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1. Lower tail of the KPZ equation

   https://doi.org/10.1215/00127094-2019-0079  

   Corwin, Ivan; Ghosal, Promit (May 2020, Duke Mathematical Journal)
2. Mesoscopic Averaging of the Two-Dimensional KPZ Equation

   https://doi.org/10.1007/s10955-023-03222-3  

   Tao, Ran (January 2024, Journal of Statistical Physics)
3. Short Time Large Deviations of the KPZ Equation

   https://doi.org/doi.org/10.1007/s00220-021-04050-w  

   Lin, Yier; Tsai, Li-Cheng (August 2021, Communications in mathematical physics)

   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 ε√ 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 52 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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4. Short Time Large Deviations of the KPZ Equation

   https://doi.org/10.1007/s00220-021-04050-w  

   Lin, Yier; Tsai, Li-Cheng (August 2021, Communications in Mathematical Physics)
5. Gaussian fluctuations from the 2D KPZ equation

   https://doi.org/10.1007/s40072-019-00144-8  

   Gu, Yu (June 2019, Stochastics and partial differential equations)