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Pointwise estimates of the fundamental solution to the fractional Kolmogorov equation

Florian Grube

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.

Pointwise estimates of the fundamental solution to the fractional Kolmogorov equation

Abstract

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

Paper Structure

This paper contains 12 sections, 19 theorems, 261 equations.

Table of Contents

  1. Introduction
  2. Outline
  3. Acknowledgements
  4. Preliminaries
  5. Derivation of the fundamental solution in Fourier-space via the method of characteristics
  6. Invariances and reduction
  7. Some specific path integrals
  8. Analysis at infinity along special paths
  9. The velocity dominant regime
  10. The spatial dominant regime
  11. Proof of the main theorem
  12. An explicit representation

Key Result

theorem 1

Let $s\in (0,1)$. There exists a constant $C=C(s)\ge 1$ such that the fundamental solution $p_s$ to eq:frac_kol at the time $t=1$ satisfies the two-sided bound for any $x,v\in \mathds{R}$.

Theorems & Definitions (41)

  • theorem 1
  • corollary 2
  • remark 3
  • lemma 4: PrZa09,ImSi20
  • remark 5
  • definition 6
  • lemma 7
  • proof
  • lemma 8
  • proof
  • ...and 31 more