The Annals of Applied Probability

Ballistic and sub-ballistic motion of interfaces in a field of random obstacles

Patrick W. Dondl and Michael Scheutzow

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Abstract

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.

Article information

Source
Ann. Appl. Probab., Volume 27, Number 5 (2017), 3189-3200.

Dates
Received: July 2016
First available in Project Euclid: 3 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1509696044

Digital Object Identifier
doi:10.1214/17-AAP1279

Mathematical Reviews number (MathSciNet)
MR3719956

Zentralblatt MATH identifier
06822215

Subjects
Primary: 34F05: Equations and systems with randomness [See also 34K50, 60H10, 93E03] 60H10: Stochastic ordinary differential equations [See also 34F05]

Keywords
Interfaces heterogeneous media random media asymptotic behavior of nonnegative solutions

Citation

Dondl, Patrick W.; Scheutzow, Michael. Ballistic and sub-ballistic motion of interfaces in a field of random obstacles. Ann. Appl. Probab. 27 (2017), no. 5, 3189--3200. doi:10.1214/17-AAP1279. https://projecteuclid.org/euclid.aoap/1509696044


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References

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