Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Fractional Laplacian with supercritical killings

Soobin Cho, Renming Song

Abstract

In this paper, we study Feynman-Kac semigroups of symmetric $α$-stable processes with supercritical killing potentials belonging to a large class of functions containing functions of the form $b|x|^{-β}$, where $b>0$ and $β>α$. We obtain two-sided estimates on the densities $p(t, x, y)$ of these semigroups for all $t>0$, along with estimates for the corresponding Green functions.

Fractional Laplacian with supercritical killings

Abstract

In this paper, we study Feynman-Kac semigroups of symmetric -stable processes with supercritical killing potentials belonging to a large class of functions containing functions of the form , where and . We obtain two-sided estimates on the densities of these semigroups for all , along with estimates for the corresponding Green functions.
Paper Structure (17 sections, 47 theorems, 342 equations)

This paper contains 17 sections, 47 theorems, 342 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some properties of $Y$
  4. Preliminary estimates for $X^\kappa$
  5. Survival probability estimates
  6. Small time estimates
  7. Small time upper heat kernel estimates
  8. Small time lower heat kernel estimates
  9. Key proposition for large time estimates
  10. The case of $d>\alpha$
  11. The case of $d=1<\alpha$
  12. The case of $d=1=\alpha$
  13. Large time estimates
  14. Large time lower heat kernel estimates
  15. Large time upper heat kernel estimates
  16. ...and 2 more sections

Key Result

Theorem 1.1

(i) (Small time estimates) Let $T>0$. There exist constants $\lambda_1=\lambda_1(\beta,b)>0$, $\lambda_2=\lambda_2(\beta,b)>0$ and $C= C(d,\alpha,\beta,b,T)>1$ such that for all $t \in (0,T]$ and $x,y \in {\mathbb R}^d_0$, and (ii) (Large time estimates) There exist comparison constants depending only on $d,\alpha,\beta$ and $b$ such that the following estimates hold for all $t \in [2,\infty)$

Theorems & Definitions (49)

  • Theorem 1.1
  • Remark 1.2
  • Definition 1.3
  • Proposition 2.1
  • Proposition 2.2
  • Corollary 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Lemma 2.6
  • Lemma 2.7
  • ...and 39 more