Plain-English statement
A Feller transition-kernel semigroup acts as a contraction semigroup of continuous linear operators on bounded continuous real observables.
Mathematical statement
If K_t maps bounded continuous functions continuously in the starting state, then f maps to integral f dK_t is linear, sup-norm contractive, has P_0 = id, and satisfies P_{s+t} = P_s P_t.
Intuition
Probability integration is linear and cannot increase the sup norm; Chapman--Kolmogorov turns iterated integration into operator composition.
Conditions
- topological measurable state space with its Borel structure
- transition-kernel Markov semigroup laws
- Feller continuity for every bounded continuous observable
Why these conditions cannot be dropped
- the Borel interface makes continuous observables measurable
- Feller continuity keeps the operator inside the bounded-continuous space
- the Markov property gives the contraction bound
Proof route
- derive boundedness from the Bochner integral norm estimate
- prove linearity using integral_add and integral_smul
- build a ContinuousLinearMap with norm at most one
- prove identity by the Dirac integral and composition by Kernel.integral_comp
Lean interface notes
- this is the missing kernel-to-normed-operator bridge
- strong continuity in time is not inferred from the Feller mapping property
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Lean statement
def continuousLinearSemigroupOfFeller {K : ℝ≥0 → Kernel E E}
(hK : FellerTransitionKernelContract K) :
ContinuousLinearSemigroup (E →ᵇ ℝ) where
op := fellerOperator hK
op_zero := fellerOperator_zero hK
op_add := fellerOperator_add hK
@[simp]Open AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/FellerSemigroup.lean:185published source at 7bcd37294df1