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Kontsevich Integral Invariants for Random Trajectories

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Ukrainian Mathematical Journal Aims and scope

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.

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References

  1. A. S. Monin and A. M. Yaglom, Statistical Hydrodynamics [in Russian], Vol. 2, Nauka, Moscow (1967).

    Google Scholar 

  2. V. Arnold and B. Khesin, Topological Methods in Hydrodynamics, Springer, New York (1998).

    MATH  Google Scholar 

  3. M. Berger and P. Roberts, “On the winding number problem with finite steps,” Adv. Appl. Probab., 20, No. 2, 261–274 (1988).

    Article  MATH  MathSciNet  Google Scholar 

  4. M. Yor, “Etude asymptotique des nombres de tours de plusieurs mouvements browniens complexes correles,” Progr. Probab., 28, 441–455 (1991).

    MathSciNet  Google Scholar 

  5. J. Pitman and M. Yor, “Asymptotic laws of planar Brownian motion,” Ann. Probab., 14, No. 3, 733–779 (1986).

    Article  MATH  MathSciNet  Google Scholar 

  6. Jean-Luc Thiffeault, “Braids of entangled particle trajectories,” Chaos, 20 (2010).

  7. Dror Bar-Natan, “Vassiliev homotopy string link invariants,” J. Knot Theory Ramifications, 4, No. 1, 13–32.

  8. M. Kontsevich, “Vassiliev’s knot invariants,” Adv. Soviet Math., 16, Pt. 2, 137–150 (1993).

    MathSciNet  Google Scholar 

  9. M. A. Berger, “Topological invariants in braid theory,” Lett. Math. Phys., 55, No. 3, 181–192 (2001).

    Article  MATH  MathSciNet  Google Scholar 

  10. J. R. Munkres, Topology: A First Course, Prentice-Hall, Englewood Cliffs, NJ (1974).

    Google Scholar 

  11. H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge Univ. Press, Cambridge (1997).

    MATH  Google Scholar 

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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 67, No. 1, pp. 57–67, January, 2014.

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Kuznetsov, V.A. Kontsevich Integral Invariants for Random Trajectories. Ukr Math J 67, 62–73 (2015). https://doi.org/10.1007/s11253-015-1065-5

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  • DOI: https://doi.org/10.1007/s11253-015-1065-5

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