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Non-local operators with low singularity kernels: regularity estimates and martingale problem

Eryan Hu, Guohuan Zhao

TL;DR

This work analyzes a class of linear non-local operators with low-singularity kernels, $\mathcal{L}u(x)=\int_{\mathbb{R}^d}(u(x+z)-u(x))a(x,z)J(z)\,dz$, arising from subordinate Brownian motions. It develops a novel analytic framework based on a generalized Orlicz-Besov scale and an intrinsic $\psi$-decomposition to obtain Schauder-type regularity for the Poisson problem $\lambda u-\mathcal{L}u=f$, including when $a$ is inhomogeneous; it also proves well-posedness of the martingale problem for $\mathcal{L}$ and derives Krylov-type estimates via Morrey-type inequalities. The gamma-subordinator example yields the logarithmic Laplacian $\mathcal{L}=-\log(I-\Delta)$ and illustrates the sharpness of the approach. By freezing coefficients and combining analytic and probabilistic methods, the paper connects regularity theory with probabilistic well-posedness for non-local operators lacking standard scaling. This provides new tools for non-local elliptic problems with low singularity kernels and for the associated jump processes."

Abstract

We consider the linear non-local operator $\mathcal{L}$ denoted by \[ \mathcal{L} u (x) = \int_{\mathbb{R}^d} \left(u(x+z)-u(x)\right) a(x,z)J(z)\,d z. \] Here $a(x,z)$ is bounded and $J(z)$ is the jumping kernel of a Lévy process, which only has a low-order singularity near the origin and does not allow for standard scaling. The aim of this work is twofold. Firstly, we introduce generalized Orlicz-Besov spaces tailored to accommodate the analysis of elliptic equations associated with $\mathcal{L}$, and establish regularity results for the solutions of such equations in these spaces. Secondly, we investigate the martingale problem associated with $\mathcal{L}$. By utilizing analytic results, we prove the well-posedness of the martingale problem under mild conditions. Additionally, we obtain a new Krylov-type estimate for the martingale solution through the use of a Morrey-type inequality for generalized Orlicz-Besov spaces.

Non-local operators with low singularity kernels: regularity estimates and martingale problem

TL;DR

This work analyzes a class of linear non-local operators with low-singularity kernels, , arising from subordinate Brownian motions. It develops a novel analytic framework based on a generalized Orlicz-Besov scale and an intrinsic -decomposition to obtain Schauder-type regularity for the Poisson problem , including when is inhomogeneous; it also proves well-posedness of the martingale problem for and derives Krylov-type estimates via Morrey-type inequalities. The gamma-subordinator example yields the logarithmic Laplacian and illustrates the sharpness of the approach. By freezing coefficients and combining analytic and probabilistic methods, the paper connects regularity theory with probabilistic well-posedness for non-local operators lacking standard scaling. This provides new tools for non-local elliptic problems with low singularity kernels and for the associated jump processes."

Abstract

We consider the linear non-local operator denoted by Here is bounded and is the jumping kernel of a Lévy process, which only has a low-order singularity near the origin and does not allow for standard scaling. The aim of this work is twofold. Firstly, we introduce generalized Orlicz-Besov spaces tailored to accommodate the analysis of elliptic equations associated with , and establish regularity results for the solutions of such equations in these spaces. Secondly, we investigate the martingale problem associated with . By utilizing analytic results, we prove the well-posedness of the martingale problem under mild conditions. Additionally, we obtain a new Krylov-type estimate for the martingale solution through the use of a Morrey-type inequality for generalized Orlicz-Besov spaces.
Paper Structure (24 sections, 26 theorems, 281 equations)

This paper contains 24 sections, 26 theorems, 281 equations.

Table of Contents

  1. Introduction
  2. Main results and examples
  3. Approach
  4. Preliminaries
  5. Orlicz space
  6. Regular variation
  7. Subordinators and jump kernels
  8. Intrinsic dyadic decomposition, function spaces and embedding theorems
  9. Intrinsic dyadic decomposition
  10. Generalized Orlicz-Besov and Orlicz-Bessel potential spaces
  11. Morrey-type inequalities
  12. Morrey-type inequality for Orlicz-Bessel potential spaces
  13. Morrey-type inequality for generalized Orlicz-Besov spaces
  14. Studies of Poisson equations
  15. Spatially homogeneous case
  16. ...and 9 more sections

Key Result

Theorem 1.1

Suppose that $\phi$ satisfies Aspt:J-1, and $a$ satisfies Aspt:a-1-Aspt:a-2. Then there exists a constant $\lambda_0>0$ such that for each $\lambda\geqslant 2\lambda_0$ and $f\in C_{\rho}^{\alpha}$, the following Poisson equation admits a unique solution in $C_{\rho}^{1+\alpha}$. Moreover, for any $\beta\in [0,\alpha]$, where $C$ only depends on $d, \psi, \rho_0, c_0$, $\alpha$ and $\beta$.

Theorems & Definitions (60)

  • Theorem 1.1: Schauder-type estimates
  • Remark 1.2
  • Corollary 1.3
  • Example 1
  • Example 2
  • Example 3
  • Theorem 1.4
  • Corollary 1.5
  • Remark 1.6
  • Definition 2.1
  • ...and 50 more