Geodesic
Representation of solutions of evolution equations on a~ramified surface by Feynman formulae
Izvestiya. Mathematics , Tome 82 (2018) no. 3, pp. 494-511.

Voir la notice de l'article provenant de la source Math-Net.Ru

We obtain solutions of parabolic second-order differential equations for functions in the class $L_1(K)$ defined on a ramified surface $K$. By using Chernoff's theorem, we prove that such solutions, whenever they exist, can be represented by Lagrangian Feynman formulae, that is, they can be written as limits of integrals over Cartesian powers of the configuration space as the number of factors tends to infinity.
Keywords: parabolic differential equation, ramified surface, Chernoff's theorem.
Mots-clés : Feynman formula 
@article{IM2_2018_82_3_a2,
     author = {V. A. Dubravina},
     title = {Representation of solutions of evolution equations on a~ramified surface by {Feynman} formulae},
     journal = {Izvestiya. Mathematics },
     pages = {494--511},
     publisher = {mathdoc},
     volume = {82},
     number = {3},
     year = {2018},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2018_82_3_a2/}
}
TY  - JOUR
AU  - V. A. Dubravina
TI  - Representation of solutions of evolution equations on a~ramified surface by Feynman formulae
JO  - Izvestiya. Mathematics 
PY  - 2018
SP  - 494
EP  - 511
VL  - 82
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_2018_82_3_a2/
LA  - en
ID  - IM2_2018_82_3_a2
ER  - 
%0 Journal Article
%A V. A. Dubravina
%T Representation of solutions of evolution equations on a~ramified surface by Feynman formulae
%J Izvestiya. Mathematics 
%D 2018
%P 494-511
%V 82
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_2018_82_3_a2/
%G en
%F IM2_2018_82_3_a2
V. A. Dubravina. Representation of solutions of evolution equations on a~ramified surface by Feynman formulae. Izvestiya. Mathematics , Tome 82 (2018) no. 3, pp. 494-511. http://geodesic.mathdoc.fr/item/IM2_2018_82_3_a2/

[1] O. G. Smolyanov, E. T. Shavgulidze, Kontinualnye integraly, 2-e pererab. i susch. dop. izd., Lenand, M., 2015, 336 pp. | MR | Zbl

[2] V. A. Dubravina, “Feynman formulas for solutions of evolution equations on ramified surfaces”, Russ. J. Math. Phys., 21:2 (2014), 285–288 | DOI | MR | Zbl

[3] A. V. Bulinskii, A. N. Shiryaev, Teoriya sluchainykh protsessov, Fizmatlit, M., 2003, 400 pp.

[4] B. Simon, The $P(\varphi)_2$ Euclidean (quantum) field theory, Princeton Series in Physics, Princeton Univ. Press, Princeton, N.J., 1974, xx+392 pp. | MR | Zbl

[5] Symplectic geometry and topology, Lecture notes from the graduate summer school program (Park City, UT, USA, 1997), IAS/Park City Math. Ser., 7, eds. Ya. Eliashberg, L. Traynor, Amer. Math. Soc., Providence, RI; Institute for Advanced Study (IAS), Princeton, NJ, 1999, xiv+430 pp. | MR | Zbl

[6] O. G. Smolyanov, A. G. Tokarev, A. Truman, “Hamiltonian Feynman path integrals via the Chernoff formula”, J. Math. Phys., 43:10 (2002), 5161–5171 | DOI | MR | Zbl

[7] K.-J. Engel, R. Nagel, One-parameter semigroups for linear evolution equations, Grad. Texts in Math., 194, Springer-Verlag, New York, 2000, xxii+586 pp. | DOI | MR | Zbl

[8] M. Kh. Numan Elsheikh, Operatory Shredingera na razvetvlennykh mnogoobraziyakh i ikh approksimatsii, Diss. ... kand. fiz.-matem. nauk, RUDN, M., 2014, 113 pp.

[9] A. S. Plyashechnik, “Feynman formulas for second-order parabolic equations with variable coefficients”, Russ. J. Math. Phys., 20:3 (2013), 377–379 | DOI | MR | Zbl

[10] O. G. Smolyanov, D. S. Tolstyga, “Feynman formulas for stochastic and quantum dynamics of particles in multidimensional domains”, Dokl. Math., 88:2 (2013), 541–544 | DOI | MR | Zbl

[11] O. G. Smolyanov, D. S. Tolstyga, H. von Weizsäcker, “Feynman description of the one-dimensional dynamics of particles whose masses piecewise continuously depend on coordinates”, Dokl. Math., 84:3 (2011), 804–807 | DOI | MR | Zbl

[12] M. Gadella, Ş. Kuru, J. Negro, “Self-adjoint Hamiltonians with a mass jump: general matching conditions”, Phys. Lett. A, 362:4 (2007), 265–268 | DOI | MR | Zbl