Multi-Occupation Field Generates the Borel-Sigma-Field of Loops

  • Yinshan Chang
Part of the Lecture Notes in Mathematics book series (LNM, volume 2123)


In this article, we consider the space of càdlàg loops on a Polish space S. The loop space can be equipped with a “Skorokhod” metric. Moreover, it is Polish under this metric. Our main result is to prove that the Borel-σ-field on the space of loops is generated by a class of loop functionals: the multi-occupation field. This result generalizes the result in the discrete case, see (Le Jan, Markov Paths, Loops and Fields, vol. 2026, Springer, Heidelberg, 2011).



The author is grateful to Professor Y. Le Jan for inspiring discussions, valuable suggestions and great help in preparation of the manuscript. The author also thanks Titus Lupu for inspiring discussions.


  1. 1.
    C. Dellacherie, P-A. Meyer, Probabilities and Potential, vol. 29 of North-Holland Mathematics Studies (North-Holland Publishing, Amsterdam, 1978)Google Scholar
  2. 2.
    S.N. Ethier, T.G. Kurtz, Markov Processes. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics (Wiley, New York, 1986). Characterization and convergenceGoogle Scholar
  3. 3.
    Y. Le Jan, Markov Paths, Loops and Fields, vol. 2026 of Lecture Notes in Mathematics (Springer, Heidelberg, 2011) Lectures from the 38th Probability Summer School held in Saint-Flour, 2008, École d’Été de Probabilités de Saint-Flour [Saint-Flour Probability Summer School]Google Scholar
  4. 4.
    Y. Le Jan, Z. Qian, Stratonovich’s signatures of Brownian motion determine Brownian sample paths. Probab. Theory Relat. Fields 157(1–2), 209–223 (2013)CrossRefzbMATHGoogle Scholar
  5. 5.
    A-S. Sznitman, Topics in Occupation Times and Gaussian Free Fields. Zurich Lectures in Advanced Mathematics (European Mathematical Society (EMS), Zürich, 2012)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Max Planck Institute for Mathematics in the SciencesLeipzigGermany

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