The Non-Gaussian to Gaussian Transition: Pointwise Heat Kernel Estimates and Optimal Convergence Rates

We establish uniform pointwise estimates for the densities of a family of $\alpha$-stable processes with respect to the index $\alpha \in [\alpha_0,2]$ for some $\alpha_0>0$. In addition, we estimate the difference between the heat kernels of non-local and local operators, showing that it is controlled by the rate $2-\alpha$. Both estimates (see Proposition 2.4) are new to the literature. Furthermore, as an application, we achieve the optimal rate $2-\alpha$ for the pointwise estimate between the transition probabilities, as well as for the (weighted) total variation and Kantorovich distances between the invariant measures, of non-Gaussian and Gaussian diffusion. These results are obtained under the assumption that the drifts are locally $\beta$-H\"older continuous, with the latter additionally requiring dissipativity. The results on transition probabilities (see Theorem 2.3) are novel, while those on invariant measures (see Theorem 2.7) significantly extend the existing literature.