An Ito Formula via Leibniz Defects in Banach Energy Spaces and PDE Generators for Stable Processes

We derive an Itô-type change-of-variables formula for symmetric γ-stable Lévy processes (γ ∈ (1,2)) within the Banach energy space framework. The classical Itô correction requires either a squared Hilbert norm or finite quadratic variation; stable processes have neither. We show that the Leibniz defect — the failure of the operator derivative to satisfy the product rule — serves as the Itô correction term. The proof proceeds by time discretization and the Banach product rule for divergences, without invoking the classical Lévy-Itô formula. As an application, we recover the fractional Laplacian (-Δ)^{γ/2} as the PDE generator through the Feynman-Kac mechanism, completing the classification: Hilbert energy spaces produce differential generators via squared norms; Banach energy spaces produce nonlocal generators via Leibniz defects.