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We consider a model in which the production of new molecules in a chemical reaction network occurs in a seemingly stochastic fashion, and can be modeled as a Poisson process with a varying arrival rate: the rate is lambdaiโ when an external Markov process J(cdot) is in state i. It is assumed that molecules decay after an exponential time with mean muโ1.
\nThe goal of this work is to analyze the distributional properties of the number of molecules in the system, under a specific time-scaling. In this scaling, the background process is sped
\nup by a factor Nalpha, for some alpha>0, whereas the arrival rates become Nlambdaiโ, for N large.
\nThe main result of this paper is a functional central limit theorem ({\\sc f-clt}) for the number of molecules, in that the number of molecules, after centering and scaling, converges to an Ornstein-Uhlenbeck process. An interesting dichotomy is observed: (i)~if alpha>1 the background process jumps faster than the arrival process, and consequently the arrival process behaves essentially as a (homogeneous) Poisson process, so that the scaling in the {\\sc f-clt} is the usual sqrtN, whereas (ii)~for alphaleq1 the background process is relatively slow, and the scaling in the {\\sc f-clt} is N1โalpha/2. In the latter regime, the parameters of the limiting Ornstein-Uhlenbeck process contain the deviation matrix associated with the background process J(cdot).
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