Functional Inequalities for Stable-Like Dirichlet Forms

2012/05/21 by Feng‐Yu Wang, Jian Wang, Wang, Feng-Yu +1 · 2 citations
Computer Science · Mathematics · #47G20 #60G52 #60J75 #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Functional Analysis (math.FA) #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.1205.4508

openalex publication_date 2012/05/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let V∈ C²(\R^d) such that μ_V(\d x):= \e^-V(x) \d x is a probability measure, and let å∈ (0,2). Explicit criteria are presented for the å-stable-like Dirichlet form \E_å,V(f,f):= ∫_\R^d×\R^d \ff|f(x)-f(y)|²|x-y|^d+α \d y \e^-V(x) \d x to satisfy Poincaré-type (i.e., Poincaré, weak Poincaré and super Poincaré) inequalities. As applications, sharp functional inequalities are derived for the Dirichlet form with V having some typical growths. Finally, the main result of \citeMRS on the Poincaré inequality is strengthened

Citations

• Some theorems on stable processes
• Logarithmic Sobolev Inequalities
• Functional Inequalities and Subordination: Stability of Nash and Poincaré inequalities

Cited by

• Rényi Differential Privacy for Heavy-Tailed SDEs via Fractional Poincaré Inequalities
• Functional inequalities for Nonlocal Dirichlet Forms With Finite Range Jumps or Large Jumps

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