Existence Theory for Correlated Random Walks on Bounded Domains
Thomas Hillen · 2009
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 sensitively 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 reactiontelegraph 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. 1