Wenjia JingYau Mathematical Sciences Center, Tsinghua University, No 1. Tsinghua Yuan, Beijing 100084, ChinaPanagiotis E. SouganidisDepartment of Mathematics, The University of Chicago, 5734 S. University Ave., Chicago, IL 60637, USAHung V. TranDepartment of Mathematics, University of Wisconsin at Madison, 480 Lincoln Drive, Madison, WI 53706, USA
Analysis of PDEsOptimization and ControlProbabilitymathscidoc:2206.03012
Discrete and Continuous Dynamical Systems - S, 11, (5), 915-939, 2018.10
We study the averaging of fronts moving with positive oscillatory normal velocity, which is periodic in space and stationary ergodic in time. The problem can be formulated as the homogenization of coercive level set Hamilton-Jacobi equations with spatio-temporal oscillations. To overcome the difficulties due to the oscillations in time and the linear growth of the Hamiltonian, we first study the long time averaged behavior of the associated reachable sets using geometric arguments. The results are new for higher than one dimensions even in the space-time periodic setting.
Guillaume BalDepartment of Applied Physics and Applied Mathematics, Columbia University, 10027 New York, USA.Wenjia JingD´epartement de Math´ematiques et Applications, Ecole Normale Sup´erieure, 45 Rue d’Ulm, 75230 Paris Cedex 05, France
Analysis of PDEsNumerical Analysis and Scientific ComputingProbabilitymathscidoc:2206.03008
This paper analyzes the random fluctuations obtained by a heterogeneous multi-scale first-order finite element method applied to solve elliptic equations with a random potential. Several multiscale numerical algorithms have been shown to correctly capture the homogenized limit of solutions of elliptic equations with coefficients modeled as stationary and ergodic random fields. Because theoretical results are available in the continuum setting for such equations, we consider here the case of a secondorder elliptic equations with random potential in two dimensions of space. We show that the random fluctuations of such solutions are correctly estimated by the heterogeneous multi-scale algorithm when appropriate fine-scale problems are solved on subsets that cover the whole computational domain. However, when the fine-scale problems are solved over patches that do not cover the entire domain, the random fluctuations may or may not be estimated accurately. In the case of random potentials with short-range interactions, the variance of the random fluctuations is amplified as the inverse of the fraction of the medium covered by the patches. In the case of random potentials with long-range interactions, however, such an amplification does not occur and random fluctuations are correctly captured independent of the (macroscopic) size of the patches. These results are consistent with those obtained in  for more general equations in the one-dimensional setting and provide indications on the loss in accuracy that results from using coarser, and hence computationally less intensive, algorithms.
The classical de Finetti theorem in probability theory relates symmetry under the permutation group with the independence of random variables. This result has application in quantum information. Here we study states that are invariant with respect to a natural action of the braid group, and we emphasize the pictorial formulation and interpretation of our results. We prove a new type of de Finetti theorem for the four-string, double-braid group acting on the parafermion algebra to braid qudits, a natural symmetry in the quon language for quantum information. We prove that a braid-invariant state is extremal if and only if it is a product state. Furthermore, we provide an explicit characterization of braid-invariant states on the parafermion algebra, including finding a distinction that depends on whether the order of the parafermion algebra is square free. We characterize the extremal nature of product states (an inverse de Finetti theorem).
We consider Pólya urns with infinitely many colours that are of a random walk type, in two related versions. We show that the colour distribution a.s., after rescaling, converges to a normal distribution, assuming only second moments on the offset distribution. This improves results by Bandyopadhyay and Thacker (2014–2017; convergence in probability), and Mailler and Marckert (2017; a.s. convergence assuming exponential moment).