Pointwise estimates of the fundamental solution to the fractional Kolmogorov equation

TL;DR

This paper derives precise two-sided estimates for the fundamental solution of the fractional Kolmogorov equation in one spatial and one temporal dimension, including an explicit form for the case involving the square root of the Laplacian.

Contribution

It provides sharp estimates and an explicit formula for the fundamental solution to the fractional Kolmogorov equation, advancing understanding of its behavior.

Findings

Sharp two-sided estimates established

Explicit fundamental solution for the square root of the Laplacian derived

Fourier methods used to obtain results

Abstract

We prove sharp two-sided estimates of the fundamental solution to the fractional Kolmogorov equation in $\mathbb{R}\times \mathbb{R}$ using Fourier methods. Additionally, we provide an explicit form of the fundamental solution in case of the square root of the Laplacian.