Invariance Principles for Hyperbolic Random Walk Systems

Reaction random walk systems are hyperbolic models for description of spatial motion (in one dimension) and reaction of particles. In contrast to reaction diiusion equations particles have nite propagation speed. For parabolic systems invariance results and maximum principles are well known. A convex set is positively invariant if at each boundary point an outer normal is a left eigenvector of the diiusion matrix, and if the vector eld deened by the pure reaction equation \points inward" at the boundary. Here we show a corresponding result for random walk systems. The model parameters are the particle speeds, the rates of change in direction and the reaction vector eld. A convex domain is invariant if at each boundary point an outer normal is a left eigenvector of the \speed matrix" and if a vector eld given by the reaction equation combined with the turning rates points inward. Finally a positivity result is shown.