chewi_lemma_1_2_2 · Samplinglib

Plain-English statement

Identity and Chapman--Kolmogorov transition-kernel laws induce the zero-time and composition laws for Markov operators.

Scope guard. This card records a compiled local declaration. Its mathematical scope is exactly the Lean statement below; the Registry note and source correspondence may describe motivation but do not strengthen it.

Mathematical statement

For measurable nonnegative observables, P_0 = id and P_s P_t = P_t P_s = P_{s+t}.

Intuition

Integrating first over one transition and then the next is exactly kernel composition; time homogeneity makes the total elapsed time the only parameter.

Conditions

measurable state space
a Markov transition kernel for every nonnegative time
identity kernel at time zero
Chapman--Kolmogorov composition

Why these conditions cannot be dropped

measurability makes kernel integration an observable
Markov kernels preserve constants
the two kernel identities are the process-level content of the semigroup law

Proof route

define P_t by lintegration against K_t
use Kernel.lintegral_id' at zero
use Kernel.lintegral_comp after Chapman--Kolmogorov
use commutativity of nonnegative-time addition

Lean interface notes

the operator law is proved from kernels rather than stored as a field
concrete process construction and time continuity remain separate

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Lean statement

theorem chewi_lemma_1_2_2 {K : ℝ≥0 → Kernel E E}
    (hK : TransitionKernelContract K) :
    markovOperator hK 0 = id ∧
      ∀ s t : ℝ≥0,
        markovOperator hK s ∘ markovOperator hK t =
            markovOperator hK (s + t) ∧
          markovOperator hK t ∘ markovOperator hK s =
            markovOperator hK (s + t) := by
  constructor
  · exact markovOperator_zero hK
  · intro s t
    constructor
    · exact markovOperator_comp hK s t
    · simpa [add_comm] using markovOperator_comp hK t s

end

end MarkovSemigroup
end StochasticProcesses
end TechnicalLemmas
end AutoSamplingTheory

Open AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/MarkovSemigroup.lean:119published source at 7bcd37294df1