The Edwards–Wilkinson Limit of the Random Heat Equation in Dimensions Three and Higher
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We consider the heat equation with a multiplicative Gaussian potential in dimensions d ≥ 3. We show that the renormalized solution converges to the solution of a deterministic diffusion equation with an effective diffusivity. We also prove that the renormalized large scale random fluctuations are described by the Edwards–Wilkinson model, that is, the stochastic heat equation (SHE) with additive white noise, with an effective variance.
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- 9.Comets, F.: Directed polymers in random environments, Lecture Notes in Mathematics, vol. 2175, Springer, Cham, 2017, Lecture notes from the 46th Probability Summer School held in Saint-Flour, (2016)Google Scholar
- 11.Dautray R., Lions, J.-L.: Mathematical Analysis and Numerical Methods in Science and Technology. Volume 3: Spectral Theory and Applicaitons, Springer, (1990)Google Scholar
- 12.Feng, Z. S.: Rescaled Directed Random Polymer in Random Environment in Dimension 1+ 2, Ph.D. thesis, University of Toronto (Canada), (2016)Google Scholar
- 14.Gu, Y., Tsai, L.-C.: Another look into the Wong–Zakai theorem for stochastic heat equation, arXiv preprint arXiv:1801.09164, (2018)
- 16.Gubinelli, M., Imkeller, P., Perkowski, N.: Paracontrolled distributions and singular PDEs, In: Forum of Mathematics, Pi, vol. 3, Cambridge University Press, p. e6. (2015)Google Scholar
- 24.Kupiainen, A.: Renormalization group and stochastic PDEs, In: Annales Henri Poincaré, vol. 17, Springer, pp. 497–535. (2016)Google Scholar
- 26.Mukherjee, C.: A central limit theorem for the annealed path measures for the stochastic heat equation and the continuous directed polymer in d ≥ 3, arXiv preprint arXiv:1706.09345, (2017)
- 27.Mukherjee, C., Shamov, A., Zeitouni, O.: Weak and strong disorder for the stochastic heat equation and continuous directed polymers in d ≥ 3. Electr. Commun. Probab. 21 (2016)Google Scholar
- 28.Otto, F., Weber, H.: Quasilinear SPDEs via rough paths, arXiv preprint arXiv:1605.09744, (2016)