Kontsevich Integral Invariants for Random Trajectories

Abstract

In connection with the investigation of the topological properties of stochastic flows, we encounter the problem of description of braids formed by several trajectories of the flow starting from different points. The complete system of invariants for braids is well known. This system is known as the system of Vasil’ev invariants and distinguishes braids to within a homotopy. We consider braids formed by the trajectories $Z_k (t) = X_k(t) + iY_k (t)$ such that $X_k, Y_k , 1 ≤ k ≤ n$, are continuous semimartingales with respect to a common filtration. For these braids, we establish a representation of the indicated invariants in the form of iterated Stratonovich integrals.