  Stationary distributions of many McKean-Vlasov diffusions with gradient-type
drifts can be obtained by solving probability measure-valued equations. We
established an existence result of a solution to this equation on a space of
probability measures endowed with weighted variation distance. After
introducing a parameter to this equation, a local Krasnoselskii bifurcation
theorem is established when distribution dependent part is an integral with
respect to the probability measure. The bifurcation point is relevant to the
phase transition point of the associated McKean-Vlasov diffusion. Regularized
determinant for the Hilbert-Schmidt operator is used to derive our criteria for
the bifurcation point. Examples, such as granular media equation and
Vlasov-Fokker-Planck equation with quadratic interaction, are given to
illustrate our results.
