an invariant measure in the operator language
An invariant (or stationary) measure is a distribution over the state space that the dynamics leaves unchanged: if the process is started with this distribution, it keeps it for all time, and the process becomes strictly stationary. In the operator language it is the natural left-eigenobject of the transition semigroup with eigenvalue 1, and it governs the long-time behaviour, ergodicity and equilibrium of the process.
Two equivalent operator characterizations. At the semigroup level, a measure mu is invariant if mu P_t = mu for all t >= 0, meaning integral (P_t f) dmu = integral f dmu for every bounded measurable f — averaging an observable forward in time does not change its mu-average. Differentiating at t = 0, this becomes the GENERATOR condition integral (A f) dmu = 0 for all f in the domain (a core), equivalently A^* mu = 0 in the sense of the formal adjoint — the invariant density is the stationary solution of the forward (Fokker-Planck) equation. The two statements are equivalent under mild conditions (a core argument), and the generator form is usually the practical one: solve A^* p = 0 with the normalization integral p = 1. When mu is invariant and the process is ergodic, time averages converge to the mu-average by the ergodic theorem, which is the bridge to Birkhoff and to MCMC.
Why it matters: invariant measures are equilibrium states; their existence and uniqueness control whether the process settles down and whether long-run sampling is meaningful. Honest cautions: invariant measures need not exist (transient processes like Brownian motion in R^d have only the non-finite Lebesgue measure as an invariant measure, which cannot be normalized to a probability), need not be unique (reducible dynamics have many), and even when a unique invariant probability exists, CONVERGENCE to it (mixing) is a separate question requiring extra hypotheses (e.g. irreducibility plus a Lyapunov/drift condition, or a spectral gap). Invariance is about being a fixed point of the dynamics; it says nothing on its own about being an attracting one.
For the Ornstein-Uhlenbeck process, A^* p = 0 reads (xp)' + p'' = 0, solved by the standard Gaussian p proportional to e^(-x^2/2). One checks mu P_t = mu directly: starting from N(0,1), X_t = e^(-t) X_0 + sqrt(1 - e^(-2t)) Z stays N(0,1) for all t. The Gaussian is the unique invariant probability and the process converges to it exponentially.
Solve A^* p = 0 and normalize to get the invariant density; invariance is mu P_t = mu, a fixed point of the flow.
Invariance (being a fixed point) is not the same as attraction: a unique invariant probability can exist without the process converging to it; convergence/mixing needs extra hypotheses like a spectral gap.