  This article refines the classical notion of a stochastic D-bifurcation to
the respective family of n-point motions for homogeneous Markovian stochastic
semiflows, such as stochastic Brownian flows of homeomorphisms, and their
generalizations. This notion essentially detects at which level $k\leq n$ the
support of the invariant measure of the k-point bifurcation has more than one
connected component. Stochastic Brownian flows and their invariant measures
which were shown by Kunita (1990) to be rigid, in the sense of being uniquely
determined by the $1$-and $2$-point motions, and hence only stochastic n-point
bifurcation of level $n=1$ or $n=2$ can occur. For general homogeneous
stochastic Markov semiflows this turns out to be false. This article constructs
minimal examples of where this rigidity is false in general on finite space and
studies the complexity of the resulting n-point bifurcations.
