IsReversible · Samplinglib

Plain-English statement

A semigroup is reversible when evolving either argument gives the same Hilbert-space pairing.

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 every t, f, and g, inner(P_t f,g) = inner(f,P_t g).

Intuition

Each time operator behaves like a self-adjoint operator. In the textbook the Hilbert space is L2(pi), so the inner product is integration against the stationary law.

Conditions

a real inner-product space
a continuous-linear semigroup indexed by nonnegative time

Why these conditions cannot be dropped

the inner product represents the L2(pi) pairing in the intended application
constructing a concrete Langevin semigroup on L2(pi) is a separate analytic theorem

Proof route

quantify the symmetry equality over all times and observables
verify the predicate on the identity semigroup as a behavior test

Lean interface notes

ContinuousLinearSemigroup supplies P_t and the semigroup laws
the definition is generic in the Hilbert space so it can later be instantiated by Mathlib Lp
no invariant measure is inferred by this declaration alone

Lean learning studio · mathematics → formal proof

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Proof architecture

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Syntax used on this page

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Lean statement

def IsReversible (S : ContinuousLinearSemigroup H) : Prop :=
  ∀ (t : ℝ≥0) (f g : H),
    inner ℝ (S.op t f) g = inner ℝ f (S.op t g)

/-- The constant identity semigroup is reversible on every real inner-product
space. -/

Open AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Reversibility.lean:29published source at 7bcd37294df1