Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An H-theorem for a conditional McKean-Vlasov process related to interacting diffusions on regular trees

Kevin Hu, Kavita Ramanan

TL;DR

The paper analyzes the κ-regular Markovian local-field equation (κ-MLFE), a conditional McKean–Vlasov evolution on κ-regular trees, establishing well-posedness and an H-theorem via the sparse free energy $\mathbb{H}_κ$ and a dissipation $\mathbb{I}_κ$. It proves that stationary distributions correspond to zeros of $\mathbb{I}_κ$ and to Cayley fixed points, with a bijection linking these to continuous Gibbs measures on the infinite tree; for κ=2, it connects $\mathbb{H}_2$ to renormalized entropies and proves exponential convergence under a logarithmic-Sobolev inequality. The results rely on a fixed-point construction, entropy methods, and sharp PDE–probabilistic arguments, and they illuminate the long-time behavior of sparse interacting diffusions, including the structure of stationary states and rates of convergence. The work bridges nonlinear Fokker–Planck dynamics with mean-field and tree-based Gibbs measures, offering a global Lyapunov framework via the sparse free energy and providing tools for quantitative convergence in the κ=2 regime.

Abstract

We study the long-time behavior of the $κ$-Markov local-field equation ($κ$-MLFE), which is a conditional McKean-Vlasov equation associated with interacting diffusions on the $κ$-regular tree. Under suitable assumptions on the coefficients, we prove well-posedness of the $κ$-MLFE. We also establish an H-theorem by identifying an energy functional, referred to as the sparse free energy, whose derivative along the measure flow of the $κ$-MLFE is given by a nonnegative functional that can be viewed as a modified Fisher information. Moreover, we show that the zeros of the latter functional coincide with the set of stationary distributions of the $κ$-MLFE and are also marginals of splitting Gibbs measures on the $κ$-regular tree. Furthermore, we show that for a natural class of initial conditions, the corresponding measure flow converges to one of the stationary distributions, thus demonstrating that the sparse free energy acts as a global Lyapunov function. Under mild additional conditions, in the case $κ= 2$ we prove that the sparse free energy arises naturally as the renormalized limit of certain relative entropies. We exploit this characterization to prove a modified logarithmic Sobolev inequality and establish an exponential rate of convergence of the $2$-MLFE measure flow to its unique stationary distribution.

An H-theorem for a conditional McKean-Vlasov process related to interacting diffusions on regular trees

TL;DR

The paper analyzes the κ-regular Markovian local-field equation (κ-MLFE), a conditional McKean–Vlasov evolution on κ-regular trees, establishing well-posedness and an H-theorem via the sparse free energy and a dissipation . It proves that stationary distributions correspond to zeros of and to Cayley fixed points, with a bijection linking these to continuous Gibbs measures on the infinite tree; for κ=2, it connects to renormalized entropies and proves exponential convergence under a logarithmic-Sobolev inequality. The results rely on a fixed-point construction, entropy methods, and sharp PDE–probabilistic arguments, and they illuminate the long-time behavior of sparse interacting diffusions, including the structure of stationary states and rates of convergence. The work bridges nonlinear Fokker–Planck dynamics with mean-field and tree-based Gibbs measures, offering a global Lyapunov framework via the sparse free energy and providing tools for quantitative convergence in the κ=2 regime.

Abstract

We study the long-time behavior of the -Markov local-field equation (-MLFE), which is a conditional McKean-Vlasov equation associated with interacting diffusions on the -regular tree. Under suitable assumptions on the coefficients, we prove well-posedness of the -MLFE. We also establish an H-theorem by identifying an energy functional, referred to as the sparse free energy, whose derivative along the measure flow of the -MLFE is given by a nonnegative functional that can be viewed as a modified Fisher information. Moreover, we show that the zeros of the latter functional coincide with the set of stationary distributions of the -MLFE and are also marginals of splitting Gibbs measures on the -regular tree. Furthermore, we show that for a natural class of initial conditions, the corresponding measure flow converges to one of the stationary distributions, thus demonstrating that the sparse free energy acts as a global Lyapunov function. Under mild additional conditions, in the case we prove that the sparse free energy arises naturally as the renormalized limit of certain relative entropies. We exploit this characterization to prove a modified logarithmic Sobolev inequality and establish an exponential rate of convergence of the -MLFE measure flow to its unique stationary distribution.

Paper Structure

This paper contains 41 sections, 29 theorems, 340 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background and Motivation.
  3. Main Contributions
  4. Well-posedness of the $\kappa$-MLFE
  5. H-theorem, sparse free energy and long-time convergence of solutions to the $\kappa$-MLFE
  6. Correspondence with stationary distributions of particle systems on $\kappa$-regular trees
  7. Renormalized relative entropies
  8. Logarithmic Sobolev inequalities and exponential convergence
  9. Structure of the Paper
  10. Notation
  11. Vectors and Gradients
  12. Probability Measures, Probability Spaces, and Equipped Functionals
  13. Function Spaces
  14. Graphs and Markov Random Fields
  15. The $\kappa$-Regular Markovian Local-Field Equation and its Well-posedness
  16. ...and 26 more sections

Key Result

Theorem 3.10

Fix $\kappa, d \in \mathbb{N}$ with $\kappa \geq 2$. Suppose $(U, W)$ satisfy Assumption as:results:lyapunov, and $\|\nabla W\|_{L^\infty(\mathbb{R}^d)} < \infty$. Then the $\kappa$-MLFE with potentials $(U, W)$ is well-posed.

Figures (2)

  • Figure 1.1: Convergence diagram for entropy renormalization. Here, $\mathcal{S}$ is the set of zeros of $\mathbb I_2$, which in our setting are also the limit points of $\mu_t$ (see Theorem \ref{['thm:results:convergence']}). By Theorem \ref{['thm:results:zeros']} and Corollary \ref{['cor:LackerZhang']}, the set $\mathcal{S}$ is also the set of root marginal distributions of continuous Gibbs measures on regular trees, and therefore can be identified as possible limit points of root marginals of $\theta^n$. This fact is reflected by the dashed arrow.
  • Figure 4.1: A comparison between $\hat{\mathbb{H}}_2(\bar{\mu}_t)$ and $\mathbb{H}_2(\mu_t)$. Here, $\mu_t$ solves the $2$-MLFE with $d = 1$, $\kappa = 2$, potentials $U(x) = 7x^2/4$ and $K =-3x^2/8$, and an initial condition that is not a 1-MRF. The left column shows the evolution of the 1-MRF renormalized limit $\hat{\mathbb H}_2(\mu_t)$ and the right column shows the evolution of the sparse free energy $\mathbb H_2(\mu_t)$. For both columns, the bottom figure shows the top figure zoomed in on the time interval $(0.2, 0.8)$. We see that $\mathbb{H}_2(\mu_t)$ is always decreasing in time while $\hat{\mathbb{H}}_2(\bar{\mu}_t)$ is not.

Theorems & Definitions (84)

  • Definition 2.1: Markov random field
  • Definition 3.1: Edge marginal
  • Definition 3.2: Symmetric probability measures
  • Remark 3.3: Exchangeability of marginals
  • Definition 3.4: $\kappa$-regular Markovian local-field equation
  • Remark 3.5
  • Remark 3.6: Local-field equation and $\kappa$-MLFE
  • Definition 3.7: Set of admissible measures
  • Remark 3.8
  • Definition 3.9: Linear growth solution and well-posedness
  • ...and 74 more