Stochastic System with Colored Noise and Absorbing States: Path Integral Solution

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The behavior of the most probable values of the order parameter x and the amplitude ϕ of conjugate force fluctuations is studied for a stochastic system with a colored multiplicative noise with absorbing states. The phase diagrams introduced as dependencies the noise self-correlation time vs temperature and noise growth velocity are defined. It is shown that phase half-plane (x,ϕ) can be split into isolated domains of large, intermediate, and small values of x. System behavior in these domains is studied by the probability represented as path integral. In the region x≪ 1, the trajectories converge to the point x = ϕ= 0 for 0 < a < 1/2 and to x = 0, ϕ→∞ for 1/2 < a ≤ 1. In the former case, the probability of realization of trajectories is finite, while in the latter case it is vanishingly small, and an absorbing state can be formed.