We examine a class of stochastic mirror descent dynamics in the context of monotone variational inequalities (including Nash equilibrium and saddle-point problems). The dynamics under study are formulated as a stochastic differential equation, driven by a (single-valued) monotone operator and perturbed by a Brownian motion. The system’s controllable parameters are two variable weight sequences, that, respectively, pre- and post-multiply the driver of the process. By carefully tuning these parameters, we obtain global convergence in the ergodic sense, and we estimate the average rate of convergence of the process. We also establish a large deviations principle, showing that individual trajectories exhibit exponential concentration around this average.

中文翻译：

---

随机镜像下降动力学及其在单调变分不等式中的收敛性

我们在单调变分不等式（包括纳什均衡和鞍点问题）的背景下研究一类随机镜像下降动力学。所研究的动力学被表述为随机微分方程，由（单值）单调算子驱动并受到布朗运动的扰动。系统的可控参数是两个可变权重序列，分别前乘和后乘过程的驱动器。通过仔细调整这些参数，我们获得了遍历意义上的全局收敛，并估计了过程的平均收敛率。我们还建立了大偏差原理，表明各个轨迹在该平均值附近表现出指数集中。