Plain-English statement
For an abstract semigroup that is strongly continuous at zero, every orbit is right-continuous after any fixed starting time.
Mathematical statement
If P_h f tends to f as h tends to zero, then P_(t+h) f tends to P_t f for every nonnegative t.
Intuition
The semigroup identity rewrites P_(t+h)f as P_t(P_h f), and the bounded linear operator P_t preserves the limit.
Conditions
- a continuous-linear semigroup on a real normed space
- strong continuity at zero for every observable
Why these conditions cannot be dropped
- the semigroup law transports the zero-time limit to time t
- continuity of P_t is needed to pass the limit through the operator
- the theorem does not prove strong continuity for a concrete Langevin process
Proof route
- apply strong continuity at zero to P_h f
- compose this convergence with the continuous linear map P_t
- rewrite P_t(P_h f) as P_(t+h)f
Lean interface notes
- the time type is the nonnegative reals
- strong continuity is a field of StronglyContinuousSemigroup and remains a concrete analytic obligation
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Lean statement
theorem StronglyContinuousSemigroup.tendsto_op_add
(S : StronglyContinuousSemigroup M) (t : ℝ≥0) (f : M) :
Tendsto (fun h : ℝ≥0 => S.op (t + h) f) (𝓝 0) (𝓝 (S.op t f)) := by
have hop : Tendsto (S.op t) (𝓝 f) (𝓝 (S.op t f)) :=
(S.op t).continuous.continuousAt
have hmapped :
Tendsto (fun h : ℝ≥0 => S.op t (S.op h f))
(𝓝 0) (𝓝 (S.op t f)) :=
hop.comp (S.stronglyContinuousAtZero f)
simpa only [ContinuousLinearSemigroup.op_add_apply] using hmapped
/-- The zero vector has generator value zero. -/Open AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/OperatorGeneratorDomain.lean:45published source at 7bcd37294df1