nLab invariant measure

Idea

An invariant measure (a.k.a. stationary or steady) is a measure which is invariant under the action of a group or monoid, often playing the role of time, or a symmetry of the system.

An action with an invariant measure is sometimes called a stationary or steady-state system.

It is an important concept in probability theory, dynamical systems, statistical physics, and representation theory. It is one of the weakest forms to mathematically formalize the idea of equilibrium. (Stronger forms of equilibrium are for example detailed balance? and ergodicity.)

Definitions

Let $X$ be a measurable space, and let $M$ be a monoid with an action on $X$ via measurable functions. For each $m\in M$, denote the action on $X$ again by $m:X\to X$.

A measure $p$ on $X$ is called invariant or stationary under the action of $M$ if and only if for all $m\in M$, the pushforward $m_*p$ is equal to $p$. That is, for each measurable subset $A\subseteq X$,

$$p(m^{-1}(A)) \;=\; p(A) .$$

A space $X$ together with an action of $M$ and a stationary measure $M$ is sometimes called a stationary system. In particular, a stochastic process whose underlying joint distribution is invariant is called a stationary process.

Note that, similarly to invariant sets, if a measure is invariant, it does not in general mean that each point in the support of the measure is an invariant point. The points may move, but their distribution overall is kept constant. For example, the Lebesgue measure on the unit circle (given by the length) is invariant under rotations, but no point of the circle is invariant.

An interpretation in terms of equilibrium is that for each measurable set $A$, some mass may move away from $A$, but an equal amount of mass moves into $A$, so that the measure of $A$ overall stays the same.

Examples

• The Haar measure on a compact topological group $G$ is invariant for the action of $G$ on itself, given by left multiplication.

• An exchangeable measure on a product space $X^\mathbb{N}$ is invariant under the action of finite permutations.

• A single measurable function $f:X\to X$ generates a monoid (the ont of natural numbers). In this case, an invariant measure $p$ simply needs to satisfy $f_*p=p$.

• In statistical physics, Liouville's theorem? says that the distribution function of the phase space is invariant under the action of Hamiltonian flow.

Actions via Markov kernels

A monoid $(M,e,\cdot)$ can also act on a measurable space $X$ via Markov kernels. That is, to each $m\in M$ we can assign a kernel $k_m:X\to X$ such that

$$k_e(A|x) \;=\; 1_A(x) \;=\; \begin{cases} 1 & x\in A ; \\ 0 & x\notin A , \end{cases}$$

and

$$k_{m\cdot n}(A|x) \;=\; \int_X k_m(A|x')\,k_n(d x'|x) .$$

In this case, a measure $p$ on $X$ is invariant or stationary if and only if for all $m\in M$, and for all measurable $A\subseteq X$,

$$\int_X k_m(A|x) \, p(d x) \;=\; p(A) ,$$

that is, each kernel $k_m$ preserves the measure $p$.

A probability space together with a measure-preserving kernel is sometimes called a stationary Markov chain.

Category-theoretic description

A group or monoid $M$ can be seen (via delooping) as a one-object category $B M$, and an action of $M$ on a measurable space $X$ can be seen as a functor $B M\to Meas$ (the category of measurable spaces?) or $B M\to Stoch$ (the category of Markov kernels). (Since every measurable function canonically defines a Markov kernel, without loss of generality we can take the category of kernels in both cases.) As a diagram, it has a single object, and a loop for each element of $M$:

An invariant probability measure is now a cone over this diagram. Specifically, since a probability measure is equivalently a Markov kernel from the one-point space, it makes the following diagram of Stoch commute for every $m\in M$:

In many situations, ergodic measures even form a limit cone.

Equivalently, a stationary measure $p$ gives an action of $M$ in the category of probability spaces and measure-preserving kernels (see for example at category of couplings).

Further properties

• An invariant probability measure is called ergodic if it is zero-one on all invariant sets.

• A stationary system (for example, Markov kernel) satisfies detailed balance?, or is called reversible? if

  $$\int_A k(B|x) \,p(dx) \;=\; \int_B k(A|x)\,p(dx) ,$$

  i.e. if $k$ is its own Bayesian inverse. This is a stronger form of equilibrium than just stationarity. (Not every stationary system satisfies this property.)

• The ergodic decomposition theorem says that, under some conditions, an invariant measure is a mixture of ergodic ones.

See also

• group, monoid, action
• cone, limit, cocone, colimit
• exchangeability, invariant set
• ergodic measure, ergodic decomposition theorem

References

• Tobias Fritz, Tomáš Gonda, Paolo Perrone, De Finetti’s theorem in categorical probability. Journal of Stochastic Analysis, 2021. (arXiv:2105.02639)

• Sean Moss, Paolo Perrone, A category-theoretic proof of the ergodic decomposition theorem, Ergodic Theory and Dynamical Systems, 2023. (arXiv:2207.07353)

• Noé Ensarguet, Paolo Perrone, Categorical probability spaces, ergodic decompositions, and transitions to equilibrium, arXiv:2310.04267

• Rob Cornish, Stochastic Neural Network Symmetrization in Markov Categories, 2024. (arXiv)

category: probability