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1. 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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2. A lower-tail limit in the weak noise theory

   https://doi.org/10.1214/23-AIHP1455  

   Lin, Yier; Tsai, Li-Cheng (May 2025, Annales de l'Institut Henri Poincaré, Probabilités et Statistiques)

   We consider the variational problem associated with the Freidlin--Wentzell Large Deviation Principle of the Stochastic Heat Equation (SHE). The logarithm of the minimizer of the variational problem gives the most probable shape of the solution of the Kardar--Parisi--Zhang equation conditioned on achieving certain unlikely values. Taking the SHE with the delta initial condition and conditioning the value of its solution at the origin at a later time, under suitable scaling, we prove that the logarithm of the minimizer converges to an explicit function as we tune the value of the conditioning to 0. Our result confirms the physics prediction Kolokolov and Korshunov (2009), Meerson, Katzav, and Vilenkin (2016), Kamenev, Meerson, and Sasorov (2016). 

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3. Sharp asymptotic estimates for expectations, probabilities, and mean first passage times in stochastic systems with small noise

   https://doi.org/10.1002/cpa.22177  

   Grafke, Tobias; Schäfer, Tobias; Vanden‐Eijnden, Eric (April 2024, Communications on Pure and Applied Mathematics)

   Freidlin‐Wentzell theory of large deviations can be used to compute the likelihood of extreme or rare events in stochastic dynamical systems via the solution of an optimization problem. The approach gives exponential estimates that often need to be refined via calculation of a prefactor. Here it is shown how to perform these computations in practice. Specifically, sharp asymptotic estimates are derived for expectations, probabilities, and mean first passage times in a form that is geared towards numerical purposes: they require solving well‐posed matrix Riccati equations involving the minimizer of the Freidlin‐Wentzell action as input, either forward or backward in time with appropriate initial or final conditions tailored to the estimate at hand. The usefulness of our approach is illustrated on several examples. In particular, invariant measure probabilities and mean first passage times are calculated in models involving stochastic partial differential equations of reaction‐advection‐diffusion type. 

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4. High moments of the SHE in the clustering regimes

   https://doi.org/10.1016/j.jfa.2024.110675  

   Tsai, Li-Cheng (January 2025, Journal of Functional Analysis)

   We analyze the high moments of the Stochastic Heat Equation (SHE) via a transformation to the attractive Brownian Particles (BPs), which are Brownian motions interacting via pairwise attractive drift. In those scaling regimes where the particles tend to cluster, we prove a Large Deviation Principle (LDP) for the empirical measure of the attractive BPs. Under the delta(-like) initial condition, we characterize the unique minimizer of the rate function and relate the minimizer to the spacetime limit shapes of the Kardar–Parisi–Zhang (KPZ) equation in the upper tails. The results of this paper are used in the companion paper [75] to prove an n-point, upper-tail LDP for the KPZ equation and to characterize the corresponding spacetime limit shape. 

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5. Rate and noise-induced tipping working in concert

   https://doi.org/10.1063/5.0129341  

   Slyman, Katherine; Jones, Christopher K. (January 2023, Chaos: An Interdisciplinary Journal of Nonlinear Science)

   Rate-induced tipping occurs when a ramp parameter changes rapidly enough to cause the system to tip between co-existing, attracting states. We show that the addition of noise to the system can cause it to tip well below the critical rate at which rate-induced tipping would occur. Moreover, it does so with significantly increased probability over the noise acting alone. We achieve this by finding a global minimizer in a canonical problem of the Freidlin–Wentzell action functional of large deviation theory that represents the most probable path for tipping. This is realized as a heteroclinic connection for the Euler–Lagrange system associated with the Freidlin–Wentzell action and we find it exists for all rates less than or equal to the critical rate. Its role as the most probable path is corroborated by direct Monte Carlo simulations. 

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