A noise-controlled dynamic bifurcation

We consider a slow passage through a point of loss of stability. If the passage is sufficiently slow, the dynamics are controlled by additive random disturbances, even if they are extremely small. We derive expressions for the `exit value' distribution when the parameter is explicitly a function of time and the dynamics are controlled by additive Gaussian noise. We derive a new expression for the small correction introduced if the noise is coloured (exponentially correlated). There is good agreement with results obtained from simulation of sample paths of the appropriate stochastic differential equations. Multiplicative noise does not produce noise-controlled dynamics in this fashion.