A Dean-Kawasaki equation for reaction diffusion systems driven by Poisson noise

We derive a stochastic partial differential equation that describes the fluctuating behaviour of reaction-diffusion systems of N particles, undergoing Markovian, unary reactions. This generalises the work of Dean [J. Phys. A: Math. and Gen., 29 (24), L613, (1996)] through the inclusion of random Poisson fields. Our approach is based on weak interactions, which has the dual benefit that the resulting equations asymptotically converge (in the N to infinity limit) on a variation of a McKean- Vlasov diffusion, whilst still being related to the case of Dean-like strong interactions via a trivial rescaling. Various examples are presented, alongside a discussion of possible extensions to more complicated reaction schemes.