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Tsai, Li-Cheng

Integrability in the weak noise theory. (English) Zbl 1521.60016

Trans. Am. Math. Soc. 376, No. 9, 6521-6572 (2023).
Summary: We consider the variational problem associated with the Freidlin-Wentzell Large Deviation Principle (LDP) for the Stochastic Heat Equation (SHE). For a general class of initial-terminal conditions, we show that a minimizer of this variational problem exists, and any minimizer solves a system of imaginary-time Nonlinear Schrödinger equations. This system is integrable. Utilizing the integrability, we prove that the formulas from the physics work (see [A. Krajenbrink and P. Le Doussal, Phys. Rev. Lett. 127, No. 6, Article ID 064101, 8 p. (2021; doi:10.1103/PhysRevLett.127.064101)]) hold for every minimizer of the variational problem. As an application, we consider the Freidlin-Wentzell LDP for the SHE with the delta initial condition. Under a technical assumption on the poles of the reflection coefficients, we prove the explicit expression for the one-point rate function that was predicted in the physics works (see [P. Le Doussal et al., Phys. Rev. Lett. 117, No. 7, Article ID 070403, 5 p. (2016; doi:10.1103/PhysRevLett.117.070403); Krajenbrink and Le Doussal, loc. cit.]). Under the same assumption, we give detailed pointwise estimates of the most probable shape in the upper-tail limit.

MSC:

60F10 Large deviations
35C15 Integral representations of solutions to PDEs
49L12 Hamilton-Jacobi equations in optimal control and differential games
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References:

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