On the exit law from saddle points
The author considers the effects of adding an asymptotically small random Brownian perturbation to a planar dynamical system with a saddle point equilibrium. He obtains a new limit law for the exit time \(\tau^\varepsilon_D\) from a neighborhood of the saddle, assuming the initial point is on the stable manifold. More precisely, if the first-order system in the plane is \(\dot x_t = b(x_t)\), with \(b(0) = 0\) and \(\Lambda = {\partial b\over \partial x} (0)\) has eigenvalues \(\lambda_2 < 0 < \lambda_1\), and the perturbated system is \[ dx^\varepsilon_t = b(x^\varepsilon_t) dt + \varepsilon^{1/2} \sigma(x^\varepsilon_t) dW_t,\qquad \sigma \text{ non singular,} \] then \(\lambda_1 \tau^\varepsilon_D - \text{Log} (\varepsilon^{-1/2})\) converges weakly to the law of the sum of two random independent variables \(H\) and \(C_v\); the law of \(H\) has a density \(k(t) = {2\over \sqrt{\pi}} e^{-(t + e^{-2t})}\), \(C_{\pm1}\) are a pair of constants and the law of \(C_v\) is \(P(C_v = C_1) = P(C_v = C_{-1}) = 1/2\). Finally the author notices that this result can be generalized to higher dimensions, but only if the unstable manifold has dimension 1.
dynamical system with a saddle point equilibrium
scientific article; zbMATH DE number 859318