Chernoff product formula for unbounded generators

Establish that the strong Chernoff/Lie–Trotter product formula for the free generator and non-Gaussian perturbation remains valid when the generators are unbounded but satisfy appropriate stability and core assumptions, using the Trotter–Kato theory of strongly continuous semigroups.

Background

The paper proves the strong Chernoff/Lie–Trotter convergence theorem only for bounded operators, where operator exponentials converge in norm and the required estimates are uniform. This bounded setting enables the accompanying Lean formalisation to remain elementary and machine-verified.

The authors indicate that extending the result to unbounded generators should require stability and core assumptions together with the Trotter–Kato theory of strongly continuous semigroups. They explicitly identify this extension as a conjecture in the accompanying Lean development and defer it to future work.

References

For unbounded generators the statement remains true under stability and core assumptions, but its proof requires the Trotter--Kato theory of $C_0$-semigroups; this is recorded as a conjecture in the accompanying Lean development and left for future work.

— Henstock--Kurzweil Gauge Integral in the Non--Gaussian Regime: A Machine--Verified Construction  (2609.10793 - Berdinsky, 9 Sep 2026) in Remark “Why boundedness matters,” Section 6.1, “Chernoff splitting for non-Gaussian dynamics”