Noise-induced transitions for one-dimensional diffusions

Abstract

We consider a one-dimensional diffusion X on a finite or infinite interval]l,r[ satisfying a stochastic differential equation with W(t) t 0 standard Brownian motion. For fixed drift b we investigate noiseinduced transitions of the boundary behavior i.e. the dependence of the behavior of X near r when σ varies. Particularly we are interested in the question whether increasing σ (for all x) may cause or prevent explosion in finite time. A negative result is that if σ is bounded and bounded away from zero then multiplying σ by a factor greater than one can never prevent explosions (Theorem 2.1). Various examples show that under different assumptions increasing the noise can have a drastic stabilizing or destabilizing effect (depending on b and σ). In the last section we study the relation between the stochastic and deterministic (σ ≡ 0) case