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A Local Bifurcation Theorem for McKean-Vlasov Diffusions

Shao-Qin Zhang

TL;DR

The work proves existence and local bifurcation results for a broad class of probability measure-valued equations that characterize stationary distributions of McKean-Vlasov diffusions with gradient-type drifts, even when coefficients are discontinuous in the distribution variable. It combines Lyapunov-based Schauder fixed-point arguments with a local Krasnosel'skii bifurcation framework, using a regularized Hilbert-Schmidt determinant to detect eigenvalue crossings. The main payoff is a concrete, finite-rank determinant criterion for bifurcation points, with applications to granular media and Vlasov-Fokker-Planck type equations; the approach also handles non-smooth interactions and degenerate SDEs. Overall, the paper provides a rigorous bridge between measure-valued fixed-point theory and phase-transition phenomena in nonlinear diffusions, offering practical criteria to locate bifurcations and multiplicity of stationary states.

Abstract

We establish an existence result of a solution to a class of probability measure-valued equations, whose solutions can be associated with stationary distributions of many McKean-Vlasov diffusions with gradient-type drifts. Coefficients of the probability measure-valued equation may be discontinuous in the weak topology and the total variation norm. Owing to that the bifurcation point of the probability measure-valued equation is relevant to the phase transition point of the associated McKean-Vlasov diffusion, we establish a local Krasnosel'skii bifurcation theorem. Regularized determinant for the Hilbert-Schmidt operator is used to derive our criteria for the bifurcation point. Concrete examples, including the granular media equation and the Vlasov-Fokker-Planck equation with quadratic interaction, are given to illustrate our results.

A Local Bifurcation Theorem for McKean-Vlasov Diffusions

TL;DR

The work proves existence and local bifurcation results for a broad class of probability measure-valued equations that characterize stationary distributions of McKean-Vlasov diffusions with gradient-type drifts, even when coefficients are discontinuous in the distribution variable. It combines Lyapunov-based Schauder fixed-point arguments with a local Krasnosel'skii bifurcation framework, using a regularized Hilbert-Schmidt determinant to detect eigenvalue crossings. The main payoff is a concrete, finite-rank determinant criterion for bifurcation points, with applications to granular media and Vlasov-Fokker-Planck type equations; the approach also handles non-smooth interactions and degenerate SDEs. Overall, the paper provides a rigorous bridge between measure-valued fixed-point theory and phase-transition phenomena in nonlinear diffusions, offering practical criteria to locate bifurcations and multiplicity of stationary states.

Abstract

We establish an existence result of a solution to a class of probability measure-valued equations, whose solutions can be associated with stationary distributions of many McKean-Vlasov diffusions with gradient-type drifts. Coefficients of the probability measure-valued equation may be discontinuous in the weak topology and the total variation norm. Owing to that the bifurcation point of the probability measure-valued equation is relevant to the phase transition point of the associated McKean-Vlasov diffusion, we establish a local Krasnosel'skii bifurcation theorem. Regularized determinant for the Hilbert-Schmidt operator is used to derive our criteria for the bifurcation point. Concrete examples, including the granular media equation and the Vlasov-Fokker-Planck equation with quadratic interaction, are given to illustrate our results.
Paper Structure (10 sections, 19 theorems, 283 equations)

This paper contains 10 sections, 19 theorems, 283 equations.

Table of Contents

  1. Introduction
  2. Existence
  3. Proof of Theorem \ref{['exis-thm1']}
  4. Bifurcation
  5. Well-definedness and regularity of $\Phi$
  6. Main result
  7. Proofs of lemmas and corollaries in Subsection 3.1
  8. Proof of Theorem \ref{['thm-bif']}
  9. Proof of Corollary \ref{['cor-finite']}
  10. Appendix: proofs of auxiliary lemmas

Key Result

Theorem 2.1

Assume that (H) holds with $F_0\in L^1(W_0\bar{\mu})$, and (W) holds with $W_1\in \mathcal{W}^{2,p_1}_{\bar{\mu}}$ for some $p_1\geq \frac{ q}{ q-1}$. Then $\hat{\mathcal{T}}\circ \mathscr{I}$ has a fixed point in $\mathcal{W}^{1,p}_{q,\bar{\mu}}\cap L^\infty\cap L^1(W\bar{\mu})$.

Theorems & Definitions (61)

  • Example 1.1
  • Theorem 2.1
  • Corollary 2.2
  • proof
  • Example 2.3
  • proof
  • Example 2.4
  • proof
  • Remark 2.1
  • Corollary 2.5
  • ...and 51 more