Stochastic dynamics for an infinite system of random closed strings: A Gibbsian point of view

We consider the stochastic dynamics of infinitely many, interacting random closed strings, and show that the law of this process can be characterized as a Gibbs state for some Hamiltonian on the path level, which is represented in terms of the interaction. This is done by means of the stochastic calculus of variations, in particular an integration by parts formula in infinite dimensions. ibbsian point of view of the stochastic dynamics allows us to characterize the reversible states as the Gibbs states for the underlying interaction. Under supplementary monotonicity conditions, there is only one stationary distribution, and we prove that there is exactly one Gibbs state.