Abstract
It is shown that under the central-limit scaling, the fluctuations of the space—time renormalized age distributions of particles (whose development is controlled by critical linear birth and death processes) around the law-of-large-numbers limit converge in a Hilbert space (containing the class of signed Radon measures with finite moment generating functionals) to a continuous Gaussian process satisfying a Langevin equation. So far, the space of rapidly decreasing functions has been considered to be the natural state space for the kind of limit theorem considered here. However, the space of rapidly decreasing functions is not suitable in the present context and we are led to define an appropriate family of Sobolev spaces. In fact, we construct a scale of Hilbert spaces based on the eigenfunctions expansions of an elliptic operator defined on a weightedL 2-space.
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This research was partially supported by an NSERC of Canada grant.
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Bose, A. A fluctuations limit for scaled age distributions and weighted Sobolev spaces. Appl Math Optim 23, 77–105 (1991). https://doi.org/10.1007/BF01442393
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DOI: https://doi.org/10.1007/BF01442393