An elementary proof of the uniqueness of invariant product measures for some infinite dimensional processesUne preuve élémentaire de l'unicité des mesures invariantes produits pour certains processus en dimension infinie |
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Authors: | Alejandro F Ram??rez |
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Institution: | Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Casilla 306-Correo 22, Santiago 6904411, Chile |
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Abstract: | Consider an infinite dimensional diffusion process with state space , where T is the circle, and defined by an infinitesimal generator L which acts on local functions f as . Suppose that the coefficients ai and bi are smooth, bounded, of finite range, have uniformly bounded second order partial derivatives, that ai are uniformly bounded from below by some strictly positive constant, and that ai is a function only of ηi. Suppose that there is a product measure ν which is invariant. Then if ν is the Lebesgue measure or if d=1,2, it is the unique invariant measure. Furthermore, if ν is translation invariant, it is the unique invariant, translation invariant measure. The proofs are elementary. Similar results can be proved in the context of an interacting particle system with state space , with uniformly positive bounded flip rates which are finite range. To cite this article: A.F. Ram??rez, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 139–144 |
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