Summary
The solutionX of a nonlinear reaction-diffusion equation on then-dimensional unit cubeS is approximated by a space-time jump Markov processX v,N (law of large numbers (LLN)).X v,N is constructed on a gridS N onS ofN cells, wherev is proportional to the initial number of particles in each cell. The deviation ofX v,N fromX is computed by a central limit theorem (CLT). The assumptions on the parametersv, N are for the LLN: υ → ∞, asN → ∞, and for the CLT:\(\frac{N}{\upsilon } \to 0\), asN → ∞. The limitY =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 ofY X in dependence ofX is also investigated.
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On leave from Universität Bremen. This work was supported by the Stiftung Volkswagenwerk and a grant from ONR
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Kotelenez, P. High density limit theorems for nonlinear chemical reactions with diffusion. Probab. Th. Rel. Fields 78, 11–37 (1988). https://doi.org/10.1007/BF00718032
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DOI: https://doi.org/10.1007/BF00718032