Invariance Principles for Hyperbolic Random Walk Systems

Reaction random walk systems are hyperbolic models for the description of spatial motion in one dimension. and reaction of particles. In contrast to reaction diffusion equations, particles have finite 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 diffusion matrix, and if the vector field defined 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 field. 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 field given by the reaction equation combined with the turning rates points inward. Finally a positivity result is shown.