Asymptotic series and exit time probabilities
The authors consider a bounded domain \(D\subset R^ n\), a stochastic equation \[ dx^ \varepsilon_ t=b^ \varepsilon(x_ t,t)dt+\varepsilon^{1/2}\sigma (x^ \varepsilon_ t)dw_ t, \] \(x_ s^ \varepsilon=x\), \(x\in D\) etc., for \(\varepsilon\to 0\), \(\tau^ \varepsilon=\inf\{t;x_ t\notin D\}\) and \(q^ \varepsilon(x,s)=P^ \varepsilon_{x,s}(\tau^ \varepsilon\leq T)\), where \(T\) is fixed and \(P^ \varepsilon_{x,s}\) is the distribution of the solution. They suppose that \(a(x)=\sigma(x)\sigma'(x)\geq\alpha_ 0\cdot 1\) for all \(x,\alpha_ 0>0\), that \(\sigma\) is \(C^ \infty\) and bounded, that \(b^ \varepsilon\) is continuous in \((s,x)\), Lipschitz in \(x\) uniformly in \(\varepsilon,s\), that \(b\cdot\nu<0\) where \(\nu\) is the exterior normal on \(\partial D\). Near 0, \(b^ \varepsilon\) is expressed as \(b+\varepsilon^{1/2}b_ 1+\cdots+\varepsilon^{m/2}b_ m+o(\varepsilon^{m/2})\) uniformly, with \(b_ i\) in \(C^ \infty\). The result is \[ q^ \varepsilon(x,s)=\exp(-u(x,s)\varepsilon^{-1}- v(x,s)\varepsilon^{-1/2}-w(x, s))(1+\varepsilon^{1/2}\varphi_ 1(x,s)+\cdots+\varepsilon^{m/2}\varphi_ m(x,s)+o( \varepsilon^{m/2})),\tag{1} \] where \(u\) appears from the application of a large deviations result in [\textit{M. I. Frejdlin} and \textit{A. D. Venttsel'}, Random perturbations of dynamical systems (1984; Zbl 0522.60055)], while \(v,w,\varphi_ i\) are solutions of partial derivative equations in which the \(s\)-derivatives appear only as \((\partial/\partial s)+\cdots\). The proof uses the fact that \(q^ \varepsilon\) satisfies the equation \(\partial q/\partial s+(\varepsilon/2)tr(a(x)D^ 2q)+b^ \varepsilon\cdot Dq=0\), \(q=1\) on \(\partial Dx[0,T)\), \(q(x,T)=0\) for \(x\in D\). The result is preceded by two sections, concerning solutions of such equations, on \(R^ n\) instead of \(D\). In the first one a result of the type \(q^ \varepsilon\to q\) is established by probabilistic methods, in the second one a result analogous to (1) but without \(\varepsilon\) under exp and for a \(\beta\) instead of \(b\). In the second section there is an extraterm \(h^ \varepsilon q\) in the equation and \(q=\exp(- \psi)\varphi^ \varepsilon\) on the boundary etc., \(\varphi^ \varepsilon\), \(h^ \varepsilon\) having expressions analogous to that of \(b\). Similar remarks concerning the first preliminary result. By the same methods, at the end of the paper, (1) is generalised to an equation on \(R^ n\) containing an extraterm \(h^ \varepsilon q/\varepsilon\), for \(q(x,T)=C_ \varepsilon\exp(-\psi^ \varepsilon/\varepsilon)\), \(\varepsilon\ln C_ \varepsilon\to 0\) etc. In fact (1) is established not on \(D\times[0,T]\), but on an open \(N\subset D\times[0,T']\), \(T'<T\), for which \(u\) is \(C^ \infty\) on \(\overline N\); \(N\) is supposed to be such that the solutions of \(\dot\xi_ t=\beta(\xi_ t,t)\), \(\xi_ s=x\), \(s<t<T\), \((x,s)\in N\), hit \(\partial N\) nontangentially when they reach it. The results of the two preliminary sections are valid on the union of \(N\) and the part of \(\partial N\) effectively hit etc. The paper starts with an introduction describing the history of the used techniques. It contains also an example to (1).
scientific article; zbMATH DE number 97398