Ergodic Theory for Stochastic Processes

Core Idea

A stochastic process is ergodic if time averages converge to ensemble averages: (1/T)∫₀ᵀ f(X(t))dt → E_π[f(X)] almost surely as T → ∞, where π is the stationary distribution. Ergodicity means a single long trajectory explores the entire state space representatively — you don't need many independent samples, just one long run. For diffusions, ergodicity follows from the existence of a unique stationary distribution and appropriate recurrence conditions.

Explainer

Ergodic theory connects the time behavior of a single trajectory to the statistical properties of the process's stationary distribution. The fundamental ergodic theorem for stochastic processes states: if X(t) is a stationary ergodic process with stationary distribution π, then (1/T)∫₀ᵀ f(X(t))dt → E_π[f(X)] almost surely as T → ∞, for any integrable function f. The time average over one long path equals the ensemble average over the stationary distribution. This is the continuous-time analogue of the ergodic theorem for measure-preserving transformations.

For diffusion processes dX = μ(X)dt + σ(X)dW, ergodicity boils down to two ingredients: the existence of a unique stationary distribution π, and positive recurrence (the process returns to compact sets in finite expected time). The stationary distribution exists when the drift is mean-reverting — pulling the process back from infinity — and the diffusion coefficient σ(x) > 0 ensures accessibility (the process can reach any state from any other state). One-dimensional diffusions are particularly well-understood: the stationary density π(x) ∝ (1/σ²(x))exp(2∫μ(x)/σ²(x)dx) exists whenever this expression is integrable, and the process is ergodic whenever the stationary distribution exists and is unique.

The distinction between stationarity and ergodicity is subtle but fundamental. A process is stationary if its statistical properties don't change over time — the distribution of X(t) is the same as X(t+s) for all s. A process is ergodic if, additionally, a single trajectory is representative of the whole distribution. The classic counterexample is a mixture: pick a random mean μ from {-1, +1} with equal probability, then run an OU process with that mean forever. The combined process is stationary (the marginal distribution at each time is the same mixture), but not ergodic — a single trajectory stays near whichever mean was chosen and never explores the other component. Ergodicity requires mixing: the process must eventually visit all parts of its stationary distribution.

The practical consequence of ergodicity is enormous for Monte Carlo methods. Computing E_π[f] by drawing independent samples from π may be impractical — the distribution might be high-dimensional and analytically intractable. Instead, simulate the process X(t) for a long time and use the time average as an estimator. Ergodicity guarantees convergence; the mixing time (how quickly the process forgets its initial condition) determines the convergence rate. This is the foundation of Markov chain Monte Carlo (MCMC): design a Markov process whose stationary distribution is the target, run it, and collect time averages. Ergodic theory provides the theoretical guarantee that this procedure works.