耦合随机动力系统多尺度行为的渐近性
Asymptotic behaviour of coupled random dynamical systems with multiscale aspects
AI总结：
本文研究了多尺度耦合随机动力系统的渐近行为，通过惩罚方法和Legendre变换条件，分析了有限时间收敛速率及大偏差原理，建立了连续时间方法与前向-后向型算法的关系。
摘要：
我们研究了一类涉及多尺度效应的随机微分包含问题，旨在解决一类广义变分不等式问题。此类问题包含受约束的凸非光滑优化问题、受约束的鞍点问题以及经济学和工程中的各种均衡问题。为尊重约束，我们采用惩罚方法，在演化系统中引入显式的时间依赖性。所得到的动力学由一个非自治随机演算方程描述，该方程由最大单调算子在漂移项中支配，并受布朗运动扰动。我们研究了渐近行为，以及以间隙函数为基准的有限时间收敛速率。用于证明收敛的条件涉及描述集合C的函数的Legendre变换，这一条件最初由Attouch和Czarnecki（J. Differ. Equations，Vol. 248，Issue 6，2010）在确定性演化方程的背景下使用。我们还建立了大偏差原理，表明个体轨迹在解集周围表现出指数集中趋势。最后，我们展示了我们的连续时间方法如何通过适当的Euler-Maruyama离散化与惩罚调节的前向-后向型算法相关联。
英文摘要：
We examine a class of stochastic differential inclusions involving multiscale effects designed to solve a class of generalized variational inequalities. This class of problems contains constrained convex non-smooth optimization problems, constrained saddle-point problems and various equilibrium problems in economics and engineering. In order to respect constraints we adopt a penalty approach, introducing an explicit time-dependency into the evolution system. The resulting dynamics are described in terms of a non-autonomous stochastic evolution equation governed by maximally monotone operators in the drift and perturbed by a Brownian motion. We study the asymptotic behavior, as well as finite time convergence rates in terms of gap functions. The condition we use to prove convergence involves a Legendre transform of the function describing the set constrained domain, a condition first used by Attouch and Czarnecki (J. Differ. Equations, Vol. 248, Issue 6, 2010) in the context of deterministic evolution equations. We also establish a large deviations principle showing that individual trajectories exhibit exponential concentration around the solution set. Finally we show how our continuous-time approach relates to penalty-regulated algorithms of forward-backward type after performing a suitable Euler-Maruyama discretization.
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