Diffusion-limited annihilating systems and the increasing convex order

Abstract

We consider diffusion-limited annihilating systems with mobile $A$-particles and stationary $B$-particles placed throughout a graph. Mutual annihilation occurs whenever an $A$-particle meets a $B$-particle. Such systems, when ran in discrete time, are also referred to as parking processes. We show for a broad family of graphs and random walk kernels that augmenting either the size or variability of the initial placements of particles increases the total occupation time by $A$-particles of a given subset of the graph. A corollary is that the same phenomenon occurs with the total lifespan of all particles in internal diffusion-limited aggregation.