2019
Том 71
№ 1

All Issues

Kuznetsov V. A.

Articles: 2
Article (Russian)

Mutual winding angles of particles in Brownian stochastic flows with zero top Lyapunov exponent

Kuznetsov V. A.

↓ Abstract

Ukr. Mat. Zh. - 2016. - 68, № 9. - pp. 1197-1228

The investigation of the geometric properties of particles moving in stochastic flows leads to the study of their mutual winding angles. The same problem for independent Brownian motions was solved in [10]. We generalize these results to the case of isotropic Brownian stochastic flows with top Lyapunov exponent equal to zero.

Article (Ukrainian)

Kontsevich Integral Invariants for Random Trajectories

Kuznetsov V. A.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2015. - 67, № 1. - pp. 57–67

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.