Stochastic two-scale convergence and Young measures. (English) Zbl 1493.74101
The paper compares the notions of stochastic two-scale convergence in the mean introduced by A. Bourgeat et al. [J. Reine Angew. Math. 456, 19–51 (1994; Zbl 0808.60056)], the quenched notion of stochastic two-scale convergence introduced by V. V. Zhikov and A. L. Pyatniskii [Izv. Math. 70, No. 1, 19–67 (2006; Zbl 1113.35021); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 70, No. 1, 23–74 (2006)] and that of stochastic unfolding introduced by the authors [Discrete Contin. Dyn. Syst., Ser. S 14, No. 1, 427–453 (2021; Zbl 1458.49013)]. Stochastic two-scale Young measures are used to compare the mean and quenched limits. The authors discuss two examples that can be analyzed via stochastic unfolding but cannot be treated by using quanched stochastic two-scale convergence.
Reviewer: Alexandre Danescu (Lyon)
MSC:
| 74Q99 | Homogenization, determination of effective properties in solid mechanics |
| 74B99 | Elastic materials |
| 74E05 | Inhomogeneity in solid mechanics |
| 74S60 | Stochastic and other probabilistic methods applied to problems in solid mechanics |
Keywords:
stochastic homogenization; unfolding; quenched two-scale convergence; Young measure; weak convergence; variance regularizationReferences:
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