Brownian motion in a euclidean space with a membrane located on a given hyperplane

Authors

DOI:

https://doi.org/10.15407/dopovidi2022.01.003

Keywords:

Brownian motion, partly permeable membrane, single layer potential, Feynman—Kac formula, local time

Abstract

For the Brownian motion in a Euclidean space, a membrane located on a given hyperplane and acting in the normal direction is constructed such that its so-called permeability coefficient can be given by an arbitrary measurable function defined on that hyperplane and taking on its values in the interval [–1, 1]. In all the previous investigations on the topic that coefficient was supposed to be a continuous function. A limit theorem for the number of crossings of the hyperplane by a discrete approximation of the process constructed is proved. A curious interpretation for the limit distribution in that theorem can be given in the case of the membrane being porous.

Downloads

Download data is not yet available.

References

Aryasova, O. V. & Portenko, M. I. (2005). One class of multidimensional stochastic differential equations having no property of weak uniqueness of a solution. Theory Stoch. Process., 11, No. 3-4, pp. 14-28.

Portenko, N. I. (1982). Generalized diffusion processes. Kyiv: Naukova Dumka (in Russian).

Portenko, N. & Yefimenko, S. (1987). On the number of crossings of a partli reflecting hyperplane by a multidimensional Wiener process. In Stochastic differential systems. Lecture notes in control and information sciences, vol. 96 (pp. 194-203). Berlin, Heidelberg: Springer. https: //doi. org/10. 1007/BFb0038935

Kopytko, B. I. & Portenko, M. I. (2002). On a multidimensional Brownian motion with a membrane located on a hyperplane and acting in an oblique direction. Ukrainian Mathematical Congress–2001. Section 9. Probability Theory and Mathematical Statistics (pp. 73-84). Kyiv.

Dynkin, E. B. (1963). Markov Processes. Moscow: Fizmatgiz (in Russian).

Skorokhod, A. V. (1961). Some limit theorems for additive functionals of a sequence of sums of independent random variables. Ukrains’kyi Matematychnyi Zhurnal, 13, No. 4, pp. 67-78 (in Russian).

Published

30.03.2022

How to Cite

Kopytko, B., & Portenko, M. (2022). Brownian motion in a euclidean space with a membrane located on a given hyperplane. Reports of the National Academy of Sciences of Ukraine, (7), 3–10. https://doi.org/10.15407/dopovidi2022.01.003