Positive speed of propagation in a semilinear parabolic interface model with unbounded random coefficients

TL;DR

This paper proves that in a semilinear parabolic interface model with unbounded random obstacles, the interface propagates at a finite speed when the driving force is sufficiently large, using discretization and supermartingale techniques.

Contribution

It establishes finite velocity propagation in a quenched Edwards-Wilkinson model with unbounded random obstacle strengths, extending previous results to more general randomness.

Findings

Interface propagates with finite velocity for large driving force.

Propagation speed is positive under certain conditions.

Method involves discretization and supermartingale estimates.

Abstract

We consider a model for the propagation of a driven interface through a random field of obstacles. The evolution equation, commonly referred to as the Quenched Edwards-Wilkinson model, is a semilinear parabolic equation with a constant driving term and random nonlinearity to model the influence of the obstacle field. For the case of isolated obstacles centered on lattice points and admitting a random strength with exponential tails, we show that the interface propagates with a finite velocity for sufficiently large driving force. The proof consists of a discretization of the evolution equation and a supermartingale estimate akin to the study of branching random walks.