Stochastic two-scale convergence and Young measures

Martin Heida; Stefan Neukamm; Mario Varga; Martin Heida; Stefan Neukamm; Mario Varga

doi:10.3934/nhm.2022004

Networks and Heterogeneous Media

2022, Volume 17, Issue 2: 227-254. doi: 10.3934/nhm.2022004

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Stochastic two-scale convergence and Young measures

1.
Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany
2.
Zentrum Mathematik, Technische Universität München, Boltzmannstr. 3, 85747 Garching bei München, Germany
3.
Fakultät Mathematik, Technische Universität Dresden, 01062 Dresden, Germany

Received: 01 June 2021 Revised: 01 January 2022 Published: 22 February 2022
Primary: 74Q05, 47J30; Secondary: 49J45

In this paper we compare the notion of stochastic two-scale convergence in the mean (by Bourgeat, Mikelić and Wright), the notion of stochastic unfolding (recently introduced by the authors), and the quenched notion of stochastic two-scale convergence (by Zhikov and Pyatnitskii). In particular, we introduce stochastic two-scale Young measures as a tool to compare mean and quenched limits. Moreover, we discuss two examples, which can be naturally analyzed via stochastic unfolding, but which cannot be treated via quenched stochastic two-scale convergence.
- stochastic homogenization,
- unfolding,
- two-scale convergence,
- Young measures,
- weak convergence
Citation: Martin Heida, Stefan Neukamm, Mario Varga. Stochastic two-scale convergence and Young measures[J]. Networks and Heterogeneous Media, 2022, 17(2): 227-254. doi: 10.3934/nhm.2022004

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Abstract

In this paper we compare the notion of stochastic two-scale convergence in the mean (by Bourgeat, Mikelić and Wright), the notion of stochastic unfolding (recently introduced by the authors), and the quenched notion of stochastic two-scale convergence (by Zhikov and Pyatnitskii). In particular, we introduce stochastic two-scale Young measures as a tool to compare mean and quenched limits. Moreover, we discuss two examples, which can be naturally analyzed via stochastic unfolding, but which cannot be treated via quenched stochastic two-scale convergence.

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