Alphabeta Math
TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14
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A Markov-chain law is determined by its initial law and kernel

Statement

Assume Choice. Two time-homogeneous Markov chains on the same measurable state space with the same initial law μ and the same transition kernel K have the same finite-dimensional distributions. Consequently their induced laws on the canonical path space equipped with its cylinder sigma-algebra are equal.

Facts & Assumptions

Given: Choice and two K-chains with initial law μ.

[F1]

Every finite-dimensional law of a Markov chain is the iterated integral determined by its initial law and iterated kernels. (Finite-dimensional laws of a Markov chain)

[F2]

A process law on countable-coordinate cylinder space is determined by its finite-dimensional distributions. (Finite-dimensional distributions determine a process law on the cylinder sigma-algebra)

Proof

1.1

For every finite increasing time list, [F1] gives the same iterated [F1] integral for both processes because their μ and K agree. This includes a single time, time zero, empty rectangle events, and the full rectangle. Therefore all their finite-dimensional distributions coincide.

F1
2.1

Push both processes forward by their path maps. The two induced [F2, step 1.1] probabilities have the finite-dimensional distributions compared in step 1.1, so [F2] makes them equal on the cylinder sigma-algebra. Choice enters through [F1]'s conditional-expectation argument; [F2] adds no new selection.

F2step 1.1∎

Depends on

Used by

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Sources