An α-stable process is non-Gaussian for any α < 2. Its probability density has heavy tails, its heat kernel differs qualitatively from the Gaussian kernel, and its invariant measures (when they exist) reflect the heavy-tailed character of the jumps. As α → 2, the process should approach Brownian motion, but “should approach” is vague. At what rate? In what sense? Uniformly in space and time, or only in some averaged sense?
The paper establishes optimal rates. The heat kernel of the α-stable process converges to the Gaussian heat kernel pointwise at rate 2 − α. The transition probabilities — including cases with low-regularity, possibly unbounded drifts — converge at the same optimal rate. The invariant measures converge in weighted total variation and Kantorovich distance, again at rate 2 − α.
The convergence is uniform across the stability parameter: the estimates hold for all α in [α₀, 2] simultaneously, not just at each individual α. This uniformity is what makes the result useful — it controls the transition from heavy-tailed to Gaussian behavior across the entire family of processes.
The rate 2 − α is the natural one: it measures the distance from Gaussianity in parameter space, and the paper shows this parameter distance translates directly to kernel distance, trajectory distance, and equilibrium distance. The non-Gaussian to Gaussian transition is not a phase transition with a sharp boundary but a continuous deformation, and the deformation rate is the simplest thing it could be — linear in the distance to the Gaussian endpoint.