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On a multidimensional Brownian motion with a membrane located on a given hyperplane

Author

Listed:
  • Kopytko, B.I.
  • Portenko, M.I.

Abstract

Brownian motion in a Euclidean space with a membrane located on a hyperplane and acting in the normal direction is constructed such that its so-called permeability coefficient can be given by an arbitrary Borel measurable function defined on that hyperplane and taking on its values from the interval [−1,1]. In all the publications on the topic, that coefficient was supposed to be a continuous function. A certain limit theorem for the number of crossings through the membrane by the consecutive values of the process constructed at the instants of time 0, 1/n, 2/n, …, [nt]/n (for fixed t>0) is proved under the assumption that n→∞. The limit distribution in that theorem can be curiously interpreted in the case of a membrane whose permeability coefficient coincides with the indicator of a measurable subset of the hyperplane.

Suggested Citation

  • Kopytko, B.I. & Portenko, M.I., 2023. "On a multidimensional Brownian motion with a membrane located on a given hyperplane," Stochastic Processes and their Applications, Elsevier, vol. 160(C), pages 371-385.
    • Handle: RePEc:eee:spapps:v:160:y:2023:i:c:p:371-385
    DOI: 10.1016/j.spa.2023.03.011
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