An invariance principle for the two-dimensional parabolic Anderson model with small potential

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. 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. 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)

    MathSciNet  Article  MATH  Google Scholar 

  2. 1.

    Allez, R., Chouk, K.: The continuous Anderson Hamiltonian in dimension two (2015), preprint arXiv:1511.02718

  3. 2.

    Bahouri, H., Chemin, J.-Y.: Danchin, Raphael, Fourier analysis and nonlinear partial differential equations. Springer, Berlin (2011)

    Google Scholar 

  4. 3.

    Bailleul, I., Bernicot, F.: Heat semigroup and singular PDEs. J. Funct. Anal. 270(9), 3344–3452 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  5. 4.

    Bailleul, Ismaël, Bernicot, F., Frey, D.: Higher order paracontrolled calculus and 3d-PAM equation, (2015), arXiv preprint arXiv:1506.08773

  6. 5.

    Biskup, M., Fukushima, R., König, W.: Eigenvalue fluctuations for lattice Anderson Hamiltonians. SIAM J. Math. Anal. 48(4), 2674–2700 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  7. 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)

    Article  MATH  Google Scholar 

  8. 7.

    Brown, B.M.: Martingale central limit theorems. Ann. Math. Statist. 42(1), 59–66 (1971)

    MathSciNet  Article  MATH  Google Scholar 

  9. 8.

    Bruned, Y.: Singular KPZ type equations, Ph.D. Thesis (2015)

  10. 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

  11. 10.

    Caravenna, F., Sun, R., Zygouras, N.: Polynomial chaos and scaling limits of disordered systems (2013), arXiv preprint arXiv:1312.3357

  12. 11.

    Carmona, R., Molchanov, S.A.: Parabolic Anderson problem and intermittency, vol. 518. American Mathematical Soc, Providence (1994)

    Google Scholar 

  13. 12.

    Catellier, R., Chouk, K.: Paracontrolled distributions and the 3-dimensional stochastic quantization equation (2013), arXiv preprint arXiv:1310.6869

  14. 13.

    Chandra, A., Shen, H.: Glauber dynamics of 2D Kac–Blume–Capel model and their stochastic PDE limits (2016), arXiv preprint arXiv:1608.06556

  15. 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

  16. 15.

    Ethier, S.N., Kurtz, T.G.: Markov Processes: Characterization and Convergence. Wiley, Hoboken (2005)

    Google Scholar 

  17. 16.

    Friz, P.K., Hairer, Martin: A Course on Rough Paths: With an Introduction to Regularity Structures. Springer, Berlin (2014)

    Google Scholar 

  18. 17.

    Gubinelli, M.: Controlling rough paths. J. Funct. Anal. 216(1), 86–140 (2004)

    MathSciNet  Article  MATH  Google Scholar 

  19. 18.

    Gubinelli, M., Imkeller, P., Perkowski, N.: Paracontrolled Distributions and Singular PDEs, p. e6. Cambridge University Press, Cambridge (2015)

    Google Scholar 

  20. 19.

    Gubinelli, M., Perkowski, N.: KPZ reloaded (2015), arXiv preprint arXiv:1508.03877

  21. 20.

    Gubinelli, M., Perkowski, N.: Lectures on singular stochastic PDEs, Ensaois Mat. 29 (2015)

  22. 21.

    Hairer, M.: Rough stochastic PDEs. Comm. Pure Appl. Math. 64(11), 1547–1585 (2011)

    MathSciNet  MATH  Google Scholar 

  23. 22.

    Hairer, M.: Solving the KPZ equation. Ann. Math. 178(2), 559–664 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  24. 23.

    Hairer, M.: A theory of regularity structures. Invent. Math. (2014). doi:10.1007/s00222-014-0505-4

  25. 24.

    Hairer, M.: The motion of a random string (2016), arXiv preprint arXiv:1605.02192

  26. 25.

    Hairer, M., Labbé, C.: Multiplicative stochastic heat equations on the whole space (2015), arXiv preprint arXiv:1504.07162

  27. 26.

    Hairer, M., Maas, J.: A spatial version of the Itô-Stratonovich correction. Ann. Probab. 40(4), 1675–1714 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  28. 27.

    Hairer, M., Maas, J., Weber, H.: Approximating rough stochastic PDEs. Comm. Pure Appl. Math. 67(5), 776–870 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  29. 28.

    Hairer, M., Matetski, K.: Discretisations of rough stochastic PDEs (2015), arXiv preprint arXiv:1511.06937

  30. 29.

    Hairer, M., Pardoux, É.: A Wong–Zakai theorem for stochastic PDEs. J. Math. Soc. Jpn. 67(4), 1551–1604 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  31. 30.

    Hairer, M., Quastel, J.: A class of growth models rescaling to KPZ (2015), arXiv preprint arXiv:1512.07845

  32. 31.

    Hairer, M., Shen, H.: A central limit theorem for the KPZ equation (2015), arXiv preprint arXiv:1507.01237

  33. 32.

    Hairer, M., Shen, H.: The dynamical sine-Gordon model. Comm. Math. Phys. 341(3), 933–989 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  34. 33.

    Hairer, M., Xu, W.: Large scale behaviour of 3D phase coexistence models (2016), arXiv preprint arXiv:1601.05138

  35. 34.

    Hoshino, M.: KPZ equation with fractional derivatives of white noise (2016), arXiv preprint arXiv:1602.04570

  36. 35.

    Hoshino, M.: Paracontrolled calculus and Funaki–Quastel approximation for the KPZ equation (2016), arXiv preprint arXiv:1605.02624

  37. 36.

    Janson, Svante: Gaussian Hilbert spaces, Cambridge Tracts in Mathematics, vol. 129. Cambridge University Press, Cambridge (1997)

    Google Scholar 

  38. 37.

    König, W.: The Parabolic Anderson Model. Random Walk in Random Potential. Birkhäuser, Basel (2016)

    Google Scholar 

  39. 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)

  40. 39.

    Kupiainen, A.: Renormalization group and stochastic PDEs. Annales Henri Poincaré 17(3), 497–535 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  41. 40.

    Kupiainen, A., Marcozzi, M.: Renormalization of generalized KPZ equation (2016), arXiv preprint arXiv:1604.08712

  42. 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)

    MathSciNet  Article  MATH  Google Scholar 

  43. 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)

    MathSciNet  Article  MATH  Google Scholar 

  44. 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)

    MathSciNet  Article  MATH  Google Scholar 

  45. 44.

    Mossel, E., O’Donnell, R., Oleszkiewicz, K.: Noise stability of functions with low influences: invariance and optimality. Ann. Math. 171, 295–341 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  46. 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

  47. 46.

    Prömel, D.J., Trabs, M.: Rough differential equations driven by signals in Besov spaces. J. Differ. Eq. 260(6), 5202–5249 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  48. 47.

    Schmidt, S.: Das parabolische Anderson–Modell mit Be- und Entschleunigung. Universität Leipzig, Leipzig (2010)

    Google Scholar 

  49. 48.

    Shen, H., Xu, W.: Weak universality of dynamical \(\phi ^4_3 \): non-Gaussian noise, (2016), arXiv preprint arXiv:1601.05724

  50. 49.

    Zhu, R., Zhu, X.: Approximating three-dimensional Navier–Stokes equations driven by space-time white noise (2014), arXiv preprint arXiv:1409.4864

  51. 50.

    Zhu, R., Zhu, X.: Lattice approximation to the dynamical \(\phi _3^4\) model (2015) arXiv preprint arXiv:1508.05613

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Nicolas Perkowski.

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

$$\begin{aligned} \lim _{N \rightarrow 0} \Vert P_N(t)\pi _Nf - \pi _N P(t) f\Vert _{L^{\infty }} = 0 \end{aligned}$$

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

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

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

Download citation

Keywords

  • Parabolic Anderson model
  • Random polymer measure
  • Random Schrödinger operator
  • Invariance principle
  • Paracontrolled distributions