Towards Matching Bounds for Large Deviations

The proposed project considers the problem of finding matching bounds for dynamical large deviations of natural models instatistical physics and fluid mechanics. In a random system with a deterministic limit, dynamical large deviations describe theexponential decay of probability that the system is close to a trajectory different from the deterministic limit; classic works by Kipnis-Olla-Varadhan and Kipnis-Landim study such bounds. However, in many interesting cases, the available upper and lower bounds onlyagree when restricted to sufficiently regular trajectories.Recent developments give the problem of investigating the matching bounds new momentum. On the positive side, Fehrman-Gessreached an important milestone in matching bounds for hydrodynamic limits by analysis of a PDE, and proved matching bounds for afamily of SPDEs related to hydrodynamic limits. On the negative side, my previous work identified a case where the naïve bound isnot sharp, related the existence of spurious solutions violating the energy identity. We will capitalise on these recent developmentsto identify cases where the expected rate function does, and does not, capture all interesting phenomena.The project will be divided into two parts, taking these recent developments as a jumping off point. The first part will concern two examples where we expect to find matching bounds, while the second part investigates (anticipated) deviations from the classical theory. In WP1, we will investigate wide-reaching generalisations of Fehrman-Gess theory, in WP1a to the more challenging case of systems of equations, and in WP1b to rougher scaling regimes via the renormalisation group. In WP2, we will investigate one model which connects the existing counterexample to the derivation of turbulence from molecular dynamics, and another model in which we hypothesise that the violation of a key technical estimate leads to behaviour different from what is predicted in the literature.