Abstract :
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.
