Geodesic
Feynman Formulas and Functional Integrals for Diffusion with Drift in a Domain on a Manifold
Matematičeskie zametki, Tome 83 (2008) no. 3, pp. 333-349.

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

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, functional integral, Riemannian manifold, Cauchy–Dirichlet problem, Weizsäcker–Smolyanov surface measure, Wiener measure, path integral
Mots-clés : Feynman formula, Feynman–Kac–Itô formula. 
@article{MZM_2008_83_3_a1,
     author = {Ya. A. Butko},
     title = {Feynman {Formulas} and {Functional} {Integrals} for {Diffusion} with {Drift} in a {Domain} on a {Manifold}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {333--349},
     publisher = {mathdoc},
     volume = {83},
     number = {3},
     year = {2008},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2008_83_3_a1/}
}
TY  - JOUR
AU  - Ya. A. Butko
TI  - Feynman Formulas and Functional Integrals for Diffusion with Drift in a Domain on a Manifold
JO  - Matematičeskie zametki
PY  - 2008
SP  - 333
EP  - 349
VL  - 83
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2008_83_3_a1/
LA  - ru
ID  - MZM_2008_83_3_a1
ER  - 
%0 Journal Article
%A Ya. A. Butko
%T Feynman Formulas and Functional Integrals for Diffusion with Drift in a Domain on a Manifold
%J Matematičeskie zametki
%D 2008
%P 333-349
%V 83
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2008_83_3_a1/
%G ru
%F MZM_2008_83_3_a1
Ya. A. Butko. Feynman Formulas and Functional Integrals for Diffusion with Drift in a Domain on a Manifold. Matematičeskie zametki, Tome 83 (2008) no. 3, pp. 333-349. http://geodesic.mathdoc.fr/item/MZM_2008_83_3_a1/

[1] O. G. Smolyanov, H. v. Weizsäcker, O. Wittich, “Brownian motion on a manifold as limit of stepwise conditioned standard Brownian motions”, Stochastic Processes, Physics and Geometry: New Interplays, II (Leipzig, 1999), CMS Conf. Proc., 29, Amer. Math. Soc., Providence, RI, 2000, 589–602 | MR | Zbl

[2] O. G. Smolyanov, H. v. Weizsäcker, O. Wittich, Chernoff's Theorem and Discrete Time Approximations of Brownian Motion on Manifolds, arXiv: math/0409155

[3] O. G. Smolyanov, H. v. Weizsäcker, O. Wittich, “Chernoff's Theorem and the construction of semigroups”, Evolution Equations: Applications to Physics, Industry, Life Sciences and Economics (Levico Terme, 2000), Progr. Nonlinear Differential Equations Appl., Birkhäuser, Basel, 2003, 349–358 | MR | Zbl

[4] O. G. Smolyanov, Kh. f. Vaitszekker, O. Vittikh, “Diffuziya na kompaktnom rimanovom mnogoobrazii i poverkhnostnye mery”, Dokl. RAN, 371:4 (2000), 442–447 | MR | Zbl

[5] O. G. Smolyanov, Kh. f. Vaitszekker, O. Vittikh, N. A. Sidorova, “Poverkhnostnye mery na traektoriyakh v rimanovykh mnogoobraziyakh, porozhdaemye diffuziyami”, Dokl. RAN, 377:4 (2001), 441–446 | MR | Zbl

[6] O. G. Smolyanov, Kh. f. Vaitszekker, O. Vittikh, N. A. Sidorova, “Poverkhnostnye mery Vinera na traektoriyakh v rimanovykh mnogoobraziyakh”, Dokl. RAN, 383:4 (2002), 458–463 | MR | Zbl

[7] R. P. Chernoff, “Note on product formulas for operator semigroups”, J. Funct. Anal., 2:2 (1968), 238–242 | DOI | MR | Zbl

[8] R. P. Chernoff, Product Formulas, Nonlinear Semigroups and Addition of Unbounded Operators, Mem. Amer. Math. Soc., 140, Amer. Math. Soc., Providence, RI, 1974 | MR | Zbl

[9] E. Nelson, “Feynman Integrals and the Schrödinger Equation”, J. Math. Phys., 5:3 (1964), 332–343 | DOI | MR | Zbl

[10] N. A. Sidorova, “The Smolyanov surface measure on trajectories in a Riemannian manifold”, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 7:3 (2004), 461–471 | DOI | MR | Zbl

[11] Dzh. Nesh, “$C_1$-izometricheskie vlozheniya”, Matematika. Sb. per., 1:2 (1957), 3–16 | MR | Zbl

[12] Dzh. Nesh, “Problema vlozhenii dlya rimanovykh mnogoobrazii”, UMN, 26:4 (1971), 173–216 | MR | Zbl

[13] L. Andersson, B. K. Driver, “Finite-dimensional approximations to Wiener measure and path integral formulas on manifolds”, J. Func. Anal., 165:2 (1999), 430–498 | DOI | MR | Zbl

[14] B. K. Driver, A Primer on Riemannian Geometry and Stochastic Analysis on Path Spaces, http://math.ucsd.edu/\allowbreakd̃river/DRIVER/Preprints/stochastic_geometry.htm

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

[16] O. G. Smolyanov, A. Trumen, “Integraly Feinmana po traektoriyam v rimanovykh mnogoobraziyakh”, Dokl. RAN, 392:2 (2003), 174–179 | MR | Zbl

[17] 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

[18] K. D. Elworthy, Stochastic Differential Equations on Manifolds, London Math. Soc. Lecture Note Ser., 70, Cambridge Univ. Press, Cambridge, 1982 | MR | Zbl

[19] Ya. A. Butko, “Representations of the solution of the Cauchy–Dirichlet problem for the heat equation in a domain on a compact Riemannian manifold by functional integrals”, Russ. J. Math. Phys., 11:2 (2004), 121–129 | MR | Zbl