Plain-English statement
The forward right time increment of a semigroup orbit converges to the semigroup applied to the generator value.
Mathematical statement
If Lf = g in the right-generator sense, then [P_{t+h}f - P_t f]/h tends to P_t g as h decreases to zero through positive times.
Intuition
The semigroup law rewrites the orbit increment as the generator quotient of P_t f.
Conditions
- a continuous-linear semigroup on a real normed space
- the right generator limit of f exists
Why these conditions cannot be dropped
- the semigroup law factors the time increment
- the one-sided filter encodes nonnegative time
Proof route
- transport the generator graph through P_t
- rewrite P_{t+h} using the semigroup law
- reuse the transported Tendsto theorem
Lean interface notes
- this is an actual one-sided Tendsto theorem
- it is not the forward equation and does not construct a Langevin process
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Lean statement
theorem kolmogorov_backward_right
(S : ContinuousLinearSemigroup M) {f g : M}
(hfg : HasRightGeneratorAt S f g) (t : ℝ≥0) :
HasRightGeneratorAt S (S.op t f) (S.op t g) ∧
Tendsto (fun h : ℝ≥0 => rightOrbitDifferenceQuotient S t h f)
(nhdsWithin 0 (Ioi 0)) (𝓝 (S.op t g)) := by
have hmap := hfg.map t
constructor
· exact hmap
· simpa only [HasRightGeneratorAt, rightOrbitDifferenceQuotient_eq] using hmap
end
end OperatorGenerator
end StochasticProcesses
end TechnicalLemmas
end AutoSamplingTheoryOpen AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/OperatorGenerator.lean:126published source at 7bcd37294df1