Invariance of the stable process under reversal of increments

On each finite grid, fixed-time continuity identifies reversed increments with the original increments on the reversed grid. Independence and permutation of product coordinates then give the stable Lévy characterization. This is the duality lemma of J. Bertoin, Lévy Processes, Cambridge University Press, 1996, Chapter II, Lemma 2.

theorem BGW.StableTrees.IsStableLevyLaw.map_reverseIncrements {θ T : ℝ} (hθ : 1 < θ ∧ θ ≤ 2) (hT : 0 < T) {P : MeasureTheory.ProbabilityMeasure (Skorokhod.D T ℝ)} (hP : IsStableLevyLaw θ T P) : P.map (Skorokhod.reverseIncrements ⋯) = P

Reversal of increments, general form: every stable Lévy law on [0,T] is invariant under reversal.

theorem BGW.StableTrees.map_reverseIncrements_stableProcessLaw {θ T : ℝ} (hθ : 1 < θ ∧ θ ≤ 2) (hT : 0 < T) : (stableProcessLaw θ T).map (Skorokhod.reverseIncrements ⋯) = stableProcessLaw θ T

The stable process law is invariant under reversal of increments.