Donsker–Varadhan Large Deviation Principle

The Donsker–Varadhan LDP for the empirical measure of an ergodic Markov process, with the DV variational rate function and its reversible (Dirichlet form) specialization.

Donsker–Varadhan Large Deviation Principle

Prerequisites

Empirical measure

Let $(X_t)_{t\ge 0}$ be an ergodic continuous-time Markov process on a state space $E$ with generator $L$ (see Markov semigroups ) and invariant probability measure $\pi$ .

The (time-averaged) empirical measure up to time $T$ is

L_T = \frac{1}{T}\int_0^T \delta_{X_t}\,dt,

a random element of the space of probability measures on $E$ .

(For a discrete-time chain $(X_n)_{n\ge0}$ , one analogously uses $L_n=\frac1n\sum_{k=0}^{n-1}\delta_{X_k}$ ; see discrete Markov chains .)

Theorem (Donsker–Varadhan LDP, empirical measure)

Under standard ergodicity and regularity assumptions ensuring a well-posed DV theory, the family $(L_T)_{T\to\infty}$ satisfies a large deviation principle with speed $T$ and a good rate function $I(\mu)$ characterized by the variational formula

I(\mu) ={} \sup_{g>0} \left\{ -\int_E \frac{Lg}{g}\,d\mu \right\} ={} -\inf_{g>0}\int_E \frac{Lg}{g}\,d\mu,

where the supremum/infimum runs over strictly positive test functions $g$ in the domain of $L$ .

Key structural properties:

$I(\mu)\ge 0$ and $I(\mu)=0$ if and only if $\mu=\pi$ (the invariant law).
$I$ is convex and lower semicontinuous, hence “good” on suitable state spaces.

Reversible specialization (Dirichlet form representation)

If the process is reversible with respect to $\pi$ (see detailed balance ), and $\mu$ is absolutely continuous with respect to $\pi$ with density $f=\frac{d\mu}{d\pi}$ , then the DV rate function can be written in terms of the Dirichlet form:

I(\mu) ={} -\left\langle \sqrt{f},\, L\sqrt{f}\right\rangle_\pi, \qquad \langle a,b\rangle_\pi=\int_E a\,b\,d\pi.

This representation makes clear that $I(\mu)$ measures how costly it is for the process to maintain atypical occupation statistics.

DV duality and cumulant generating functions

A central application is the asymptotic log-moment generating function of additive functionals. For a bounded measurable “potential” $V:E\to\mathbb{R}$ , define

\Lambda(V) ={} \lim_{T\to\infty}\frac{1}{T}\log \mathbb{E}_\pi\!\left[ \exp\!\left(\int_0^T V(X_t)\,dt\right) \right].

DV theory identifies $\Lambda(V)$ as the Legendre–Fenchel dual of $I$ :

\Lambda(V) ={} \sup_{\mu} \left\{ \int_E V\,d\mu - I(\mu) \right\},

an instance of Fenchel duality (often combined with Varadhan-type lemmas).

Context in statistical mechanics

The DV functional plays the role of an “entropic cost” for sustaining a non-typical time-averaged state in nonequilibrium models (compare nonequilibrium steady states ).
In interacting particle systems, DV-type LDPs underpin variational principles for dynamical free energies and connect to hydrodynamic and macroscopic fluctuation theories (see hydrodynamic limits ).