 2015Patrik L. Ferrari, Herbert Spohn and Thomas Weiss
Scaling Limit for Brownian Motions with Onesided Collisions
Ann. Appl. Probab., 25(3): 13491382 2015
http://dx.doi.org/10.1214/14AAP1025
Abstract: We consider Brownian motions with onesided collisions, meaning that each particle is reflected at its right neighbour. For a finite number of particles a Schütztype formula is derived for the transition probability. We investigate an infinite system with periodic initial configuration, i.e., particles are located at the integer lattice at time zero. The joint distribution of the positions of a finite subset of particles is expressed as a Fredholm determinant with a kernel defining a signed determinantal point process. In the appropriate large time scaling limit, the fluctuations in the particle positions are described by the Airy\(_1\) process. 
 
 Patrik L. Ferrari and Balint Vető
TracyWidom asymptotics for qTASEP
Ann. Inst. H. Poincaré Probab. Statist., 51(4): 14651485 2015
http://dx.doi.org/10.1214/14AIHP614
Abstract: We consider the qTASEP, that is a qdeformation of the totally asymmetric simple exclusion process (TASEP) on \(\mathbb{Z}\) for \(q \in [0,1)\) where the jump rates depend on the gap to the next particle. For step initial condition, we prove that the current fluctuation of \(q\)TASEP at time \(\tau\) are of order \(\tau^{1/3}\) and asymptotically distributed as the GUE TracyWidom distribution. 
 
 Patrik L. Ferrari, Herbert Spohn and Thomas Weiss
Brownian motions with onesided collisions: the stationary case
Electronic Journal of Probability, 20(Art. 69): 141 2015
http://dx.doi.org/10.1214/EJP.v204177
Abstract: We consider an infinite system of Brownian motions which interact through a given Brownian motion being reflected from its left neighbor. Earlier we studied this system for deterministic periodic initial configurations. In this contribution we consider initial configurations distributed according to a Poisson point process with constant intensity, which makes the process spacetime stationary. We prove convergence to the Airy process for stationary the case. As a byproduct we obtain a novel representation of the finitedimensional distributions of this process. Our method differs from the one used for the TASEP and the KPZ equation by removing the initial step only after the large time limit. This leads to a new universal crossover process. 
 
 Patrik L. Ferrari and Peter Nejjar
Shock fluctuations in flat TASEP under critical scaling
J. Stat. Phys., 160(4): 9851004 2015
http://arxiv.org/abs/1408.4850
Abstract: We consider TASEP with two types of particles starting at every second site. Particles to the left of the origin have jump rate $1$, while particles to the right have jump rate $\alpha$. When $\alpha<1$ there is a formation of a shock where the density jumps to $(1\alpha)/2$. For $\alpha<1$ fixed, the statistics of the associated height functions around the shock is asymptotically (as time $t\to\infty$) a maximum of two independent random variables as shown in [arXiv:1306.3336]. In this paper we consider the critical scaling when $1\alpha=a t^{1/3}$, where $t\gg 1$ is the observation time. In that case the decoupling does not occur anymore. We determine the limiting distributions of the shock and numerically study its convergence as a function of $a$. We see that the convergence to product $F_{\rm GOE}^2$ occurs quite rapidly as $a$ increases. The critical scaling is analogue to the one used in the last passage percolation to obtain the BBP transition processes. 
 
 Benedict Geihe and Martin Rumpf
A posteriori error estimates for sequential laminates in shape optimization
In DCDSS Special issue on HomogenizationBased Numerical Methods 2015
http://arxiv.org/abs/1501.07461
Abstract: A posteriori error estimates are derived in the context of twodimensional structural elastic shape optimization under the compliance objective. It is known that the optimal shape features are microstructures that can be constructed using sequential lamination. The descriptive parameters explicitly depend on the stress. To derive error estimates the dual weighted residual approach for control problems in PDE constrained optimization is employed, involving the elastic solution and the microstructure parameters. Rigorous estimation of interpolation errors ensures robustness of the estimates while local approximations are used to obtain fully practical error indicators. Numerical results show sharply resolved interfaces between regions of full and intermediate material density. 
 
 Nicola Gigli, Tapio Rajala and KarlTheodor Sturm
Optimal maps and exponentiation on finite dimensional spaces with Ricci curvature bounded from below
J. Geom. Anal. 2015
http://arxiv.org/abs/1305.4849
Abstract: We prove existence and uniqueness of optimal maps on RCD∗(K,N) spaces under the assumption that the starting measure is absolutely continuous. We also discuss how this result naturally leads to the notion of exponentiation and to the localtoglobal property of RCD∗(K,N) bounds. 
 
 Michael Griebel, Christian Rieger and Barbara Zwicknagl
Multiscale approximation and reproducing kernel Hilbert space methods
SIAM Journal on Numerical Analysis, 53(2): 852873 2015
http://dx.doi.org/10.1137/130932144

 
 Michael Griebel, Alexander Hullmann and Oeter Oswald
Optimal scaling parameters for sparse grid discretizations
Numerical Linear Algebra with Applications, 22(1): 76100 2015
http://dx.doi.org/10.1002/nla.1939
Abstract: We apply iterative subspace correction methods to elliptic PDE problems discretized by generalized sparse grid systems. The involved subspace solvers are based on the combination of all anisotropic full grid spaces that are contained in the sparse grid space. Their relative scaling is at our disposal and has significant influence on the performance of the iterative solver. In this paper, we follow three approaches to obtain closetooptimal or even optimal scaling parameters of the subspace solvers and thus of the overall subspace correction method. We employ a Linear Program that we derive from the theory of additive subspace splittings, an algebraic transformation that produces partially negative scaling parameters which result in improved asymptotic convergence properties, and finally we use the OptiCom method as a variable nonlinear preconditioner. 
 
 Lisa B. Hartung and Anton Klimovsky
The glassy phase of the complex branching Brownian motion energy model
Electron. Commun. Probab., 20(Art. 78): 115 2015
http://dx.doi.org/10.1214/ECP.v204360

 
 Stefanie Heyden, Bo Li, Kerstin Weinberg, Sergio Conti and Michael Ortiz
A micromechanical damage and fracture model for polymers based on fractional straingradient elasticity
J. Mech. Phys. Solids, 74: 175195 2015
http://dx.doi.org/10.1016/j.jmps.2014.08.005

 
 Stefanie Heyden, Sergio Conti and Michael Ortiz
A nonlocal model of fracture by crazing in polymers
Mech. Materials, 90: 131139 2015
http://dx.doi.org/10.1016/j.mechmat.2015.02.006
Abstract: We derive and numerically verify scaling laws for the macroscopic fracture energy of poly mers undergoing crazing from a micromechanical model of damage. The model posits a local energy density that generalizes the classical network theory of polymers so as to account for chain failure and a nonlocal regularization based on straingradient elasticity. We specifically consider periodic deformations of a slab subject to prescribed opening dis placements on its surfaces. Based on the growth properties of the energy densities, scaling relations for the local and nonlocal energies and for the specific fracture energy are derived. We present finiteelement calculations that bear out the heuristic scaling relations. 
 
 Aicke Hinrichs, Lev Markhasin, Jens Oettershagen and Tino Ullrich
Optimal quasiMonte Carlo rules on higher order digital nets for the numerical integration of multivariate periodic functions
2015
http://arxiv.org/pdf/1501.01800v1

 
 Martin Huesmann
Transport estimates for random measures in dimension one
ArXiv eprint 2015
http://arxiv.org/abs/1510.03601

 
 Juhi Jang, Juan J. L. Velázquez and Hyung Ju Hwang
On the structure of the singular set for the kinetic FokkerPlanck equations in domains with boundaries
2015
http://arxiv.org/abs/1509.03366

 
 Christian Ketterer
Evolution variational inequality and Wasserstein control in variable curvature context
ArXiv eprints 2015
http://arxiv.org/abs/1509.02178

 
 Christian Ketterer
On the geometry of metric measure spaces with variable curvature bounds
ArXiv eprints 2015
http://arxiv.org/abs/1506.03279

 
 Christian Ketterer
Cones over metric measure spaces and the maximal diameter theorem
J. Math. Pures Appl. (9), 103(5): 12281275 2015
http://dx.doi.org/10.1016/j.matpur.2014.10.011

 
 Christian Ketterer
Obata's Rigidity Theorem for Metric Measure Spaces
Anal. Geom. Metr. Spaces, 3(Art. 16): 278295 2015
http://dx.doi.org/10.1515/agms20150016

 
 Arthur H. M. Kierkels and Juan J. L. Velázquez
On the transfer of energy towards infinity in the theory of weak turbulence for the nonlinear Schrödinger equation
J. Stat. Phys., 159(3): 668712 2015
http://dx.doi.org/10.1007/s1095501511940
Abstract: We study the mathematical properties of a kinetic equation which describes the long time behaviour of solutions to the weak turbulence equation associated to the cubic nonlinear Schrödinger equation.In particular, we give a precise definition of weak solutions and prove global existence of solutions for all initial data with finite mass. We also prove that any nontrivial initial datum yields the instantaneous onset of a condensate, i.e.~a Dirac mass at the origin for any positive time. Furthermore we show that the only stationary solutions with finite total measure are Dirac masses at the origin. We finally construct solutions with finite energy, which is transferred to infinity in a selfsimilar manner. 
 
 Herbert Koch and Nikolai Nadirashvili
Partial analyticity and nodal sets for nonlinear elliptic systems
2015
http://arxiv.org/abs/1506.06224

 