(3), 1212–1256 (2014)
- 1.
Allez, R., Chouk, K.: The continuous Anderson Hamiltonian in dimension two (2015), preprint arXiv:1511.02718
- 2.
Bahouri, H., Chemin, J.-Y.: Danchin, Raphael, Fourier analysis and nonlinear partial differential equations. Springer, Berlin (2011)
- 3.
Bailleul, I., Bernicot, F.: Heat semigroup and singular PDEs. J. Funct. Anal. 270(9), 3344–3452 (2016)
- 4.
Bailleul, Ismaël, Bernicot, F., Frey, D.: Higher order paracontrolled calculus and 3d-PAM equation, (2015), arXiv preprint arXiv:1506.08773
- 5.
Biskup, M., Fukushima, R., König, W.: Eigenvalue fluctuations for lattice Anderson Hamiltonians. SIAM J. Math. Anal. 48(4), 2674–2700 (2016)
- 6.
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)
- 7.
Brown, B.M.: Martingale central limit theorems. Ann. Math. Statist. 42(1), 59–66 (1971)
- 8.
Bruned, Y.: Singular KPZ type equations, Ph.D. Thesis (2015)
- 9.
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
- 10.
Caravenna, F., Sun, R., Zygouras, N.: Polynomial chaos and scaling limits of disordered systems (2013), arXiv preprint arXiv:1312.3357
- 11.
Carmona, R., Molchanov, S.A.: Parabolic Anderson problem and intermittency, vol. 518. American Mathematical Soc, Providence (1994)
- 12.
Catellier, R., Chouk, K.: Paracontrolled distributions and the 3-dimensional stochastic quantization equation (2013), arXiv preprint arXiv:1310.6869
- 13.
Chandra, A., Shen, H.: Glauber dynamics of 2D Kac–Blume–Capel model and their stochastic PDE limits (2016), arXiv preprint arXiv:1608.06556
- 14.
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
- 15.
Ethier, S.N., Kurtz, T.G.: Markov Processes: Characterization and Convergence. Wiley, Hoboken (2005)
- 16.
Friz, P.K., Hairer, Martin: A Course on Rough Paths: With an Introduction to Regularity Structures. Springer, Berlin (2014)
- 17.
Gubinelli, M.: Controlling rough paths. J. Funct. Anal. 216(1), 86–140 (2004)
- 18.
Gubinelli, M., Imkeller, P., Perkowski, N.: Paracontrolled Distributions and Singular PDEs, p. e6. Cambridge University Press, Cambridge (2015)
- 19.
Gubinelli, M., Perkowski, N.: KPZ reloaded (2015), arXiv preprint arXiv:1508.03877
- 20.
Gubinelli, M., Perkowski, N.: Lectures on singular stochastic PDEs, Ensaois Mat. 29 (2015)
- 21.
Hairer, M.: Rough stochastic PDEs. Comm. Pure Appl. Math. 64(11), 1547–1585 (2011)
- 22.
Hairer, M.: Solving the KPZ equation. Ann. Math. 178(2), 559–664 (2013)
- 23.
Hairer, M.: A theory of regularity structures. Invent. Math. (2014). doi:10.1007/s00222-014-0505-4
- 24.
Hairer, M.: The motion of a random string (2016), arXiv preprint arXiv:1605.02192
- 25.
Hairer, M., Labbé, C.: Multiplicative stochastic heat equations on the whole space (2015), arXiv preprint arXiv:1504.07162
- 26.
Hairer, M., Maas, J.: A spatial version of the Itô-Stratonovich correction. Ann. Probab. 40(4), 1675–1714 (2012)
- 27.
Hairer, M., Maas, J., Weber, H.: Approximating rough stochastic PDEs. Comm. Pure Appl. Math. 67(5), 776–870 (2014)
- 28.
Hairer, M., Matetski, K.: Discretisations of rough stochastic PDEs (2015), arXiv preprint arXiv:1511.06937
- 29.
Hairer, M., Pardoux, É.: A Wong–Zakai theorem for stochastic PDEs. J. Math. Soc. Jpn. 67(4), 1551–1604 (2015)
- 30.
Hairer, M., Quastel, J.: A class of growth models rescaling to KPZ (2015), arXiv preprint arXiv:1512.07845
- 31.
Hairer, M., Shen, H.: A central limit theorem for the KPZ equation (2015), arXiv preprint arXiv:1507.01237
- 32.
Hairer, M., Shen, H.: The dynamical sine-Gordon model. Comm. Math. Phys. 341(3), 933–989 (2016)
- 33.
Hairer, M., Xu, W.: Large scale behaviour of 3D phase coexistence models (2016), arXiv preprint arXiv:1601.05138
- 34.
Hoshino, M.: KPZ equation with fractional derivatives of white noise (2016), arXiv preprint arXiv:1602.04570
- 35.
Hoshino, M.: Paracontrolled calculus and Funaki–Quastel approximation for the KPZ equation (2016), arXiv preprint arXiv:1605.02624
- 36.
Janson, Svante: Gaussian Hilbert spaces, Cambridge Tracts in Mathematics, vol. 129. Cambridge University Press, Cambridge (1997)
- 37.
König, W.: The Parabolic Anderson Model. Random Walk in Random Potential. Birkhäuser, Basel (2016)
- 38.
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)
- 39.
Kupiainen, A.: Renormalization group and stochastic PDEs. Annales Henri Poincaré 17(3), 497–535 (2016)
- 40.
Kupiainen, A., Marcozzi, M.: Renormalization of generalized KPZ equation (2016), arXiv preprint arXiv:1604.08712
- 41.
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)
- 42.
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)
- 43.
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)
- 44.
Mossel, E., O’Donnell, R., Oleszkiewicz, K.: Noise stability of functions with low influences: invariance and optimality. Ann. Math. 171, 295–341 (2010)
- 45.
Mourrat, J.-C., Weber, H.: Convergence of the two-dimensional dynamic Ising-Kac model to \(\phi ^4_2\) (2014), arXiv preprint arXiv:1410.1179
- 46.
Prömel, D.J., Trabs, M.: Rough differential equations driven by signals in Besov spaces. J. Differ. Eq. 260(6), 5202–5249 (2016)
- 47.
Schmidt, S.: Das parabolische Anderson–Modell mit Be- und Entschleunigung. Universität Leipzig, Leipzig (2010)
- 48.
Shen, H., Xu, W.: Weak universality of dynamical \(\phi ^4_3 \): non-Gaussian noise, (2016), arXiv preprint arXiv:1601.05724
- 49.
Zhu, R., Zhu, X.: Approximating three-dimensional Navier–Stokes equations driven by space-time white noise (2014), arXiv preprint arXiv:1409.4864
- 50.
Zhu, R., Zhu, X.: Lattice approximation to the dynamical \(\phi _3^4\) model (2015) arXiv preprint arXiv:1508.05613
Author information
Affiliations
Corresponding author
Additional information
We are very grateful to Massimiliano Gubinelli for countless discussions on the subject matter and in particular for suggesting the random operator approach that helped resolve the technically most challenging problem of the paper. We would also like to thank Hao Shen and Hendrik Weber for helpful discussions and suggestions, Jörg Martin, Rongchan Zhu, and Xiangchan Zhu for pointing out small mistakes in an earlier version of the paper, and the anonymous referees for their careful reading and for their corrections and suggestions which helped to improve the presentation.
Nicolas Perkowski: Financial support by the DFG via Research Unit FOR 2402 is gratefully acknowledged.
A criterion for the weak convergence of Markov processes
A criterion for the weak convergence of Markov processes
Lemma 8.1
([16], Theorem 2.11 in Chap. 4) Let E and \((E_N)_{N \in \mathbb {N}}\) be metric spaces such that E is compact and separable and assume that for all N we are given a measurable map \(\psi _N :E_N \rightarrow E\) and a semigroup \((P_N(t))_{t \in [0,T]}\) of a Markov process \(Y_N\) on \(E_N\), such that \(X_N = \psi _N (Y_N)\) has sample paths in D([0, T], E). Assume also that there exists a Feller semigroup \((P(t))_{t \in [0,T]}\) such that
for every \(f\in C(E, \mathbb {R})\), where \(\pi _N :L^{\infty }(E)\rightarrow L^{\infty }(E_N)\) is defined by the relation \(\pi _N f (x) =f(\psi _N(x))\), \(x \in E_N\). Then if \(X_N(0)\) has a limiting probability distribution \(\nu \) on E, the process \((X_N)\) converges in distribution in D([0, T], E) to the Markov process X starting at \(\nu \) with semigroup \((P(t))_{t \in [0,T]}\).
Rights and permissions
About this article
Cite this article
Chouk, K., Gairing, J. & Perkowski, N. An invariance principle for the two-dimensional parabolic Anderson model with small potential. Stoch PDE: Anal Comp 5, 520–558 (2017). https://doi.org/10.1007/s40072-017-0096-3
Received:
Published:
Issue Date:
Keywords
- Parabolic Anderson model
- Random polymer measure
- Random Schrödinger operator
- Invariance principle
- Paracontrolled distributions