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Volume 18,     Number 1,     Spring 2010

 

EXISTENCE THEORY FOR CORRELATED
RANDOM WALKS ON BOUNDED DOMAINS
T. HILLEN

Abstract. In this paper we present a comprehensive existence theory for linear and nonlinear reaction random walk systems. The methods are based on semigroup theory for solutions of differential equations on Banach spaces. The solution properties on a bounded domain depend on the choice of boundary conditions. For Neumann or for periodic boundary conditions, singularities are transported along characteristics and the solutions form a group. Surprisingly, for Dirichlet boundary conditions, singularities are washed out, the problem regularizes in finite time, and the solution operator forms a semigroup.
    Furthermore, we study the relation to damped wave equations and reaction-telegraph equations. The relation between random walk models and telegraph equations for Neumann and periodic boundary conditions requires a compatibility condition of the initial condition. For Dirichlet boundary conditions, however, there is no direct relation between the random walk model and the telegraph equation.

 

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