On the effect of multiplicative noise in a supercritical pitchfork bifurcation

The most important characteristic of {\em multiplicative noise} is that its effects of system's dynamics depends on the recent system's state. Consideration of multiplicative noise on self-referential systems including biological and economical systems therefore is of importance. In this note we study an elementary example. While in a deterministic super critical pitchfork bifurcation with positive bifurcation parameter $\lambda$ the positive branch $\sqrt{\lambda}$ is stable, multiplicative white noise $\lambda_t ={\lambda} + \sigma \zeta_t$ on the unique parameter reduces stability in that the system's state tends to 0 almost surely, even for ${\lambda}>0$, while for 'small' noise $\sigma < \sqrt{2 \lambda}$ the point $\sqrt{\lambda-\sigma^2/2}$ is a meta-stable state. In this case, correspondingly, the system will 'die out', i.e. $X_t \to 0$ within finite time.