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Fokker--Planck equations and n--dimensional Poincaré inequalities for isotropic densities

G. Furioli, A. Pulvirenti, E. Terraneo, G. Toscani

TL;DR

The paper develops a bridge between n-dimensional Fokker--Planck equations with variable diffusion and weighted Poincaré inequalities for isotropic densities. By expressing FP dynamics in gradient form and identifying the stationary state $f_\infty$, the authors derive diffusion weights $K(|\mathbf x|)$ that induce sharp or near-sharp Poincaré-type inequalities for several isotropic density families, including Gaussian, Cauchy-type, exponential, and Barenblatt densities. They extend known 1D results to product and radially symmetric multi-dimensional settings, provide refined and hybrid inequalities, and illustrate the sharpness and limitations of the weights in various cases. The results illuminate how convergence to equilibrium in FP models is governed by the interplay between diffusion structure and the tail behavior of stationary densities, with potential applications in kinetic theory and socio-physical modeling.

Abstract

We consider new connections between the problem of trend to equilibrium for the n-dimensional Fokker--Planck equation of statistical physics, and weighted Poincaré inequality. To this aim we consider a class of n-dimensional Fokker--Planck equations with variable isotropic coefficient of diffusion and drift, inspired by the analogous one-dimensional Fokker--Planck equation appearing when studying the evolution of wealth distribution.

Fokker--Planck equations and n--dimensional Poincaré inequalities for isotropic densities

TL;DR

The paper develops a bridge between n-dimensional Fokker--Planck equations with variable diffusion and weighted Poincaré inequalities for isotropic densities. By expressing FP dynamics in gradient form and identifying the stationary state , the authors derive diffusion weights that induce sharp or near-sharp Poincaré-type inequalities for several isotropic density families, including Gaussian, Cauchy-type, exponential, and Barenblatt densities. They extend known 1D results to product and radially symmetric multi-dimensional settings, provide refined and hybrid inequalities, and illustrate the sharpness and limitations of the weights in various cases. The results illuminate how convergence to equilibrium in FP models is governed by the interplay between diffusion structure and the tail behavior of stationary densities, with potential applications in kinetic theory and socio-physical modeling.

Abstract

We consider new connections between the problem of trend to equilibrium for the n-dimensional Fokker--Planck equation of statistical physics, and weighted Poincaré inequality. To this aim we consider a class of n-dimensional Fokker--Planck equations with variable isotropic coefficient of diffusion and drift, inspired by the analogous one-dimensional Fokker--Planck equation appearing when studying the evolution of wealth distribution.

Paper Structure

This paper contains 14 sections, 10 theorems, 158 equations.

Table of Contents

  1. Introduction
  2. Basic properties of Fokker--Planck type equations
  3. Fokker--Planck equilibria and weight in Poincaré--type inequalities
  4. A first Poincaré--type result for isotropic densities
  5. A Poincaré--type inequality for product probability densities
  6. Poincaré--type inequalities for radially symmetric densities in $\mathbb R^n$.
  7. Examples.
  8. The case of a Gaussian density function
  9. Poincaré type inequalities for Cauchy-type densities
  10. Poincaré--type inequalities for exponentially decaying densities
  11. Poincaré inequalities for Barenblatt type densities
  12. A refined Poincaré--type result for isotropic densities
  13. Another hybrid inequality for isotropic densities
  14. Appendix

Key Result

Theorem 1

FPTT Let $f_0$ be the initial data of a Cauchy problem for equation (FP-k) coupled with boundary conditions (bou) and let us assume ∫_ I [( f_0( x)f_∞( x))^2 - 1 ] f_∞( x) d x <+∞ e where $f_\infty$ is the stationary state and satisfies the Poincaré inequality (Poi) with weight $w(|{\bf x}|) = c K

Theorems & Definitions (27)

  • Theorem 1
  • Remark 2
  • Theorem 3
  • Remark 4
  • Theorem 5
  • Theorem 6
  • Remark 7
  • Theorem 8
  • Remark 9
  • proof
  • ...and 17 more