Symmetries and Zero Modes in Sample Path Large Deviations

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 along minimizers or instantons, either forward backward in time. Previous works this direction often rely existence of isolated positive definite second variation. By adopting techniques from field theory and explicitly evaluating prefactors as determinant ratios using Forman's theorem, we extend approach to general systems where degenerate submanifolds exist. The key technique is a boundary-type regularization variation operator. This extension particularly relevant if system possesses continuous symmetries that are broken instantons. We find removing vanishing eigenvalues associated zero modes possible within formulation amounts modifying initial final conditions evaluation matrices. apply our results multiple examples including dynamical phase transition average surface height short-time deviations one-dimensional Kardar-Parisi-Zhang equation flat profile.