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

Abstract

We establish the first positive Bakry-'Emery curvature bound for a fractional Laplacian -- with and without drift -- resolving a case of the open problem of extending the $Γ_2$-calculus to non-local operators (Garofalo 2019, Spener-Weber-Zacher 2020).Our approach relies on a novel identification: the Fourier-space carr'e du champ of the $γ$-stable generator $L_γ= -(-Δ)^{γ/2}$ coincides with the covariance kernel of fractional Brownian motion with Hurst parameter $H = γ/2$. On the torus $\mathbb{T}$, this reduces the curvature bound to a generalized eigenvalue problem. For the Cauchy process ($γ= 1$), the curvature spectrum consists of the odd integers, yielding the global bound $CD(1, \infty)$. Furthermore, we show $γ=1$ is the unique stability index admitting a strictly positive curvature bound $κ\geq 1$.Adding a confining potential $V(x) = -ω^2 \cos x$, we analyze the L'evy-Fokker-Planck operator $L = -(-Δ)^{1/2} - V'(x)\partial_x$. We prove the drift acts as a scalar shift on the curvature spectrum, yielding the global bound $CD(1 - ω^2/2, \infty)$ for $ω^2 < 2$. This provides a Poincar'e inequality and gradient estimate for a non-diagonal operator with unknown spectrum, complementing known negative results that $(-Δ)^{γ/2}$ on $\mathbb{R}^d$ fails $CD(κ, N)$ for all finite $N$.

Theorems & Definitions (38)

• Remark 1.1: On the depth of the identification
• Remark 2.1: Role of compactness
• Definition 2.2
• Definition 2.3
• Theorem 3.1: Stable--fBM correspondence
• proof
• Remark 3.2: Scope of the identification
• Remark 3.3: Three literatures, one formula
• Proposition 3.4
• proof