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The minimum-intensity reaction coupling preserves both marginals

Statement

Assume Countable Choice and Dependent Choice. Let $X$ and $Y$ be finite-state reaction systems with the same $R$ channel labels, respective increments $ν_{j}^{X}, ν_{j}^{Y}$ and propensities $a_{j}^{X} (x), a_{j}^{Y} (y)$ , and fixed initial states $(x_{0}, y_{0})$ . Run Minimum-intensity coupling of reaction networks with an unlimited candidate budget. Then each coordinate path has exactly the law of its own reaction process run by the Gillespie direct method. At each pair state $(x, y)$ and channel $j$ , shared events have rate $\min (a_{j}^{X} (x), a_{j}^{Y} (y))$ , $X$ -only events have rate $(a_{j}^{X} (x) - a_{j}^{Y} (y))_{+}$ , and $Y$ -only events have rate $(a_{j}^{Y} (y) - a_{j}^{X} (x))_{+}$ . The expected total number of paired candidates on $[0, T]$ is $M T$ , where $M = \sum_{j} M_{j}$ is defined in the algorithm; hence the coupled construction has finitely many events almost surely. If $M = 0$ , both paths are constant.

Facts & Assumptions

Given: Countable Choice, Dependent Choice, two finite-state reaction systems, initial states, and the independent exponential and uniform candidate streams.

[F1]

Each system has finite channels, admissible increments, and finite propensities; its reaction rate and marked transition convention are those in the reaction-system definition. (Finite-channel reaction propensity systems and their generator)

[F2]

The coupling algorithm uses a rate- $M$ candidate stream, fixed disjoint channel blocks of lengths $M_{j} / M$ , and within-block nested acceptance intervals of lengths $a_{j}^{X} (x) / M$ and $a_{j}^{Y} (y) / M$ . (Minimum-intensity coupling of reaction networks)

[F3]

Dominating thinning has the prescribed first-event and full path laws for pairwise-disjoint mark sets with the required channel measures and product-measurable time-mark regions. The sets induced by this algorithm are fixed intervals for each coordinate state, so their time-mark regions are product-measurable. (Dominating thinning has the prescribed reaction law)

[A1]

Countable Choice is assumed by the exponential, Lebesgue-measure and countable-product existence results used to realize the candidate inputs in [F2] and [F3]. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[A2]

Dependent Choice is used in the countable-product construction of the independent input sequence. (The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain)

Proof

technique · direct

1.1A1A2F2givenalgebra

If $M = 0$ , every $M_{j} = 0$ , so all propensities in both systems vanish and the constant paths are exact. Suppose $M > 0$ . The candidate process has rate $M$ . If $M_{j} = 0$ , both channel- $j$ propensities vanish everywhere and its empty block is omitted. If $M_{j} > 0$ , its mark block has length $M_{j} / M$ , and the $X$ -acceptance portion has length $(M_{j} / M) (a_{j}^{X} (x) / M_{j}) = a_{j}^{X} (x) / M$ ; the corresponding $Y$ portion has length $a_{j}^{Y} (y) / M$ . These portions are disjoint across channels for either coordinate and depend only on that coordinate's own current state.

2.1F1F2F3step 1.1given

For fixed $x$ , the disjoint $X$ mark sets have measures $a_{j}^{X} (x) / M$ and the total accepted mark measure is $\sum_{j} a_{j}^{X} (x) / M \leq 1$ . Apply the generalized mark-set case of [F3] with constant rates $a_{j}^{X} (x)$ and envelope $M$ : it gives the exact first-event time and channel law from $x$ and, after each event, the same conditional law at the new state. Thus the $X$ coordinate has the exact marginal path law. The same argument with the $Y$ mark sets, whose measures are $a_{j}^{Y} (y) / M$ , proves the exact $Y$ marginal.

3.1F2F3step 1.1algebra∎

Within channel block $j$ , the acceptance intervals for $X$ and $Y$ share their left endpoint. Their intersection has length $\min (a_{j}^{X} (x), a_{j}^{Y} (y)) / M$ , the $X$ -only difference has length $(a_{j}^{X} (x) - a_{j}^{Y} (y))_{+} / M$ , and the $Y$ -only difference has length $(a_{j}^{Y} (y) - a_{j}^{X} (x))_{+} / M$ . Multiplying each mark length by candidate rate $M$ gives the three stated event intensities. Candidate events are a Poisson process of finite mean $M T$ , so the total number of candidates is finite almost surely and bounds all shared and residual events.

Depends on

Used by

Minimum-intensity coupling of two birth-death chains Example

Dependency tree · two levels

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Sources

David F. Anderson, Introduction to Stochastic Processes with Applications to Biology, Chapters 6–7 (standard reference, not scraped)
David F. Anderson and Thomas G. Kurtz, Continuous Time Markov Chain Models for Chemical Reaction Networks (standard reference, not scraped)