Geodesic
On a probabilistic method of solving a one-dimensional initial-boundary value problem
Teoriâ veroâtnostej i ee primeneniâ, Tome 58 (2013) no. 2, pp. 255-281 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We obtain an analogue of probabilistic representation of a solution of an initial-boundary value problem for the equation $\partial u/\partial t+(\sigma^2/2)\partial^2u/\partial x^2+f(x)u=0$, where $\sigma$ is a complex number.
Keywords: random processes; evolution equation; limit theorems; Feynman–Kac formula; Feynman integral; Feynman measure. 
@article{TVP_2013_58_2_a3,
     author = {I. A. Ibragimov and N. V. Smorodina and M. M. Faddeev},
     title = {On a probabilistic method of solving a one-dimensional initial-boundary value problem},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {255--281},
     year = {2013},
     volume = {58},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_2013_58_2_a3/}
}
TY  - JOUR
AU  - I. A. Ibragimov
AU  - N. V. Smorodina
AU  - M. M. Faddeev
TI  - On a probabilistic method of solving a one-dimensional initial-boundary value problem
JO  - Teoriâ veroâtnostej i ee primeneniâ
PY  - 2013
SP  - 255
EP  - 281
VL  - 58
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TVP_2013_58_2_a3/
LA  - ru
ID  - TVP_2013_58_2_a3
ER  - 
%0 Journal Article
%A I. A. Ibragimov
%A N. V. Smorodina
%A M. M. Faddeev
%T On a probabilistic method of solving a one-dimensional initial-boundary value problem
%J Teoriâ veroâtnostej i ee primeneniâ
%D 2013
%P 255-281
%V 58
%N 2
%U http://geodesic.mathdoc.fr/item/TVP_2013_58_2_a3/
%G ru
%F TVP_2013_58_2_a3
I. A. Ibragimov; N. V. Smorodina; M. M. Faddeev. On a probabilistic method of solving a one-dimensional initial-boundary value problem. Teoriâ veroâtnostej i ee primeneniâ, Tome 58 (2013) no. 2, pp. 255-281. http://geodesic.mathdoc.fr/item/TVP_2013_58_2_a3/

[1] Venttsel A. D., Kurs teorii sluchainykh protsessov, Nauka, M., 1975, 319 pp.

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

[3] Chung K. L., Zhao Z. X, From Brownian Motion to Schrödinger's Equation, Springer-Verlag, Berlin–Heidelberg, 1995, 287 pp. | MR | Zbl

[4] Freidlin M., Functional Integration and Partial Differential Equations, Princeton Univ. Press, Princeton, 1985, 545 pp. | MR | Zbl

[5] Daletskii Yu. L., Fomin S. V., Mery i differentsialnye uravneniya v beskonechnomernykh prostranstvakh, Nauka, M., 1983, 383 pp.

[6] Glimm Dzh., Dzhaffe A., Matematicheskie metody kvantovoi fiziki: Podkhod s ispolzovaniem funktsionalnykh integralov, Mir, M., 1984, 445 pp. | MR | Zbl

[7] Rid M., Saimon B., Metody sovremennoi matematicheskoi fiziki, v. 2, Mir, M., 1978, 396 pp.

[8] Doss H., “Sur une résolution stochastique de l'équation de Schrödinger à coefficients analytiques”, Comm. Math. Phys., 73:4 (1980), 247–264 | DOI | MR | Zbl

[9] Albeverio S., Høegh-Krohn R. J., Mazzucchi S., “Mathematical theory of Feynman path integrals”, Lecture Notes in Math., 523, 2008, 1–176 | DOI | MR

[10] Thaler H., “Solution of Schrödinger equations on compact Lie groups via probabilistic methods”, Potential Anal., 18:2 (2003), 119–140 | DOI | MR | Zbl

[11] Maslov V. P., Kompleksnye markovskie tsepi i kontinualnyi integral Feinmana, Nauka, M., 1976, 191 pp.

[12] Maslov V. P., Chebotarev A. M., “Opredelenie kontinualnogo integrala Feinmana v $p$-predstavlenii”, Dokl. AN SSSR, 229:1 (1976), 37–38 | MR | Zbl

[13] Chebotarev A. M., “O predstavlenii resheniya uravneniya Shredingera v vide matematicheskogo ozhidaniya funktsionalov skachkoobraznogo protsessa”, Matem. zametki, 24:5 (1978), 699–706 | MR | Zbl

[14] Albeverio S., Smorodina N., Faddeev M., “The probabilistic representation of the exponent of a class of pseudo-differential operators”, J. Global Stoch. Anal., 1:2 (2011), 123–148

[15] Ibragimov I. A., Smorodina N. V., Faddeev M. M., “Veroyatnostnaya approksimatsiya reshenii zadachi Koshi dlya nekotorykh evolyutsionnykh uravnenii”, Zap. nauchn. sem. POMI, 396, 2011, 111–143

[16] Gelfand I. M., Shilov G. E., Obobschennye funktsii i deistviya nad nimi, Fizmatgiz, M., 1958, 439 pp.

[17] Gikhman I. I., Skorokhod A. V., Teoriya sluchainykh protsessov, v. 1, Nauka, M., 1971, 664 pp.