Small Fluctuations in Systems with Multiple Steady States

We consider the effect of small random perturbations on deterministic systems with multiple steady states. The deterministic systems of interest may have two stable nodes and a saddle point. As parameters vary, two or three of the steady states coalesce. This work is concerned with the long time behavior of the system, when it starts near the deterministic separatrix. The separatrix is surrounded by a tube that may contain two stable steady states. The quantity of basic interest is the conditional probability of first exit from the tube through a specified boundary, conditioned on initial position. We use the diffusion approximation, so that the conditional probability satisfies the Kolmogorov backward equation. Formal asymptotic solutions of the backward equation are constructed. The solutions are obtained by a generalized “ray method” and are given in terms of various incomplete special functions. In the appendices, we construct asymptotic solutions of the equation satisfied by the mean value of the fir...