Precise large deviations for S(P)DEs: theory, numerics, applications

In this talk, I will first review the classical theory of Laplace asymptotics for small-noise diffusions in R^n and extensions to stochastic partial differential equations (SPDEs), with particular emphasis on sharp asymptotics. Going beyond the usual log-asymptotics of large deviation theory, we focus on prefactors and thus Hessian contributions that are essential for accurate probability estimation.

I will then describe how these theoretical results can be turned into a practical computational framework for estimating rare event probabilities without sampling. The approach is designed to scale to high-dimensional state spaces, as required for SPDEs, and involves solving ODE/PDE-constrained optimization problems to identify optimal paths, as well as computing Hessian determinants around minimizers. Modern tools, such as automatic differentiation, make these steps tractable.

As an application, I will present results for intermittency in the stochastic 1D viscous Burgers equation, as a canonical toy model of fluid turbulence. Time permitting, I will conclude with an outlook on future improvements based on ideas from the functional renormalization group.

Based on joint work with Tobias Grafke, Rainer Grauer, Georg Stadler, Shanyin Tong, in Stat. Comput. 33(6), 137 (2023), arXiv:2502.20114, and arXiv:2512.03841.