Bifurcations of one-dimensional stochastic differential equations

We consider families of random dynamical systems induced by parametrized one-dimensional stochastic differential equations. We give necessary and sufficient conditions on the invariant measures of the associated Markov semigroups which ensure a stochastic bifurcation. This leads to sufficient conditions on drift and diffusion coefficients for a stochastic pitchfork and transcritical bifurcation of the family of random dynamical systems.