Problems susceptible to be mathematically represented by stochastic Langevin equations including a multiplicative noise abound not only in physics, but also in biology, ecology, economy, or social sciences. In a broad sense a Langevin equation is said to be multiplicative if the noise amplitude depends on the state variables themselves. In this sense, problems exhibiting absorbing states, i.e. fluctuation-less states in which the system can be trapped, are described by equations whose noise amplitude is proportional to the square-root of the (space and time dependent) activity density, vanishing at the absorbing state. Systems within this class are countless: propagating epidemics, autocatalytic reactions, reaction-diffusion problems, self-organized criticality, pinning of lux lines in superconductors, etc..
We focused on the mean-field solution of Langevin equations with multiplicative noise. Three different regimes depending on noise-intensity (weak, intermediate, and strong-noise) can be identified by performing a self-consistent calculation on a fully connected lattice. The most interesting, strong-noise, regime is found 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. These results shed question 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. This could also have implications on the existence of a finite upper critical dimension for multiplicative noise and Kardar-Parisi-Zhang problems.
Mean-field limit of systems with multiplicative noise
M. A. Muñoz, F. Colaiori and C. Castellano, Phys. Rev. E 72, 056102 (2005).