Symmetries and zero modes in sample path large deviations

TL;DR

This paper extends large deviation analysis for stochastic differential equations to systems with symmetries and zero modes by using Riccati equations and field theory techniques, enabling the treatment of degenerate minimizers.

Contribution

It introduces a boundary regularization method for the second variation operator, allowing analysis of systems with continuous symmetries and zero modes in large deviations.

Findings

Extended large deviation estimates to systems with symmetries.

Applied the method to KPZ equation for surface height fluctuations.

Demonstrated the handling of zero modes via Riccati matrix modifications.

Abstract

Sharp large deviation estimates for stochastic differential equations with small noise, based on minimizing the Freidlin-Wentzell action functional under appropriate boundary conditions, can be obtained by integrating certain matrix Riccati differential equations along the large deviation minimizers or instantons, either forward or backward in time. Previous works in this direction often rely on the existence of isolated minimizers with positive definite second variation. By adopting techniques from field theory and explicitly evaluating the large deviation prefactors as functional determinant ratios using Forman's theorem, we extend the approach to general systems where degenerate submanifolds of minimizers exist. The key technique for this is a boundary-type regularization of the second variation operator. This extension is particularly relevant if the system possesses continuous symmetries that are broken by the instantons. We find that removing the vanishing eigenvalues associated with the zero modes is possible within the Riccati formulation and amounts to modifying the initial or final conditions and evaluation of the Riccati matrices. We apply our results in multiple examples including a dynamical phase transition for the average surface height in short-time large deviations of the one-dimensional Kardar-Parisi-Zhang equation with flat initial profile.