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

TL;DR

This paper explores Bakry-Emery curvature for fractional Laplacians by linking it to fractional Brownian motion covariance, leading to a matrix eigenvalue problem that reveals curvature bounds and spectral shifts.

Contribution

It introduces a novel matrix formulation of Bakry-Emery curvature for fractional Laplacians using fractional Brownian covariance, with explicit eigenstructure analysis on the torus.

Findings

Eigenstructure computed explicitly for gamma=1 (Brownian case)

Curvature bounds derived from eigenvalues of covariance matrices

Adding drift results in a scalar shift of the curvature spectrum

Abstract

We study Bakry-Emery curvature for fractional Laplacian generators using a Fourier representation of the carr\'e du champ operator. For the stable generator of order gamma, the associated kernel on same-sign frequencies coincides with the covariance kernel of fractional Brownian motion with Hurst parameter equal to gamma divided by two. This observation allows the curvature inequality to be reformulated as a generalized eigenvalue problem for covariance matrices. On the one dimensional torus we analyze this matrix formulation for trigonometric polynomials. In the Cauchy case (gamma equal to one), corresponding to Brownian covariance, the eigenstructure can be computed explicitly and yields a Bakry-Emery curvature bound on the corresponding Fourier subspaces. We also study the effect of adding a confining drift to the Cauchy generator and show that the curvature spectrum undergoes a simple scalar shift. These results provide a matrix formulation of Bakry-Emery curvature for certain nonlocal operators and highlight a structural connection between fractional Laplacians and fractional Brownian covariance kernels.