Geodesic
Initial-boundary value problems in a bounded domain: probabilistic representations of solutions and limit theorems. II
Teoriâ veroâtnostej i ee primeneniâ, Tome 62 (2017) no. 3, pp. 446-467 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The paper puts forward a new method of construction of a probabilistic representation of solutions to initial-boundary value problems for a number of evolution equations (in particular, for the Schrödinger equation) in a bounded subdomain of $\mathbb R^2$ with smooth boundary. Our method is based on the construction of a special extension of the initial function from the domain to the entire plane. For problems with Neumann boundary condition, this method produces a new approach to the construction of a Wiener process “reflected from the boundary,” which was first introduced by A. V. Skorokhod.
Keywords: initial-boundary value problems, Schrödinger equation, limit theorems, Skorokhod problem, Feynman integral, Feynman measure.
Mots-clés : evolution equations 
@article{TVP_2017_62_3_a1,
     author = {I. A. Ibragimov and N. V. Smorodina and M. M. Faddeev},
     title = {Initial-boundary value problems in a~bounded domain: probabilistic representations of solutions and limit {theorems.~II}},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {446--467},
     year = {2017},
     volume = {62},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_2017_62_3_a1/}
}
TY  - JOUR
AU  - I. A. Ibragimov
AU  - N. V. Smorodina
AU  - M. M. Faddeev
TI  - Initial-boundary value problems in a bounded domain: probabilistic representations of solutions and limit theorems. II
JO  - Teoriâ veroâtnostej i ee primeneniâ
PY  - 2017
SP  - 446
EP  - 467
VL  - 62
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/TVP_2017_62_3_a1/
LA  - ru
ID  - TVP_2017_62_3_a1
ER  - 
%0 Journal Article
%A I. A. Ibragimov
%A N. V. Smorodina
%A M. M. Faddeev
%T Initial-boundary value problems in a bounded domain: probabilistic representations of solutions and limit theorems. II
%J Teoriâ veroâtnostej i ee primeneniâ
%D 2017
%P 446-467
%V 62
%N 3
%U http://geodesic.mathdoc.fr/item/TVP_2017_62_3_a1/
%G ru
%F TVP_2017_62_3_a1
I. A. Ibragimov; N. V. Smorodina; M. M. Faddeev. Initial-boundary value problems in a bounded domain: probabilistic representations of solutions and limit theorems. II. Teoriâ veroâtnostej i ee primeneniâ, Tome 62 (2017) no. 3, pp. 446-467. http://geodesic.mathdoc.fr/item/TVP_2017_62_3_a1/

[1] N. Ikeda, Sh. Watanabe, Stochastic differential equations and diffusion processes, North-Holland Math. Library, 24, North-Holland Publishing Co., Amsterdam–New York; Kodansha, Ltd., Tokyo, 1981, xiv+464 pp. | MR | MR | Zbl | Zbl

[2] G. N. Watson, A treatise on the theory of Bessel functions, 2nd ed., Cambridge Univ. Press, Cambridge; The Macmillan Co., New York, 1944, vi+804 pp. | MR | Zbl

[3] I. M. Gelfand, G. E. Schilow, Verallgemeinerte Funktionen (Distributionen), v. I, Hochschulbücher für Math., 47, Funktionen und das Rechnen mit ihnen, VEB Deutscher Verlag der Wissenschaften, Berlin, 1960, 364 pp. | MR | MR | Zbl | Zbl

[4] J. L. Doob, “A probabilistic approach to the heat equation”, Trans. Amer. Math. Soc., 80 (1955), 216–280 | DOI | MR | Zbl

[5] I. A. Ibragimov, N. V. Smorodina, M. M. Faddeev, “A limit theorem on convergence of random walk functionals to a solution of the Cauchy problem for the equation $\frac{\partial u}{\partial t}=\frac{\sigma^2}2\,\Delta u$ with complex $\sigma$”, J. Math. Sci. (N. Y.), 206:2 (2015), 171–180 | DOI | MR | Zbl

[6] I. A. Ibragimov, N. V. Smorodina, M. M. Faddeev, “A complex analogue of the central limit theorem and a probabilistic approximation of the Feynman integral”, Dokl. Math., 90:3 (2014), 724–726 | DOI | DOI | MR | Zbl

[7] 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

[8] K. Itô, H. P. McKean, Jr., Diffusion processes and their sample paths, Grundlehren Math. Wiss., 125, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin–New York, 1965, xvii+321 pp. | MR | Zbl | Zbl

[9] O. A. Ladyzhenskaya, N. N. Ural'tseva, Linear and quasilinear elliptic equations, Academic Press, New York–London, 1968, xviii+495 pp. | MR | MR | Zbl | Zbl

[10] J.-L. Lions, E. Magenes, Problèmes aux limites non homogènes et applications, v. 1, Travaux et Recherches Mathématiques, 17, Dunod, Paris, 1968, xx+372 pp. | MR | MR | Zbl | Zbl

[11] A. Pilipenko, An introduction to stochastic differential equations with reflection, Lectures in Pure Appl. Math., 1, Universitätsverlag, Potsdam, 2014, ix+75 pp. | Zbl

[12] V. I. Smirnov, A course of higher mathematics, v. III, Complex variables. Special functions, Part 2, Pergamon Press, Oxford–Edinburgh–New York–Paris–Frankfurt; Addison-Wesley Publishing Co., Inc., 1964, x+700 pp. | MR | Zbl

[13] A. V. Skorokhod, “Stochastic equations for diffusion processes in a bounded region”, Theory Probab. Appl., 6:3 (1961), 264–274 | DOI | MR | Zbl

[14] K. Sato, H. Tanaka, “Local times on the boundary for multidimensional reflecting diffusion”, Proc. Japan Acad., 38:10 (1962), 699–702 | DOI | MR | Zbl

[15] E. C. Titchmarsh, Eigenfunction expansions associated with second-order differential equations, v. II, Clarendon Press, Oxford, 1958, xi+404 pp. | MR | Zbl | Zbl

[16] D. K. Faddeev, B. Z. Vulikh, N. N. Uraltseva i dr., Izbrannye glavy analiza i vysshei algebry, Izd-vo Leningr. un-ta, L., 1981, 200 pp. | Zbl