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

Xianming Liu, Chongyang Ren, Mingyan Wu

Abstract

We establish uniform pointwise estimates for the densities of a family of $α$-stable processes with respect to the stability index $α\in [α_0,2]$ for some $α_0>0$, and we estimate the difference between the heat kernels of non-local and local operators, showing that it is controlled by the rate $2-α$. Both estimates (see Proposition 1.5) are new to the literature. Furthermore, as an application, we achieve the optimal rate $2-α$ for the pointwise estimate between the transition probabilities, as well as for the (weighted) total variation and Kantorovich distances between the invariant measures, of anomalous and normal diffusion with drifts that are low in regularity and possibly unbounded. The results on transition probabilities (see Theorem 1.3) are novel, while those on invariant measures (see Theorem 1.8) significantly extend the existing literature.

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

Abstract

We establish uniform pointwise estimates for the densities of a family of -stable processes with respect to the stability index for some , and we estimate the difference between the heat kernels of non-local and local operators, showing that it is controlled by the rate . Both estimates (see Proposition 1.5) are new to the literature. Furthermore, as an application, we achieve the optimal rate for the pointwise estimate between the transition probabilities, as well as for the (weighted) total variation and Kantorovich distances between the invariant measures, of anomalous and normal diffusion with drifts that are low in regularity and possibly unbounded. The results on transition probabilities (see Theorem 1.3) are novel, while those on invariant measures (see Theorem 1.8) significantly extend the existing literature.
Paper Structure (25 sections, 25 theorems, 223 equations, 1 table)

This paper contains 25 sections, 25 theorems, 223 equations, 1 table.

Table of Contents

  1. Introduction
  2. Heat kernel estimates: from non-local to local
  3. Assumptions and deterministic regularized flows
  4. Main results
  5. $\alpha$-dependence of invariant measures
  6. Assumptions and distances between measures
  7. Main results
  8. Conventions and notations
  9. Preliminaries
  10. Isotropic $\alpha$-stable processes with $\alpha \in (0,2]$
  11. The subordination representation
  12. Infinitesimal generators and Lévy measures
  13. Some basic inequalities of $\varrho^{(k)}_{\gamma_1,\gamma_2}(t,x)$
  14. Some uniform estimates of densities
  15. Heat kernels: uniform estimates and $\alpha$-continuity
  16. ...and 10 more sections

Key Result

Lemma 1.1

Under $(\mathbf{H}_b^\beta)$, for each $\epsilon>0$ and $s,t \geq 0$, the mapping $x\mapsto \theta^{(\epsilon)}_{s,t}(x)$ is a $C^1$-diffeomorphism and its inverse is given by $x\mapsto \theta^{(\epsilon)}_{s,t}(x)$. Moreover, for all $s,r,t\geq 0$, For any $T>0$, there exists a constant $c=c(d,T,\kappa_0)\geq 1$ such that for all $s,t \in [0, T]$ and $x,y\in {\mathbb R}^d$,

Theorems & Definitions (54)

  • Lemma 1.1: Deterministic flow
  • Remark 1.2
  • Theorem 1.3: Heat kernel estimates
  • Proposition 1.4
  • Remark 1.5
  • Example 1.6: On the distance between laws of solutions to SDEs
  • Theorem 1.7: Optimal convergence rate
  • Remark 1.8
  • Remark 1.9
  • Lemma 2.1
  • ...and 44 more