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Mat. Zametki, 2008, Volume 83, Issue 3, Pages 333–349 (Mi mz3772)  

This article is cited in 17 scientific papers (total in 17 papers)

Feynman Formulas and Functional Integrals for Diffusion with Drift in a Domain on a Manifold

Ya. A. Butkoab

a M. V. Lomonosov Moscow State University
b N. E. Bauman Moscow State Technical University

Abstract: We obtain representations for the solution of the Cauchy–Dirichlet problem for the diffusion equation with drift in a domain on a compact Riemannian manifold as limits of integrals over the Cartesian powers of the domain; the integrands are elementary functions depending on the geometric characteristics of the manifold, the coefficients of the equation, and the initial data. It is natural to call such representations Feynman formulas. Besides, we obtain representations for the solution of the Cauchy–Dirichlet problem for the diffusion equation with drift in a domain on a compact Riemannian manifold as functional integrals with respect to Weizsäcker–Smolyanov surface measures and the restriction of the Wiener measure to the set of trajectories in the domain; such a restriction of the measure corresponds to Brownian motion in a domain with absorbing boundary. In the proof, we use Chernoff's theorem and asymptotic estimates obtained in the papers of Smolyanov, Weizsäcker, and their coauthors.

Keywords: diffusion with drift, Feynman formula, functional integral, Riemannian manifold, Cauchy–Dirichlet problem, Weizsäcker–Smolyanov surface measure, Wiener measure, path integral, Feynman–Kac–Itô formula

DOI: https://doi.org/10.4213/mzm3772

Full text: PDF file (559 kB)
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English version:
Mathematical Notes, 2008, 83:3, 301–316

Bibliographic databases:

UDC: 517.987.4
Received: 28.06.2005
Revised: 15.03.2007

Citation: Ya. A. Butko, “Feynman Formulas and Functional Integrals for Diffusion with Drift in a Domain on a Manifold”, Mat. Zametki, 83:3 (2008), 333–349; Math. Notes, 83:3 (2008), 301–316

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Butko Ya.A., Grothaus M., Smolyanov O.G., “Feynman formula for a diffusion of particles with a variable mass in a domain”, 5th International Symposium on Quantum Theory and Symmetries QTS5, J. Phys.: Conf. Ser., 128, no. 1, IOP Publishing Ltd, 2008, 012050  crossref  mathscinet  isi  scopus
    2. Butko Ya.A., Smolyanov O.G., Schilling R.L., “Feynman formulae for Feller semigroups”, Dokl. Math., 82:2 (2010), 679–683  crossref  mathscinet  zmath  isi  elib  scopus
    3. Butko Ya.A., Grothaus M., Smolyanov O.G., “Lagrangian Feynman formulas for second-order parabolic equations in bounded and unbounded domains”, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 13:3 (2010), 377–392  crossref  mathscinet  zmath  isi  elib  scopus
    4. Butko Ya.A., Duryagin A.V., “77-30569/251251 formuly Feinmana dlya semeistva parabolicheskikh uravnenii, sootvetstvuyuschikh tau-kvantovaniyu kvadratichnoi funktsii Gamiltona”, Nauka i obrazovanie: elektronnoe nauchno-tekhnicheskoe izdanie, 2011, no. 11, 60–60  elib
    5. “77-30569/239563 formula Feinmana dlya polugrupp s multiplikativno vozmuschennymi generatorami”, Nauka i obrazovanie: elektronnoe nauchno-tekhnicheskoe izdanie, 2011, no. 10, 69–69  elib
    6. Böttcher B., Butko Ya.A., Schilling R.L., Smolyanov O.G., “Feynman formulas and path integrals for some evolution semigroups related to $\tau$-quantization”, Russ. J. Math. Phys., 18:4 (2011), 387–399  crossref  mathscinet  zmath  isi  elib  scopus
    7. Butko Ya.A., Schilling R.L., Smolyanov O.G., “Hamiltonian Feynman-Kac and Feynman formulae for dynamics of particles with position-dependent mass”, Internat. J. Theoret. Phys., 50:7 (2011), 2009–2018  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    8. Butko Ya.A., Schilling R.L., Smolyanov O.G., “Lagrangian and Hamiltonian Feynman formulae for some Feller semigroups and their perturbations”, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 15:3 (2012), 1250015, 26 pp.  crossref  zmath  isi  elib  scopus
    9. Buzinov M.S., Butko Ya.A., “Formuly feinmana dlya parabolicheskogo uravneniya s bigarmonicheskim differentsialnym operatorom na konfiguratsionnom prostranstve”, Nauka i obrazovanie: elektronnoe nauchno-tekhnicheskoe izdanie, 2012, no. 08, 9–9  elib
    10. A. K. Kravtseva, “Infinite-Dimensional Schrödinger Equations with Polynomial Potentials and Representation of Their Solutions via Feynman Integrals”, Math. Notes, 94:5 (2013), 824–828  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    11. Laetsch T., “An Approximation to Wiener Measure and Quantization of the Hamiltonian on Manifolds with Non-Positive Sectional Curvature”, J. Funct. Anal., 265:8 (2013), 1667–1727  crossref  mathscinet  zmath  isi  elib  scopus
    12. Kravtseva A.K., Smolyanov O.G., Shavgulidze E.T., “Asymptotic expansions of Feynman integrals of exponentials with polynomial exponent”, Russ. J. Math. Phys., 23:4 (2016), 491–509  crossref  mathscinet  zmath  isi  scopus
    13. Remizov I.D., “Quasi-Feynman formulas – a method of obtaining the evolution operator for the Schrödinger equation”, J. Funct. Anal., 270:12 (2016), 4540–4557  crossref  mathscinet  zmath  isi  elib  scopus
    14. Butko Ya.A., Grothaus M., Smolyanov O.G., “Feynman formulae and phase space Feynman path integrals for tau-quantization of some Lévy-Khintchine type Hamilton functions”, J. Math. Phys., 57:2 (2016), 023508  crossref  mathscinet  zmath  isi  elib  scopus
    15. Remizov I.D., “New Method For Constructing Chernoff Functions”, Differ. Equ., 53:4 (2017), 566–570  crossref  mathscinet  zmath  isi  scopus
    16. Butko Ya.A., “Chernoff Approximation of Subordinate Semigroups”, Stoch. Dyn., 18:3 (2018), 1850021  crossref  mathscinet  zmath  isi  scopus
    17. Butko Ya.A., “Chernoff Approximation For Semigroups Generated By Killed Feller Processes and Feynman Formulae For Time-Fractional Fokker-Planck-Kolmogorov Equations”, Fract. Calc. Appl. Anal., 21:5 (2018), 1203–1237  crossref  mathscinet  isi  scopus
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