We consider Brownian motion perturbed by the exponential of an action. The action is the sum of an external, one-body potential and a two-body interaction potential which depends only on the increments. Under suitable conditions on these potentials, we establish existence and uniqueness of the corresponding Gibbs measure. We also provide an example where uniqueness fails because of a slow decay in the interaction potential.
Hirofumi Osada. Herbert Spohn. "Gibbs Measures Relative to Brownian Motion." Ann. Probab. 27 (3) 1183 - 1207, July 1999. https://doi.org/10.1214/aop/1022677444