Geodesic
Probabilistic representations for initial-boundary value problem solutions to the non-stationary Schrödinger equation in $d$-hyperball
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 27, Tome 474 (2018), pp. 149-170 
P. N. Ievlev. Probabilistic representations for initial-boundary value problem solutions to the non-stationary Schrödinger equation in $d$-hyperball. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 27, Tome 474 (2018), pp. 149-170. http://geodesic.mathdoc.fr/item/ZNSL_2018_474_a10/
@article{ZNSL_2018_474_a10,
     author = {P. N. Ievlev},
     title = {Probabilistic representations for initial-boundary value problem solutions to the non-stationary {Schr\"odinger} equation in $d$-hyperball},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {149--170},
     year = {2018},
     volume = {474},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2018_474_a10/}
}
TY  - JOUR
AU  - P. N. Ievlev
TI  - Probabilistic representations for initial-boundary value problem solutions to the non-stationary Schrödinger equation in $d$-hyperball
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2018
SP  - 149
EP  - 170
VL  - 474
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2018_474_a10/
LA  - ru
ID  - ZNSL_2018_474_a10
ER  - 
%0 Journal Article
%A P. N. Ievlev
%T Probabilistic representations for initial-boundary value problem solutions to the non-stationary Schrödinger equation in $d$-hyperball
%J Zapiski Nauchnykh Seminarov POMI
%D 2018
%P 149-170
%V 474
%U http://geodesic.mathdoc.fr/item/ZNSL_2018_474_a10/
%G ru
%F ZNSL_2018_474_a10

Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

We extend the construction of probabilistic representations for initial-boundary value problem solutions to the non-stationary Schrödinger equation in d-hyperball first obtained in the works by I. Ibragimov, N. Smorodina and M. Faddeev to a multidimensional case. Further on, we show that in these representations the Wiener process could be replaced by a random walk approximation. The $L_2$-convergence rates are obtained.

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

[2] 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

[3] 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

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

[5] I. A. Ibragimov, N. V. Smorodina, M. M. Faddeev, “Ob odnoi predelnoi teoreme, svyazannoi s veroyatnostnym predstavleniem resheniya zadachi Koshi dlya uravneniya Shredingera”, Zap. nauch. semin. POMI, 454, 2016, 158–176

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

[7] I. A. Ibragimov, N. V. Smorodina, M. M. Faddeev, “Kompleksnyi analog tsentralnoi predelnoi teoremy i veroyatnostnaya approksimatsiya integrala Feinmana”, Doklady RAN, 459:3 (2014), 400–402 | DOI | Zbl

[8] A. Pilipenko, An introduction to stochastic differential equations with reflection, Universitätsverlag, Potsdam, 2014 | Zbl

[9] James Emil Avery, John Scales Avery, Hyperspherical Harmonics and Their Physical Applications, World Scientific, 2017 | MR

[10] E. M. Stein, G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, 1971 | MR | Zbl

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

[12] E. Ch. Titchmarsh, Razlozheniya po sobstvennym funktsiyam, svyazannye s differentsialnymi uravneniyami vtorogo poryadka, Izdatelstvo inostrannoi literatury, M., 1961

[13] G. N. Vatson, Teoriya besselevykh funktsii, v. I, Izdatelstvo inostrannoi literatury, M., 1949

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

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

[16] Dzh. Kingman, Puassonovskie protsessy, Izdatelstvo MTsNMO, M., 2007