Stochastic homogenization of Λ\Lambda-convex gradient flows

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.Comment: arXiv admin note: text overlap with arXiv:1805.0954

Stochastic two-scale convergence and Young measures

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 unfolding and homogenization

The notion of periodic two-scale convergence and the method of periodic unfolding are prominent and useful tools in multiscale modeling and analysis of PDEs with rapidly oscillating periodic coefficients. In this paper we are interested in the theory of stochastic homogenization for continuum mechanical models in form of PDEs with random coefficients, describing random heterogeneous materials. The notion of periodic two-scale convergence has been extended in different ways to the stochastic case. In this work we introduce a stochastic unfolding method that features many similarities to periodic unfolding. In particular it allows to characterize the notion of stochastic two-scale convergence in the mean by mere weak convergence in an extended space. We illustrate the method on the (classical) example of stochastic homogenization of convex integral functionals, and prove a new result on stochastic homogenization for a non-convex evolution equation of Allen-Cahn type. Moreover, we discuss the relation of stochastic unfolding to previously introduced notions of (quenched and mean) stochastic two-scale convergence. The method described in the present paper extends to the continuum setting the notion of discrete stochastic unfolding, as recently introduced by the second and third author in the context of discrete-to-continuum transition.Comment: 46 page

Stochastic homogenization of Lambda-convex gradient flows

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 Λ-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, ∞). 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 (Λ-)convex functionals

Stochastic unfolding and homogenization

The notion of periodic two-scale convergence and the method of periodic unfolding are prominent and useful tools in multiscale modeling and analysis of PDEs with rapidly oscillating periodic coefficients. In this paper we are interested in the theory of stochastic homogenization for continuum mechanical models in form of PDEs with random coefficients, describing random heterogeneous materials. The notion of periodic two-scale convergence has been extended in different ways to the stochastic case. In this work we introduce a stochastic unfolding method that features many similarities to periodic unfolding. In particular it allows to characterize the notion of stochastic two-scale convergence in the mean by mere convergence in an extended space. We illustrate the method on the (classical) example of stochastic homogenization of convex integral functionals, and prove a stochastic homogenization result for an non-convex evolution equation of Allen-Cahn type. Moreover, we discuss the relation of stochastic unfolding to previously introduced notions of (quenched and mean) stochastic two-scale convergence. The method introduced in this paper extends extitdiscrete stochastic unfolding, as recently introduced by the second and third author in the context of discrete-to-continuum transition

Ancient Bacterial Genomes Reveal a High Diversity of Treponema pallidum Strains in Early Modern Europe

Syphilis is a globally re-emerging disease, which has marked European history with a devastating epidemic at the end of the 15th century. Together with non-venereal treponemal diseases, like bejel and yaws, which are found today in subtropical and tropical regions, it currently poses a substantial health threat worldwide. The origins and spread of treponemal diseases remain unresolved, including syphilis' potential introduction into Europe from the Americas. Here, we present the first genetic data from archaeological human remains reflecting a high diversity of Treponema pallidumin early modern Europe. Our study demonstrates that a variety of strains related to both venereal syphilis and yaws-causing T. pallidum subspecies were already present in Northern Europe in the early modern period. We also discovered a previously unknown T. pallidum lineage recovered as a sister group to yaws- and bejel-causing lineages. These findings imply a more complex pattern of geographical distribution and etiology of early treponemal epidemics than previously understood.Peer reviewe

Ancient Bacterial Genomes Reveal a High Diversity of Treponema pallidum Strains in Early Modern Europe

Syphilis is a globally re-emerging disease, which has marked European history with a devastating epidemic at the end of the 15th century. Together with non-venereal treponemal diseases, like bejel and yaws, which are found today in subtropical and tropical regions, it currently poses a substantial health threat worldwide. The origins and spread of treponemal diseases remain unresolved, including syphilis' potential introduction into Europe from the Americas. Here, we present the first genetic data from archaeological human remains reflecting a high diversity of Treponema pallidumin early modern Europe. Our study demonstrates that a variety of strains related to both venereal syphilis and yaws-causing T. pallidum subspecies were already present in Northern Europe in the early modern period. We also discovered a previously unknown T. pallidum lineage recovered as a sister group to yaws- and bejel-causing lineages. These findings imply a more complex pattern of geographical distribution and etiology of early treponemal epidemics than previously understood

Stochastic homogenization of Lambda-convex gradient flows

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 Λ-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, ∞). 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 (Λ-)convex functionals

Stochastic two-scale convergence and Young measures

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