A Local Bifurcation Theorem for McKean-Vlasov Diffusions
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.
Cite
@article{arxiv.2401.14784,
title = {A Local Bifurcation Theorem for McKean-Vlasov Diffusions},
author = {Shao-Qin Zhang},
journal= {arXiv preprint arXiv:2401.14784},
year = {2025}
}