This short survey article stems from recent progress on critical cases of stochastic evolution equations in variational formulation with additive, multiplicative or gradient noises. Typical examples appear as the limit cases of the stochastic porous medium equation, stochastic fast- and super fast-diffusion equations, self-organized criticality, stochastic singular $p$-Laplace equations, and the stochastic total variation flow, among others. We present several different notions of solutions, results on convergence of solutions depending on a parameter, and homogenization. Furthermore, we provide some references hinting at the recent progress in regularity results, long-time behavior, ergodicity, and numerical analysis.

翻译：本短篇综述文章源于近期在变分形式下具有加性、乘性或梯度噪声的随机演化方程临界情形研究的最新进展。典型示例包括随机多孔介质方程、随机快速与超快速扩散方程、自组织临界性、随机奇异$p$-Laplace方程以及随机全变分流等方程的极限情形。我们介绍了若干不同的解概念，给出了依赖于参数的解收敛性结果及均匀化理论。此外，本文还提供了关于正则性结果、长期行为、遍历性及数值分析等领域最新进展的参考文献指引。