Bakry–Emery Curvature of the Fractional Laplacian via Fractional Brownian Covariance

We prove that the carré du champ of the symmetric γ-stable generator on the torus coincides with the covariance kernel of fractional Brownian motion with Hurst parameter H = γ/2. We exploit this correspondence to develop the first positive Bakry–Émery curvature theory for the fractional Laplacian. At γ = 1 (Cauchy process), the curvature operator has eigenvalues {1, 3, 5, …, 2N−1}, giving curvature κ = 1. Cross-sign vanishing makes γ = 1 the unique stability index with κ ≥ 1 on real trigonometric polynomials. These results complement the negative result of Spener–Weber–Zacher on ℝᵈ.