High density limit theorems for nonlinear chemical reactions with diffusion (Q1093268)

The solution X of a nonlinear reaction-duffusion equation on the n- dimensional unit cube S is approximated by a space-time jump Markov process \(X_{v,N}\) (law of large numbers (LLN)). \(X_{v,N}\) is constructed on a grid \(S_ n\) on S of N cells, where v is proportional to the initial number of particles in each cell. The deviation of \(X_{v,N}\) from X is computed by a central limit theorem (CLT). The assumptions on the parameters v,N for the LLN are: \(v\to \infty\), as \(N\to \infty\), and for the CLT: N/v\(\to 0\), as \(N\to \infty\). The limit \(Y=Y^ X\) in the CLT, which is a generalized Ornstein-Uhlenbeck process, is represented as the mild solution of a linear stochastic partial differential equation (SPDE) and its best possible state spaces are described. The problem of stationary solutions of \(Y^ X\) in dependence of X is also investigated.