Short Time Large Deviations of the KPZ Equation

Abstract

We establish the Freidlin–Wentzell Large Deviation Principle (LDP) for the Stochastic Heat Equation with multiplicative noise in one spatial dimension. That is, we introduce a small parameter \( \sqrt{\varepsilon } \) to the noise, and establish an LDP for the trajectory of the solution. Such a Freidlin–Wentzell LDP gives the short-time, one-point LDP for the KPZ equation in terms of a variational problem. Analyzing this variational problem under the narrow wedge initial data, we prove a quadratic law for the near-center tail and a \( \frac{5}{2} \) law for the deep lower tail. These power laws confirm existing physics predictions (Kolokolov and Korshunov in Phys Rev B 75(14):140201, 2007, Phys Rev E 80(3):031107, 2009; Meerson et al. in Phys Rev Lett 116(7):070601, 2016; Le Doussal et al. in Phys Rev Lett 117(7):070403, 2016; Kamenev et al. in Phys Rev E 94(3):032108, 2016).

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Acknowledgements

We thank Ivan Corwin for suggesting this problem to us and for useful discussions, thank Sayan Das, Amir Dembo, Promit Ghosal, Konstantin Matetski, and Shalin Parekh for useful discussions, and thank Martin Hairer, Hao Shen, and Hendrik Weber for clarifying some points about the literature. We thank the anonymous referees for useful comments that improve the presentation of this paper. The research of YL is partially supported by the Fernholz Foundation’s “Summer Minerva Fellow” program and also received summer support from Ivan Corwin’s NSF Grant DMS-1811143. The research of LCT is partially supported by the NSF through DMS-1953407.

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Lin, Y., Tsai, LC. Short Time Large Deviations of the KPZ Equation. Commun. Math. Phys. (2021). https://doi.org/10.1007/s00220-021-04050-w

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