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High density limit theorems for nonlinear chemical reactions with diffusion
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  • Published: March 1988

High density limit theorems for nonlinear chemical reactions with diffusion

  • Peter Kotelenez1 

Probability Theory and Related Fields volume 78, pages 11–37 (1988)Cite this article

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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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Authors and Affiliations

  1. Department of Mathematics and Statistics, Case Western Reserve University, 44106, Cleveland, OH, USA

    Peter Kotelenez

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  1. Peter Kotelenez
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Additional information

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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  • Received: 14 January 1987

  • Revised: 10 December 1987

  • Issue Date: March 1988

  • DOI: https://doi.org/10.1007/BF00718032

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Keywords

  • State Space
  • Stochastic Process
  • Stationary Solution
  • Mathematical Biology
  • Central Limit Theorem
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