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 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.
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