Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the cutoff phenomenon for fast diffusion and porous medium equations

Djalil Chafaï, Max Fathi, Nikita Simonov

TL;DR

This work establishes a sharp cutoff phenomenon for the high-dimensional fast diffusion and porous medium Fokker–Planck equations by leveraging the exact Barenblatt self-similar solutions. It analyzes three distances—$W_2^2$, the relative entropy $\mathrm{H}_m$, and the Fisher information $\mathrm{I}_m$—and demonstrates that the distance to equilibrium collapses abruptly at a critical time $t_d$ of order $\log d$ as $d\to\infty$, with precise thresholds determined by the parameter $\alpha$ (linked to $m$). The results recover the Ornstein–Uhlenbeck linear case when $\alpha=1/2$ and connect to Sobolev-type inequalities at $\alpha=1$, revealing a universal cutoff mechanism across distances. The analysis combines exact solvability, moment/elliptic-structure calculations, and affine-transformation techniques to obtain explicit high-dimensional asymptotics and offers a bridge between probabilistic cutoff phenomena and nonlinear PDE dynamics, with potential extensions to curved geometries and other diffusion operators.

Abstract

The cutoff phenomenon, conceptualized at the origin for finite Markov chains, states that for a parametric family of evolution equations, started from a point, the distance towards a long time equilibrium may become more and more abrupt for certain choices of initial conditions, when the parameter tends to infinity. This threshold phenomenon can be seen as a critical competition between trend to equilibrium and worst initial condition. In this note, we investigate this phenomenon beyond stochastic processes, in the context of the analysis of nonlinear partial differential equations, by proving cutoff for the fast diffusion and porous medium Fokker-Planck equations on the Euclidean space, when the dimension tends to infinity. We formulate the phenomenon using quadratic Wasserstein distance, as well as using specific relative entropy and Fisher information. Our high dimensional asymptotic analysis uses the exact solvability of the model involving Barenblatt profiles. It includes the Ornstein-Uhlenbeck dynamics as a special linear case.

On the cutoff phenomenon for fast diffusion and porous medium equations

TL;DR

This work establishes a sharp cutoff phenomenon for the high-dimensional fast diffusion and porous medium Fokker–Planck equations by leveraging the exact Barenblatt self-similar solutions. It analyzes three distances—, the relative entropy , and the Fisher information —and demonstrates that the distance to equilibrium collapses abruptly at a critical time of order as , with precise thresholds determined by the parameter (linked to ). The results recover the Ornstein–Uhlenbeck linear case when and connect to Sobolev-type inequalities at , revealing a universal cutoff mechanism across distances. The analysis combines exact solvability, moment/elliptic-structure calculations, and affine-transformation techniques to obtain explicit high-dimensional asymptotics and offers a bridge between probabilistic cutoff phenomena and nonlinear PDE dynamics, with potential extensions to curved geometries and other diffusion operators.

Abstract

The cutoff phenomenon, conceptualized at the origin for finite Markov chains, states that for a parametric family of evolution equations, started from a point, the distance towards a long time equilibrium may become more and more abrupt for certain choices of initial conditions, when the parameter tends to infinity. This threshold phenomenon can be seen as a critical competition between trend to equilibrium and worst initial condition. In this note, we investigate this phenomenon beyond stochastic processes, in the context of the analysis of nonlinear partial differential equations, by proving cutoff for the fast diffusion and porous medium Fokker-Planck equations on the Euclidean space, when the dimension tends to infinity. We formulate the phenomenon using quadratic Wasserstein distance, as well as using specific relative entropy and Fisher information. Our high dimensional asymptotic analysis uses the exact solvability of the model involving Barenblatt profiles. It includes the Ornstein-Uhlenbeck dynamics as a special linear case.

Paper Structure

This paper contains 14 sections, 7 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. The cutoff phenomenon
  3. Fast diffusion, porous medium, Barenblatt profile, Fokker--Planck
  4. Distances and divergences, and main result on the cutoff phenomenon
  5. Further comments
  6. Linearization
  7. Linear case or diffusion limit
  8. Choice of initial data
  9. Cutoff universality and functional inequalities
  10. Further developments
  11. About methods for cutoffs
  12. Useful lemmas about Wasserstein distance
  13. Full-space Barenblatt and proof of Theorem \ref{['th:main']} (fast diffusion case)
  14. Compactly supported Barenblatt and proof of Theorems \ref{['th:main']} (porous medium case) and \ref{['th:fixed_exp']}

Key Result

Theorem 1.1

Let $\alpha\in(0,1/2)\cup(1/2,+\infty)$ and $r > 0$, and set $m=\frac{(d-2)\alpha +1}{d\alpha}$. Then for all $\mathrm{dist}(\cdot,\cdot)\in\{\mathrm{W}_2^2(\cdot,\cdot),\mathrm{H}_m(\cdot\mid\cdot),\mathrm{I}_m(\cdot\mid\cdot)\}$, and for all $\varepsilon>0$, In particular, the cutoff critical time is of order $\log(d)$.

Theorems & Definitions (16)

  • Theorem 1.1: Cutoff phenomenon
  • Theorem 1.2: Cutoff phenomenon for fixed exponent
  • Lemma 2.1: Wasserstein for position-scale transformation
  • proof
  • Lemma 2.2: Wasserstein for elliptic families
  • proof
  • Lemma 3.1: Full-space Barenblatt profile
  • proof
  • proof : Proof of Theorem \ref{['th:main']} when $\alpha>1/2$ (fast diffusion case)
  • Remark 3.2: Full-space Barenblatt profile and multivariate Student t-distribution
  • ...and 6 more