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Reflecting Brownian Motion in the d-Ball

Following the works of I. A. Ibrahimov, N. V. Smorodina and M. M. Faddeev, we develop a new construction of the Brownian motion with reflection in the d-ball. The main advantage of our new approach is that it allows one to construct reflecting Levy processes, whereas previous constructions are limited to diffusion processes. In our upcoming work, we shall extend the results to symmetric Lévy processes in a smooth domain.

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References

  1. 1.

    S. Asmussen, Applied Probability and Queues, Vol. 51, Springer Science & Business Media (2008).

  2. 2.

    F. Bass Richard and P. Hsu, “Some potential theory for reflecting Brownian motion in Hölder and Lipschitz domains,” Ann. Probab., 19, 486–508 (1991).

    MathSciNet  MATH  Google Scholar 

  3. 3.

    R. Bekker and B. Zwart, “On an equivalence between loss rates and cycle maxima in queues and dams,” Probab. Engin. Inform. Sci., 19, 241–255 (2005).

    MathSciNet  Article  Google Scholar 

  4. 4.

    B. Krzysztof, B. Krzysztof, and Z.-Q. Chen, “Censored stable processes,” Probab. Theory and Related Fields, 127, 89–152 (2003).

    MathSciNet  Article  Google Scholar 

  5. 5.

    Z.-Q. Chen and T. Kumagai, “Heat kernel estimates for stable-like processes on d-sets,” Stochastic Processes and Their Applications, 108, 27–62 (2003).

    MathSciNet  Article  Google Scholar 

  6. 6.

    K. L. Chung and R. J. Williams, Introduction to Stochastic Integration, 2nd ed., Birkhäuser, Boston (1990).

  7. 7.

    J. W. Cohen, The Single Server Queue, 2nd ed., North-Holland, Amsterdam (1982).

  8. 8.

    W. L. Cooper, V. Schmidt, and R. F. Serfozo, “Skorohod–Loynes characterizations of queueing, fluid, and inventory processes,” Queueing Systems, 37, 233–257 (2001).

    MathSciNet  Article  Google Scholar 

  9. 9.

    D. J. Daley, “Single-server queueing systems with uniformly limited queueing time,” J. Austral. Math. Soc., 4, 489–505 (1964).

    MathSciNet  Article  Google Scholar 

  10. 10.

    J. E. Avery and J. C. Avery, Hyperspherical Harmonics and Their Physical Applications, World Scientific (2017).

  11. 11.

    M. Fukushima, “A constuction of reflecting barrier Brownian motions for bounded domains,” Osaka J. Math., 4, 183–215 (1967).

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Qing-Yang Guan and Zhi-Ming Ma, “Reflected symmetric α-stable processes and regional fractional Laplacian,” Probability Theory and Related Fields, 134, 649–694 (2006).

    MathSciNet  Article  Google Scholar 

  13. 13.

    P.-L. Lions and A.-S. Sznitman, “Stochastic differential equations with reflecting boundary conditions,” Comm. Pure Applied Math., 37, 511–537 (1984).

    MathSciNet  Article  Google Scholar 

  14. 14.

    A. P. Moran, The Theory of Storage (1959).

  15. 15.

    A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer Science & Business Media (2012).

  16. 16.

    A. Pilipenko, An Introduction to Stochastic Differential Equations with Reflection, Vol. 1, Universitätsverlag Potsdam (2014).

  17. 17.

    M. Reed and B. Simon, Analysis of Operators, Vol. 4, Elsevier (1978).

  18. 18.

    W. Stadje, “A new look at the Moran dam,” J. Applied Probab., 30, 489–495 (1993).

    MathSciNet  Article  Google Scholar 

  19. 19.

    E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean Spaces Princeton Math. Series, Vol. 32, Princeton Univ. Press (2016).

  20. 20.

    D. W. Stroock and S. R. Srinivasa Varadhan, “Diffusion processes with boundary conditions,” Comm. Pure and Appl. Math., 24, 147–225 (1971).

    MathSciNet  Article  Google Scholar 

  21. 21.

    G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge Univ. Press (1995).

  22. 22.

    M. Sh. Birman and M. Z. Solomyak, Spectral Theory of Self-adjoint Operators in Hilbert Space [in Russian], Leningrad (1980).

  23. 23.

    N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes [Russian translation], Moscow (1986).

  24. 24.

    I. I. Gikhman and A. V. Skorokhod, Stochastic Differential Equations [in Russian], Kiev (1968).

  25. 25.

    I. A. Ibragimov, N. V. Smorodina, and M. M. Faddeev, “A limit theorem on the convergence of random walk functionals to a solution of the Cauchy problem for the equation \( \frac{\partial u}{\partial t}={\sigma}^2\Delta u \) with complex σ,” Zap. Nauchn. Semin. POMI, 420, 88–108 (2013); English transl. J. Math. Sci., 206, No. 2, 171–180 (2015).

  26. 26.

    I. A. Ibragimov, N. V. Smorodina, and M. M. Faddeev, “A complex analog of the central limit theorem and a probabilistic approximation of the Feynman integral,” Dokl. RAN, 459, 400–402 (2014).

    MathSciNet  MATH  Google Scholar 

  27. 27.

    I. A. Ibragimov, N. V. Smorodina, and M. M. Faddeev, “Initial boundary value problems in a bounded domain: probabilistic respesentations of solutions and limit theorems. I,” Teor. Veroyatn. Primen., 61, 733–752 (2016).

    Article  Google Scholar 

  28. 28.

    I. A. Ibragimov, N. V. Smorodina, and M. M. Faddeev, “On a limit theorem related to a probabilistic representation of the solution of the Cauchy problem for the Schrödinger equation,” Zap. Nauchn. Semin. POMI, 454, 158–175 (2016); English transl. J. Math. Sci., 229, No. 6, 702–713 (2018).

  29. 29.

    I. A. Ibragimov, N. V. Smorodina, and M. M. Faddeev, “Initial boundary value problems in a bounded domain: probabilistic representations of solutions and limit theorem,” Teor. Veroyatn. Primen., 62, 446–467 (2017).

    Article  Google Scholar 

  30. 30.

    I. A. Ibragimov, N. V. Smorodina, and M. M. Faddeev, “Reflecting Lévy processes and the families of linear operators generated by them,” Teor. Veroyatn. Primen., 64, 417–441 (2019).

    MathSciNet  Article  Google Scholar 

  31. 31.

    P. N. Ievlev, “Probabilistic representation of the solution of the Cauchy problem for the multidimensional Schr¨odinger equation,” Zap. Nauchn. Semin. POMI, 466, 145–158 (2017); English transl. J. Math. Sci., 244, No. 5, 796–804 (2020).

  32. 32.

    P. N. Ievlev, “Probabilistic representations of initial boundary value problems for the Schrödinger equation in a d-simensional ball,” Zap. Nauchn. Semin. POMI, 474, 149–170 (2018); English transl. J. Math. Sci., 251, No. 1, 96–110 (2020).

  33. 33.

    T. Kato, Perturbation Theory for Linear Operators [Russian translation], Moscow (1972).

  34. 34.

    J. F. C. Kingman, Poisson Processes [Russian translation], Moscow (2007).

  35. 35.

    O. A. Ladyzhenskaya and N. N. Uraltseva, Linear and Quasilinear Elliptic Equations [in Russian], Moscow (1973).

  36. 36.

    A. V. Skorokhod, “Stochastic equations for diffusion processes with a boundary,” Teor. Veroyatn. Primen., 6, 287–298 (1961).

    MathSciNet  MATH  Google Scholar 

  37. 37.

    E. C. Titchmarsh, Eigenfunction Expansions Associated with Second-order Differential Equations [Russian translation], Moscow (1961).

  38. 38.

    D. K. Faddeev, B. Z. Vulikh, and N. N. Uraltseva, Selected Chapters of Analysis and Higher Algebra [in Russian], Leningrad Univ. Press (1981).

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Correspondence to P. N. Ievlev.

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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 486, 2019, pp. 158–177.

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Ievlev, P.N. Reflecting Brownian Motion in the d-Ball. J Math Sci 258, 845–858 (2021). https://doi.org/10.1007/s10958-021-05584-z

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