Stochastic plots and universal cover of the loop space (Q5951877)

Let \(M\) be a simply connected compact manifold, \(L_x(M)\) be a continuous loop space of mappings \(\gamma\) from \(S^1\) into \(M\) such that \(\gamma_0=x\). Let \(\widetilde L_x(M)\) be the universal cover for a uniform distance of \(L_x(M)\) and \(\pi\) be a projection map \(\pi:\widetilde L_x(M)\to L_x(M)\). The author proves that for a stochastic closed 1-form \(\sigma\) over \(L_x(M)\) there exists a unique functional \(\widetilde F\) over \(\widetilde L_x(M)\) such that \(d\widetilde F=\pi^*\sigma\) which is defined up to a constant.