In this paper we present a stochastic homogenization result for a class of Hilbert space evolutionary gradient systems driven by a quadratic dissipation potential and a $ \Lambda $-convex energy functional featuring random and rapidly oscillating coefficients. Specific examples included in the result are Allen-Cahn type equations and evolutionary equations driven by the $ p $-Laplace operator with $ p\in (1, \infty) $. The homogenization procedure we apply is based on a stochastic two-scale convergence approach. In particular, we define a stochastic unfolding operator which can be considered as a random counterpart of the well-established notion of periodic unfolding. The stochastic unfolding procedure grants a very convenient method for homogenization problems defined in terms of ($ \Lambda $-)convex functionals.

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随机均质化 \ begin {document} $ \ Lambda $ \ end {document}凸梯度流

在本文中，我们给出了一类希尔伯特空间演化梯度系统的随机均质化结果，该系统由二次耗散势和具有随机和快速振荡系数的λ凸能量函数驱动。结果中包括的特定示例是Allen-Cahn型方程和由$ p $ -Laplace运算符以$ p \ in（1，\ infty）$驱动的演化方程。我们应用的均质化程序基于随机两尺度收敛方法。特别是，我们定义了一个随机展开算子，可以将其视为已建立的周期性展开概念的随机对应物。随机展开过程为根据（$ \ Lambda $-）凸泛函定义的均质化问题提供了一种非常方便的方法。