Abstract.
We study a Gibbs measure over Brownian motion with a pair potential which depends only on the increments. Assuming a particular form of this pair potential, we establish that in the infinite volume limit the Gibbs measure can be viewed as Brownian motion moving in a dynamic random environment. Thereby we are in a position to use the technique of Kipnis and Varadhan and to prove a functional central limit theorem.
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Betz, V., Spohn, H. A central limit theorem for Gibbs measures relative to Brownian motion. Probab. Theory Relat. Fields 131, 459–478 (2005). https://doi.org/10.1007/s00440-004-0381-8
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DOI: https://doi.org/10.1007/s00440-004-0381-8