Preprints - IAM Publications


	2014 Véronique Gayrard and Adéla Švejda Convergence of clock processes on infinite graphs and aging in Bouchaud's asymmetric trap model on $\mathbbZ^d$ Lat. Am. J. Probab. Math. Stat., 11(2): 78-822 2014 http://alea.impa.br/articles/v11/11-35.pdf Abstract: Using a method developed by Durrett and Resnick, [23], we establish general criteria for the convergence of properly rescaled clock processes of random dynamics in random environments on infinite graphs. This extends the results of Gayrard, [27], Bovier and Gayrard, [20], and Bovier, Gayrard, and Svejda, [21], and gives a unified framework for proving convergence of clock processes. As a first application we prove that Bouchaud's asymmetric trap model on $\mathbb{Z}^d$ exhibits a normal aging behavior for all $d \geq 2$. Namely, we show that certain two-time correlation functions, among which the classical probability to find the process at the same site at two time points, converge, as the age of the process diverges, to the distribution function of the arcsine law. As a byproduct we prove that the fractional kinetics process ages.

	Herbert Koch, Hart F. Smith and Daniel Tataru Sharp $L^p$ bounds on spectral clusters for Lipschitz metrics Amer. J. Math., 136(6): 1629-1663 2014 http://dx.doi.org/10.1353/ajm.2014.0039 Abstract: We establish Lp bounds on L2 normalized spectral clusters for self-adjoint elliptic Dirichlet forms with Lipschitz coefficients. In two dimensions we obtain best possible bounds for all 2 ≤ p ≤ ∞, up to logarithmic losses for 6 < p ≤ 8. In higher dimensions we obtain best possible bounds for a limited range of p.

	Stefan Müller, Lucia Scardia and Caterina Ida Zeppieri Geometric rigidity for incompatible fields and an application to strain-gradient plasticity Indiana Univ. Math. J., 63(5): 1365-1396 2014 http://dx.doi.org/10.1512/iumj.2014.63.5330

	Shin-ichi Ohta and Karl-Theodor Sturm Bochner-Weitzenböck formula and Li-Yau estimates on Finsler manifolds Adv. Math., 252: 429-448 2014 http://dx.doi.org/10.1016/j.aim.2013.10.018

	Tapio Rajala and Karl-Theodor Sturm Non-branching geodesics and optimal maps in strong CD (K,$\backslash$ infty)-spaces Calculus of Variations and Partial Differential Equations, 50(3-4): 831--846 2014 http://dx.doi.org/10.1007/s00526-013-0657-x

	Marc A. Schweitzer and Sa Wu Numerical Integration of on-the-fly-computed Enrichment Functions in the PUM In M. Griebel and M. A. Schweitzer, editor, Meshfree Methods for Partial Differential Equations VII, Volume 100 of Lecture Notes in Computational Science and Engineering Chapter 13, page 247-267. Publisher: Springer International 2014 http://dx.doi.org/10.1007/978-3-319-06898-5_13

	Marc A. Schweitzer and Albert Ziegenhagel Dispersion Properties of the Partition of Unity Method & Explicit Dynamics In M. Griebel and M. A. Schweitzer, editor, Meshfree Methods for Partial Differential Equations VII, Volume 100 of Lecture Notes in Computational Science and Engineering Chapter 14, page 269-292. Publisher: Springer International 2014 http://dx.doi.org/10.1007/978-3-319-06898-5_14

	Karl-Theodor Sturm Gradient Flows for Semiconvex Functions on Metric Measure Spaces - Existence, Uniqueness and Lipschitz Continuity ArXiv e-prints 2014 http://arxiv.org/abs/1410.3966

	Karl-Theodor Sturm Ricci Tensor for Diffusion Operators and Curvature-Dimension Inequalities under Conformal Transformations and Time Changes ArXiv e-prints 2014 http://arxiv.org/abs/1401.0687 Abstract: Within the Γ2-calculus of Bakry and Ledoux, we define the Ricci tensor induced by a diffusion operator, we deduce precise formulas for its behavior under drift transformation, time change and conformal transformation, and we derive new transformation results for the curvature-dimension conditions of Bakry-Emery as well as for those of Lott-Sturm-Villani. Our results are based on new identities and sharp estimates for the N-Ricci tensor and for the Hessian. In particular, we obtain Bochner's formula in the general setting.

	Karl-Theodor Sturm A Monotone Approximation to the Wasserstein Diffusion In M. Griebel, editor, Singular Phenomena and Scaling in Mathematical Models, Volume 1 Chapter 2, page 25-48. Publisher: Springer International 2014 http://dx.doi.org/10.1007/978-3-319-00786-1_2

	Yoshio Sugiyama, Yohei Tsutsui and Juan J. L. Velázquez Global solutions to a chemotaxis system with non-diffusive memory J. Math. Anal. Appl., 410(2): 908-917 2014 http://dx.doi.org/10.1016/j.jmaa.2013.08.065 Abstract: In this article, an existence theorem of global solutions with small initial data belonging to L1∩Lp, (n

	2013 Patrik L. Ferrari, Tomohiro Sasamoto and Herbert Spohn Coupled Kardar-Parisi-Zhang Equations in One Dimension J. Stat. Phys., 153(3): 377-399 2013 http://dx.doi.org/10.1007/s10955-013-0842-5 Abstract: Over the past years our understanding of the scaling properties of the solutions to the one-dimensional KPZ equation has advanced considerably, both theoretically and experimentally. In our contribution we export these insights to the case of coupled KPZ equations in one dimension. We establish equivalence with nonlinear fluctuating hydrodynamics for multi-component driven stochastic lattice gases. To check the predictions of the theory, we perform Monte Carlo simulations of the two-component AHR model. Its steady state is computed using the matrix product ansatz. Thereby all coefficients appearing in the coupled KPZ equations are deduced from the microscopic model. Time correlations in the steady state are simulated and we confirm not only the scaling exponent, but also the scaling function and the non-universal coefficients.

	Martin Huesmann and Karl-Theodor Sturm Optimal transport from Lebesgue to Poisson The Annals of Probability, 41(4): 2426-2478 2013 http://dx.doi.org/10.1214/12-AOP814 Abstract: This paper is devoted to the study of couplings of the Lebesgue measure and the Poisson point process. We prove existence and uniqueness of an optimal coupling whenever the asymptotic mean transportation cost is finite. Moreover, we give precise conditions for the latter which demonstrate a sharp threshold at d=2d=2. The cost will be defined in terms of an arbitrary increasing function of the distance. The coupling will be realized by means of a transport map (“allocation map”) which assigns to each Poisson point a set (“cell”) of Lebesgue measure 1. In the case of quadratic costs, all these cells will be convex polytopes.

	Kazumasa Kuwada and Karl-Theodor Sturm Monotonicity of time-dependent transportation costs and coupling by reflection Potential Analysis, 39(3): 231-263 2013 http://dx.doi.org/10.1007/s11118-012-9327-4 Abstract: Based on a study of the coupling by reflection of diffusion processes, a new monotonicity in time of a time-dependent transportation cost between heat distribution is shown under Bakry-Émery’s curvature-dimension condition on a Riemannian manifold. The cost function comes from the total variation between heat distributions on spaceforms. As a corollary, we obtain a comparison theorem for the total variation between heat distributions. In addition, we show that our monotonicity is stable under the Gromov-Hausdorff convergence of the underlying space under a uniform curvature-dimension and diameter bound.

	Marc A. Schweitzer Multilevel Partition of Unity Method for Elliptic Problems with Strongly Discontinuous Coefficients, Meshfree Methods for Partial Differential Equations VI In M. Griebel and M. A. Schweitzer, editor, Lecture Notes in Computational Science and Engineering, Volume 89, Meshfree Methods for Partial Differential Equations VI of Lecture Notes in Computational Science and Engineering Chapter 6, page 93-110. Publisher: Springer International 2013 http://dx.doi.org/10.1007/978-3-642-32979-1_6 Abstract: In this paper, we study the robustness of a multilevel partition of unity method. To this end, we consider a scalar diffusion equation in two and three space dimensions with large jumps in the diffusion coefficient or material properties. Our main focus in this investigation is if the use of simple enrichment functions is sufficient to attain a robust solver independent of the geometric complexity of the material interface.

	Marc A. Schweitzer Variational Mass Lumping in the Partition of Unity Method SIAM Journal on Scientific Computing, 35(2): A1073-A1097 2013 http://dx.doi.org/10.1137/120895561 Abstract: This paper is concerned with the construction of a variational mass lumping scheme for the partition of unity methods. The presented approach is applicable to arbitrary local approximation spaces and any nonnegative partition of unity. We give improved error bounds for the partition of unity method using a nonnegative partition of unity and show that our lumped mass matrix is conservative at least for any $f \in V^{\rm PU}$ such that $f\|_{\Omega\cap\omega_i} \in V_i(\omega_i)$ for all patches $\omega_i$. We present numerical results using smooth, higher order, discontinuous, and singular local approximation spaces confirming our theoretical results.

	2012 Sven Beuchler, Veronika Pillwein, Joachim Schöberl and Sabine Zaglmayr Sparsity optimized high order finite element functions on simplices In Numerical and symbolic scientific computing, Texts Monogr. Symbol. Comput. page 21--44. Publisher: SpringerWienNewYork, Vienna 2012 http://dx.doi.org/10.1007/978-3-7091-0794-2_2

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