Asymptotic behavior of the first exit time of randomly perturbed dynamical systems with a repulsive equilibrium point
Let \(X^\varepsilon (t,x)\) \((t\geq 0,x\in R^d, \varepsilon>0)\) be the solution of the following stochastic differential equation: \[ dX^\varepsilon (t,x)=b \bigl(X^\varepsilon (t,x)\bigr)dt + \varepsilon^{1/2} \sigma \bigl(X^\varepsilon (t,x)\bigr) dW(t), \quad X^\varepsilon (0,x)=x, \] where \(b(\cdot) =(b^i(\cdot))^d_{i=1}: R^d \mapsto R^d\) is Lipschitz continuous, where \(\sigma (\cdot)= (\sigma^{ij} (\cdot))^d_{i,j=1}: R^d \mapsto M_d(R)\) is bounded, Lipschitz continuous, and uniformly nondegenerate, and where \(W(\cdot)\) is a \(d\)-dimensional Wiener process. Let \(D\) \((\subset R^d)\) be a bounded domain which contains the origin \(o\), with \(C^2\)-boundary \(\partial D\). Suppose that \(b(x)=o\) if and only if \(x=o\), and that there exists a positive constant \(l\) such that \(| b(x) |\) is of order \(| x|^{l+1}\) as \(x\to o\), and that \(\langle x,b(x) \rangle\) is positive in some neighborhood of \(o\), and that for \(x\in\overline D\setminus \{o\}\), there exists \(s= s(x)\geq 0\) such that \(X^0(t,x) \not\in \overline D\) for \(t>s\) and such that \(X^0 (t,x) \in D\) for \(t<s\) and \(X^0(t,x) \mapsto o\) as \(t\to -\infty\). The asymptotic behavior of the first exit time \(\tau_D^\varepsilon (o)\) of \(X^\varepsilon (t,o)\) from \(D\), as \(\varepsilon\to 0\), is studied, \(\tau^\varepsilon_D (o)\) is of order \(\varepsilon^{-l/(l+2)}\), as \(\varepsilon\to 0\).
scientific article; zbMATH DE number 1185917