Large deviation principle

A sequence $(Z_n)$ satisfies a large deviation principle with speed $n$ and rate function $I$ when probabilities of measurable sets $A$ decay exponentially between the bounds

A rate function is a lower-semicontinuous map $I$ into $[0,\infty ]$. It is good when every sublevel set $\{x:I(x)\le c\}$ is compact.

For a small-noise diffusion $dX=a(X)dt+\sqrt{ϵ}\sigma (X)dW$, the Freidlin--Wentzell action of a smooth path is

On a zero-Hamiltonian infinite-time fluctuation path, the Freidlin--Wentzell action can be parametrized independently of time as $I=\int \Vert a\Vert ds-a\cdot d\varphi$ in the diffusion metric.