  A general approach to consider spatially extended stochastic systems with
correlations between additive and multiplicative noises subject to nonlinear
damping is developed. Within modified cumulant expansion method, we derive an
effective Fokker-Planck equation whose stationary solutions describe a
character of ordered state. We find that fluctuation cross-correlations lead to
a symmetry breaking of the distribution function even in the case of the
zero-dimensional system. In general case, continuous, discontinuous and
reentrant noise induced phase transitions take place. It is appeared the
cross-correlations play a role of bias field which can induce a chain of phase
transitions being different in nature. Within mean field approach, we give an
intuitive explanation of the system behavior through an effective potential of
thermodynamic type. This potential is written in the form of an expansion with
coefficients defined by temperature, intensity of spatial coupling, auto- and
cross-correlation times and intensities of both additive and multiplicative
noises.
