A note on small random perturbations of dynamical systems
One considers a family of one-dimensional diffusion processes \(X_{\epsilon}(\cdot)\) satisfying: \[ dX_{\epsilon}(t)=b(X_{\epsilon}(t))dt+\epsilon^{1/2} dW_ t\quad (0<\epsilon <1), \] where \((W_ t)_ t\) is a standard Brownian motion on the line with \(W_ 0=0\). The main goal is the study of its escape from an unstable, completely degenerated equilibrium point of the underlying deterministic system \((\epsilon =0).\) Fixing a particular case where \(b=-U'\), with U being a double well potential with a ``completely degenerated'' local maximum at 0 (unique in some neighborhood) and two local minima \((\pm m^*)\), and with \(\mu_{\epsilon}(t)\) the law of \(X_{\epsilon}(t)\) when \(X_{\epsilon}(0)=0\), it is proven that for any \(t>0:\) \[ \mu_{\epsilon}(\epsilon^{-1}| \ell n \epsilon |^{-1} t)\to a(t)\delta_ 0+2^{-1}(1-a(t))(\delta_{- m^*}+\delta_{m^*}),\quad \epsilon \to 0, \] where a(t) is a continuous, decreasing function with \(a(0)=1\), \(a(+\infty)=0\). From the proof we understand that the picture holds in general for degenerated systems, but the scaling \(\epsilon^{-1}| \ell n \epsilon |^{- 1}\) depends on the shape of U ``at zero''. The motivation comes from the study of transient bimodality in interacting particle systems.
escape from an unstable, completely degenerated equilibrium point
scientific article; zbMATH DE number 4163842