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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 $λ$ the positive branch $\sqrtλ$ is stable, multiplicative white noise $λ_t =λ + σζ_t$ on the unique parameter reduces stability in that the system's state tends to 0 almost surely, even for $λ>0$, while for 'small' noise $σ< \sqrt{2 λ}$ the point $\sqrt{λ-σ^2/2}$ is a meta-stable state. In this case, correspondingly, the system will 'die out', i.e. $X_t \to 0$ within finite time.