Reflecting Brownian motion in $d$-ball
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 28, Tome 486 (2019), pp. 158-177 Cet article a éte moissonné depuis la source Math-Net.Ru

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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 $d$-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 the symmetric Levy processes in a smooth domain.
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P. N. Ievlev. Reflecting Brownian motion in $d$-ball. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 28, Tome 486 (2019), pp. 158-177. http://geodesic.mathdoc.fr/item/ZNSL_2019_486_a8/

[1] S. Asmussen, Applied probability and queues, Stochastic Modelling and Applied Probability, 51, Springer Science Business Media, 2008 | MR

[2] F. Bass Richard, Pei Hsu, “Some potential theory for reflecting brownian motion in holder and lipschitz domains”, Ann. Probab., 19:2 (1991), 486–508 | DOI | MR | Zbl

[3] R. Bekker, B, Zwart, “On an equivalence between loss rates and cycle maxima in queues and dams”, Probability in the Engineering and Informational Sciences, 19:2 (2005), 241–255 | DOI | MR | Zbl

[4] B. Krzysztof, B. Krzysztof, Zhen-Qing Chen, “Censored stable processes”, Probability theory and related fields, 127:1 (2003), 89–152 | DOI | MR | Zbl

[5] Zhen-Qing Chen, Takashi Kumagai, “Heat kernel estimates for stable-like processes on $d$-sets”, Stochastic Processes and their applications, 108:1 (2003), 27–62 | DOI | MR | Zbl

[6] Kai Lai Chung, R. J. Williams, Introduction to stochastic integration, v. 2, Springer, 1990 | MR

[7] J. W. Cohen, A. Browne, The single server queue, v. 8, North-Holland, Amsterdam, 1982 | MR | Zbl

[8] W. L. Cooper, V. Schmidt, R. F. Serfozo, “Skorohod–loynes characterizations of queueing, fluid, and inventory processes”, Queueing Systems, 37:1–3 (2001), 233–257 | DOI | MR | Zbl

[9] D. J. Daley, “Single-server queueing systems with uniformly limited queueing time”, J. Australian Math. Soc., 4:4 (1964), 489–505 | DOI | MR | Zbl

[10] Avery James Emil, Avery John Scales, Hyperspherical harmonics and their physical applications, World Scientific, 2017 | MR

[11] M. Fukushima, “A constuction of reflecting barrier brownian motions for bounded domains”, Osaka J. Math., 4 (1967), 183–215 | MR | Zbl

[12] Qing-Yang Guan, Zhi-Ming Ma, “Reflected symmetric $\alpha$-stable processes and regional fractional laplacian”, Probability theory and related fields, 134:4 (2006), 649–694 | DOI | MR | Zbl

[13] P.-L. Lions, A.-S.Sznitman, “Stochastic differential equations with reflecting boundary conditions”, Communications on Pure and Applied Mathematics, 37:4 (1984), 511–537 | DOI | MR | Zbl

[14] P. A. P. Moran, The theory of storage, 1959 | MR | Zbl

[15] Amnon Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, 44, Springer Science Business Media, 2012 | MR

[16] A. Pilipenko, An introduction to stochastic differential equations with reflection, v. 1 \newblock Universitätsverlag Potsdam, 2014 | Zbl

[17] M. Reed, B. Simon, Analysis of Operators, v. 4, Elsevier, 1978 | MR

[18] W. Stadje, “A new look at the moran dam”, J. Applied Probability, 30:2 (1993), 489–495 | DOI | MR | Zbl

[19] E. M. Stein, G. Weiss, Introduction to Fourier analysis on Euclidean spaces, PMS, 32, Princeton university press, 2016 | MR

[20] D. W. Stroock, SR Srinivasa Varadhan, “Diffusion processes with boundary conditions”, Communications on Pure and Applied Mathematics, 24:2 (1971), 147–225 | DOI | MR | Zbl

[21] G. N. Watson, A treatise on the theory of Bessel functions, Cambridge university press, 1995 | MR | Zbl

[22] M. Sh. Birman, M. Z. Solomyak, Spektralnaya teoriya samosopryazhennykh operatorov v gilbertovom prostranstve, Izdatelstvo Leningradskogo universiteta, 1980

[23] S. Vatanabe, N. Ikeda, Stokhasticheskie differentsialnye uravneniya i diffuzionnye protsessy, Nauka, 1986

[24] I. I. Gikhman, A. V. Skorokhod, Stokhasticheskie differentsialnye uravneniya, Izdatelstvo Nauka Dumka, 1968 | MR

[25] I. A. Ibragimov, N. V. Smorodina, M. M. Faddeev, “Predelnaya teorema o skhodimosti funktsionalov ot sluchainogo bluzhdaniya k resheniyu zadachi Koshi dlya uravneniya {$\frac{\partial u}{\partial t}=\sigma^2 \Delta u$} s kompleksnym parametrom $\sigma$”, Zap. nauchn. semin. POMI, 420, 2013, 88–102

[26] I. A. Ibragimov, N. V. Smorodina, M. M. Faddeev, “Kompleksnyi analog tsentralnoi predelnoi teoremy i veroyatnostnaya approksimatsiya integrala Feinmana”, Doklady Akademii nauk, 459 (2014), 400–400 | DOI | Zbl

[27] I. A Ibragimov, N. V. Smorodina, M. M. Faddeev, “Nachalno-kraevye zadachi v ogranichennoi oblasti: veroyatnostnye predstavleniya reshenii i predelnye teoremy, I”, Teoriya veroyatn. i ee primen., 61:4 (2016), 733–752 | DOI

[28] I. A. Ibragimov, N. V. Smorodina, M. M. Faddeev, “Ob odnoi predelnoi teoreme, svyazannoi s veroyatnostnym predstavleniem resheniya zadachi Koshi dlya uravneniya Shredingera”, Zap. nauchn. semin. POMI, 454, 2016, 158–175

[29] I. A. Ibragimov, N. V. Smorodina, M. M. Faddeev, “Nachalno-kraevye zadachi v ogranichennoi oblasti: veroyatnostnye predstavleniya reshenii i predelnye teoremy, II”, Teoriya veroyatn. i ee primen., 62:3 (2017), 446–467 | DOI

[30] I. A. Ibragimov, N. V. Smorodina, M. M. Faddeev, “Otrazhayuschiesya protsessy Levi i porozhdaemye imi semeistva lineinykh operatorov”, Teoriya veroyatn. i ee primen., 64:3 (2019), 417–441 | DOI | MR | Zbl

[31] P. N. Ievlev, “Veroyatnostnoe predstavlenie resheniya zadachi Koshi dlya mnogomernogo uravneniya Shredingera”, Zap. nauchn. semin. POMI, 466, 2017, 145–158

[32] P. N. Ievlev, “Veroyatnostnye predstavleniya dlya reshenii nachalno-kraevykh zadach dlya uravneniya Shredingera v $d$-mernom share”, Zap. nauchn. semin. POMI, 474, 2018, 149–170 | MR

[33] T. Kato, Teoriya vozmuschenii lineinykh operatorov, Mir, 1972

[34] Dzh. Kingman, Puassonovskie protsessy, 2007

[35] O. A. Ladyzhenskaya, N. N. Uraltseva, Lineinye i kvazilineinye uravneniya ellipticheskogo tipa, Nauka, M., 1973 | MR

[36] A. V. Skorokhod, “Stokhasticheskie uravneniya dlya protsessov diffuzii s granitsami”, Teoriya veroyatn. i ee primen., 6:3 (1961), 287–298 | MR | Zbl

[37] E. Ch. Titchmarsh, V. B. Lidskii, Razlozhenie po sobstvennym funktsiyam, svyazannye s differentsialnymi uravneniyami vtorogo poryadka, Izdatelstvo inostrannoi literatury, M., 1961

[38] D. K. Faddeev, B. Z. Vulikh, N. N. Uraltseva, i dr., Izbrannye glavy analiza i vysshei algebry, Izdatelstvo Leningradskogo universiteta, 1981