Exponential Leveling for Stochastically Perturbed Dynamical Systems
Exponential Leveling for Stochastically Perturbed Dynamical Systems
复制标题
随机扰动动力系统的指数整平
DOI：
10.1137/0513035
复制
发表时间：
1982
影响因子：
2
通讯作者：
M. Day
中科院分区：
文献类型：
--
作者：
M. Day
This paper considers solutions of $0 = \varepsilon \sum_{i,j} {a_{i,j}^\varepsilon } (x)u_{x_i x_j }^\varepsilon + \sum_i {b_i^\varepsilon } (x)u_i^\varepsilon $ in a bounded domain $\Omega $ for which $\sup _\Omega | {u^\varepsilon } |$ is bounded in $\varepsilon > 0$. We assume that $a^\varepsilon \to a^0 $, $b^\varepsilon \to b^0 $ and that all solutions of the ODE $\dot x = b^0 (x),x(0) \in \Omega $ converge to a single linearly asymptotically stable critical point in $\Omega $ without leaving $\Omega $. We give a proof, based on the standard probabilistic interpretation of $u^\varepsilon $, of an exponential leveling property: $\sup _{x,y \in K} | {u^\varepsilon (x) - u^\varepsilon (y)} | \leq e^{{ - \delta }/{\varepsilon }} $ for some $\delta > 0$ which depends on the compact set $K \subseteq \Omega $.
