Stabilizing role of multiplicative noise in nonconfining potentials

We provide a simple framework for the study of parametric (multiplicative) noise, making use of scale parameters. We focus on a large class of one-dimensional stochastic differential equations in which the deterministic drift pushes trajectories toward infinity. We show that increasing the multiplicative noise intensity surprisingly causes the mass of the stationary probability distribution to become increasingly concentrated around the points of minimal multiplicative noise strength. Under quite general conditions the trajectory exhibits intermittent burstlike jumps away from these minima. Our framework relies on first-term expansions, which become more accurate for larger noise intensities. In this work we show that the full width at half maximum in addition to the maximum is appropriate for quantifying the stationary probability distribution (instead of the mean and variance, which are often undefined). We define a corresponding kind of weak-sense stationarity. We end by applying these results to the problem of a double-well potential with multiplicative noise, where noise stabilizes unstable fixed points.