DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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Discrete generator of a countable-state transition matrix

Definition

Let $S$ be countable with sigma-algebra $2^{S}$ , and let $p = (p (x, y))_{x, y \in S}$ be a transition matrix. For bounded $f : S \to R$ , write $P f (x) = \sum_{y \in S} p (x, y) f (y)$ and define the discrete generator $L f (x) = P f (x) - f (x) = \sum_{y \in S} p (x, y) (f (y) - f (x)) .$ The last series is absolutely convergent, since $\sum_{y} p (x, y) ∣ f (y) - f (x) ∣ \leq 2 ∥ f ∥_{\infty} \sum_{y} p (x, y) = 2 ∥ f ∥_{\infty} .$ All functions on $(S, 2^{S})$ are measurable. Constant functions have generator zero; in particular $L 0 = L 1 = 0$ . On an empty $S$ every assertion is vacuous. No choice is used.

Depends on

Measure kernel and probability kernel

Used by

Bounded harmonic functions yield Markov-chain martingales Corollary
Simple random-walk transition kernel Example
Countable-state martingale-problem characterization Theorem

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Roch, Markov Chains: Martingale Methods, Note 24, Section 1 (standard reference, not scraped)