# The Fractional Laplacian has Infinite Dimension

## TL;DR

This paper demonstrates that the fractional Laplacian operator on Euclidean space does not satisfy any finite-dimensional curvature-dimension inequality, indicating an infinite-dimensional behavior in this context.

## Contribution

It establishes that the fractional Laplacian fails to meet the Bakry-Émery $CD(
abla)$ inequality for all finite dimensions, revealing its inherently infinite-dimensional nature.

## Key findings

- Fractional Laplacian does not satisfy $CD(
abla)$ for any finite $N$.
- Shows the infinite-dimensional aspect of the fractional Laplacian.
- Provides insight into the geometric properties of non-local operators.

## Abstract

We show that the fractional Laplacian on $\mathbb{R}^d$ fails to satisfy the Bakry-\'Emery curvature-dimension inequality $CD(\kappa,N)$ for all curvature bounds $\kappa\in \mathbb{R}$ and all finite dimensions $N>0$.