Large deviations for invariant measures of perturbed flows

We study a flow $\phi_t$ on a compact manifold $M$ with a unique invariant measure $\mu$, and ask how empirical averages concentrate. The central object is the rate function $I$ governing

$$\mathbb{P}\!\left( \tfrac{1}{T}\int_0^T \delta_{\phi_t x}\,dt \approx \nu \right) \asymp e^{-T\, I(\nu)}, \qquad T \to \infty.$$

A geometric form of the rate function

When the perturbation is gradient-like, $I$ admits a clean variational form,

$$I(\nu) = \sup_{u \in C^1(M)} \int_M \left( \mathcal{L}u - \tfrac{1}{2}\,\Gamma(u,u) \right) d\nu,$$

where $\Gamma$ is the carré du champ operator. The geometry enters through $\Gamma$: curvature lower bounds translate, via Bakry–Émery, into quantitative concentration.

The slogan: curvature controls rare events.

This note collects the estimates I’m currently trying to make uniform in the perturbation strength $\varepsilon$.