Reaction–Diffusion models: From particle systems to SDE’s

Let V be any finite set and p(⋅,⋅) a transition kernel on it. We present a construction of a family of Reaction–Diffusion models that converge after scaling to the solution to the |V|-dimensional SDE: dζt=[Δpζt−β⋅ζtk]dt+α⋅ζtℓdBtwith arbitrary initial condition ζ0∈RV. Here, Δp is the diffusion on V corresponding to p, α, β are positive real numbers and k, ℓ are positive integers.