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Probabilistic Basin of Attraction and Its Estimation Using Two Lyapunov Functions
We study stability for dynamical systems specifed by autonomous stochastic diferential equations of the form dX(t) = f(X(t))dt +
g(X(t))dW(t), with (X(t))tâ„0 an Rd
-valued Ito process and Ë (W(t))tâ„0 an RQ-valued Wiener process, and the functions f : Rd â Rd
and g : Rd â RdĂQ are Lipschitz and vanish at the origin, making it an equilibrium for the system. Te concept of asymptotic
stability in probability of the null solution is well known and implies that solutions started arbitrarily close to the origin remain
close and converge to it. Te concept therefore pertains exclusively to system properties local to the origin. We wish to address
the matter in a more practical manner: Allowing for a (small) probability that solutions escape from the origin, how far away can
they then be started? To this end we defne a probabilistic version of the basin of attraction, the y-BOA, with the property that any
solution started within it stays close and converges to the origin with probability at least y. We then develop a method using a local
Lyapunov function and a nonlocal one to obtain rigid lower bounds on y-BOA.This work was supported by The Icelandic Research Fund,
Grant no. 152429-051.Peer Reviewe