Interacting diffusions in a random medium: comparison and longtime behavior

We consider a collection of linearly interacting diffusions (indexed by a countable space) in a random medium. The diffusion coefficients are the product of a space-time dependent random field (the random medium) and a function depending on the local state. The main focus of the present work is to establish a comparison technique for systems in the same medium but with different state dependence in the diffusion terms. The technique is applied to generalize statements on the longtime behavior, previously known only for special choices of the diffusion function. One of these special choices, which we employ as a reference model, is that of interacting Fisher-Wright diffusions in a catalytic medium where duality was used to obtain detailed results. The other choice is that of interacting Feller's branching diffusions in a catalytic medium which is itself an (autonomous) branching process and where infinite divisibility was used as the main tool.