Open Access
October 2017 Ballistic and sub-ballistic motion of interfaces in a field of random obstacles
Patrick W. Dondl, Michael Scheutzow
Ann. Appl. Probab. 27(5): 3189-3200 (October 2017). DOI: 10.1214/17-AAP1279


We consider a discretized version of the quenched Edwards–Wilkinson model for the propagation of a driven interface through a random field of obstacles. Our model consists of a system of ordinary differential equations on a $d$-dimensional lattice coupled by the discrete Laplacian. At each lattice point, the system is subject to a constant driving force and a random obstacle force impeding free propagation. The obstacle force depends on the current state of the solution, and thus renders the problem nonlinear. For independent and identically distributed obstacle strengths with an exponential moment, we prove ballistic propagation (i.e., propagation with a positive velocity) of the interface if the driving force is large enough. For a specific case of dependent obstacles, we show that no stationary solution exists, but still the propagation of the front is not ballistic.


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Patrick W. Dondl. Michael Scheutzow. "Ballistic and sub-ballistic motion of interfaces in a field of random obstacles." Ann. Appl. Probab. 27 (5) 3189 - 3200, October 2017.


Received: 1 July 2016; Published: October 2017
First available in Project Euclid: 3 November 2017

zbMATH: 06822215
MathSciNet: MR3719956
Digital Object Identifier: 10.1214/17-AAP1279

Primary: 34F05 , 60H10

Keywords: asymptotic behavior of nonnegative solutions , heterogeneous media , Interfaces , Random media

Rights: Copyright © 2017 Institute of Mathematical Statistics

Vol.27 • No. 5 • October 2017
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