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

The behavior of the most probable values of the order parameter $x$ and the amplitude $\varphi$ 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 $(x,\varphi )$ 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 $x\ll 1$,thetrajectoriesconvergetothepoint$x = \phi = 0$for$0 < a < 1/2$ and to $x=0$, $\varphi \to \mathrm{\infty }$ for $1\mathrm{/}2<a\le 1$. 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.