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Gibbs measures for self-interacting Wiener paths

M. Gubinelli
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Abstract

In this note we study a class of specifications over $d$-dimensional Wiener measure which are invariant under uniform translation of the paths. This degeneracy is removed by restricting the measure to the $σ$-algebra generated by the increments of the coordinate process. We address the problem of existence and uniqueness of Gibbs measures and prove a central limit theorem for the rescaled increments. These results apply to the study of the ground state of the Nelson model of a quantum particle interacting with a scalar boson field.

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M. Gubinelli. 2006-03-07. Gibbs measures for self-interacting Wiener paths. https://arxiv.org/abs/math-ph/0511068

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