Density-dependent limits for a nonlinear reaction-diffusion model
A density-dependent stochastic reaction-diffusion model describing a chemical reaction is studied. The model is constructed by partitioning \([0,1]^p\), \(p \geq 1\), into \(N^p\) congruent cells, distributing approximately \(N^{p_l}\) particles in the system and allowing particles to diffuse between cells by rate \(N^2\) random walks and to react within cells as density dependent birth and death processes. Dividing cell numbers by \(l\) one gets a step-function-valued Markov process \(X_{N,l} (t,r)\), \(t \geq 0\), \(r \in [0,1]^p\). Following Kotelenez in comparing \(X_{N,l}\) with its deterministic counterpart \(\psi\) two laws of large numbers for \(X_{N,l}\), when \(l \to \infty\) as \(N \to \infty\) and for constant \(l\), are proved. Moreover, in the case \(l = N\), \(p = 1\), a central limit theorem for \(X_{N,l}\) is obtained, thus completing the case \(N/l \to 0\) considered earlier by \textit{P. Kotelenez} [Probab. Theory Relat. Fields 78, No. 1, 11-37 (1988; Zbl 0628.60108)] and by the author [Stochastic Processes Appl. 45, No. 2, 193-207 (1993; Zbl 0773.60052)].
stochastic partial differential equation
scientific article; zbMATH DE number 828376