nLab time-reversible stochastic process

Idea

A stochastic process is time-reversible if it is symmetric under time reversal.

This symmetry is meant in a stochastic sense: it says that given any trajectory, its reverse trajectory has the same probability of happening.

Definition

A stationary stochastic process $(X_t)_{t\in T}$ is called time-reversible if and only if for all finite $t_1 \lt \dots \lt t_n$, we have that the finite marginals

$$(X_{t_1},\dots,X_{t_n}) \qquad and \qquad (X_{t_n},\dots,X_{t_1})$$

have the same joint distribution.

(The definition outside the stationary case does not seem to appear in the literature.)

Examples

• A Markov process? or Markov chain is time-reversible if and only if it satisfies detailed balance. In particular, every idempotent Markov chain? is time-reversible.
• Every exchangeable process is time-reversible, in particular Bernoulli processes are.

Related entries

• probability theory, categorical probability
• invariant measure, invariant set
• equilibrium, detailed balance
• Markov process?, Markov kernel, Markov category
• time-reversal symmetry

References

• Wikipedia, Time reversibility

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

category: probability