Hamilton-Jacobi Theory of Colored Noise Problem and Probability Hole in the Bistable System

The Hamilton-Jacobi theory in nonpotential systems developed by Maier and Stein is applied to colored noise-driven systems. The behavior of the most probable escape path (MPEP) is discussed. Singularity occurs for the MPEP in the large correlation time τ limit, which is closely related to the well-known probability hole problem. The solution along the MPEP is derived. The behavior near the fixed point is analyzed, and a critical value τ_c, is found, which is in agreement with that obtained by other authors.