Path integral solution of the system with coloured multiplicative noise

The behaviour of most probable values of the order parameter x and the amplitude of the conjugate force fluctuations φ is studied for a stochastic system with a coloured multiplicative noise and absorbing state. The phase diagrams introduced as dependencies of the noise self-correlation time vs. temperature and noise growth velocity are defined. It is shown that phase plane (x,φ) is divided into isolated domains of large, intermediate and small values of x. The behaviour of the system in these domains is studied in terms of the path integral. In the region x⪡1, the trajectories converge to the point x=φ=0 at 0<a<12 and to x=0, φ→∞ at 12<a⩽1. In the former case, the probability of realization of trajectories is finite, while in the latter case it is vanishing small, and an absorbing state can be formed.