  A detailed study of the mean-field solution of Langevin equations with
multiplicative noise is presented. Three different regimes depending on
noise-intensity (weak, intermediate, and strong-noise) are identified by
performing a self-consistent calculation on a fully connected lattice. The most
interesting, strong-noise, regime is shown to be intrinsically unstable with
respect to the inclusion of fluctuations, as a Ginzburg criterion shows. On the
other hand, the self-consistent approach is shown to be valid only in the
thermodynamic limit, while for finite systems the critical behavior is found to
be different. In this last case, the self-consistent field itself is broadly
distributed rather than taking a well defined mean value; its fluctuations,
described by an effective zero-dimensional multiplicative noise equation,
govern the critical properties. These findings are obtained analytically for a
fully connected graph, and verified numerically both on fully connected graphs
and on random regular networks. The results presented here shed some doubt on
what is the validity and meaning of a standard mean-field approach in systems
with multiplicative noise in finite dimensions, where each site does not see an
infinite number of neighbors, but a finite one. The implications of all this on
the existence of a finite upper critical dimension for multiplicative noise and
Kardar-Parisi-Zhang problems are briefly discussed.
