the Freidlin-Wentzell theory
/ FRAYD-lin VENT-sel /
The Freidlin-Wentzell theory is the large deviations theory of small-noise diffusions: stochastic differential equations whose noise vanishes in a limit. It answers the questions that dominate applied stochastics — how does a system driven by tiny random forcing escape from a stable equilibrium, which path does it take when it does, and how long does it typically wait? These rare escapes underlie chemical reaction rates, metastable phase transitions, climate tipping, and the failure of noisy engineered systems, all of which are exponentially rare on the natural timescale.
Consider the SDE dX = b(X) dt + sqrt(epsilon) sigma(X) dB with epsilon -> 0; at epsilon = 0 the solution is the deterministic flow dx = b(x) dt. Freidlin-Wentzell states that the laws of X^epsilon satisfy an LDP on path space with speed 1/epsilon and good rate function, the action functional, I(phi) = (1/2) integral of (phi'(t) - b(phi(t)))^T a(phi(t))^(-1) (phi'(t) - b(phi(t))) dt over absolutely continuous phi (with a = sigma sigma^T the diffusion matrix), and +infinity otherwise. The integrand measures, at each instant, how hard the noise must push to make the path deviate from the drift b; the cheapest deviation is the noise that exactly fills the gap between phi' and the drift. This rate function is obtained by contracting Schilder's Brownian LDP through the (continuous) solution map of the SDE.
From the action functional the theory extracts the quasipotential V(x_0, x) = inf of I(phi) over paths from x_0 to x in any time, which governs everything. The expected exit time from the basin of a stable point grows like e^(V/epsilon) (the Eyring-Kramers / Arrhenius law, with the prefactor needing finer analysis), the most probable exit point is where V is smallest on the boundary, and in multi-well landscapes the long-time hopping between wells becomes a Markov jump process with rates set by the quasipotential barriers. An essential caveat: the clean LDP requires non-degenerate (uniformly elliptic) diffusion a, or hypoellipticity; in degenerate cases the rate function and the exit asymptotics are far more delicate.
For a particle in a double-well potential U, dX = -U'(X) dt + sqrt(epsilon) dB, the action of a path equals (1/2) integral (phi' + U'(phi))^2 dt. The cheapest escape over the barrier follows the time-reversed downhill path, giving quasipotential V = 2 * (barrier height), so the mean transition time scales like e^(2 Delta U / epsilon) — Kramers' reaction-rate law, derived as a large deviation.
Freidlin-Wentzell: escape costs the quasipotential; exit time ~ e^(V/epsilon).
The LDP gives only the exponential order e^(V/epsilon) of exit times; the polynomial/subexponential prefactor (the Eyring-Kramers correction) needs separate analysis. And the action functional requires the diffusion to be non-degenerate (or hypoelliptic); degenerate noise breaks the clean inverse-a^(-1) form.