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.
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References
O. G. Smolyanov, H. v. Weizsäcker, and O. Wittich, “Brownian motion on a manifold as limit of stepwise conditioned standard Brownian motions,” in Stochastic Processes, Physics, and Geometry: New Interplays, CMS Conf. Proc., II, Leipzig, 1999 (Amer. Math. Soc., Providence, RI, 2000), Vol. 29, pp. 589–602.
O. G. Smolyanov, H. v. Weizsäcker, and O. Wittich, Chernoff’s Theorem and Discrete Time Approximations of Brownian Motion on Manifolds, arXiv: math/0409155.
O. G. Smolyanov, H. v. Weizsäcker, and O. Wittich, “Chernoff’s Theorem and the construction of semigroups,” in Evolution Equations: Applications to Physics, Industry, Life Sciences, and Economics, Progr. Nonlinear Differential Equations Appl., Levico Terme, 2000 (Birkhäuser, Basel, 2003), pp. 349–358.
O. G. Smolyanov, H. v. Weizsäcker, and O. Wittich, “Diffusion on compact Riemannian manifolds and surface measures,” Dokl. Ross. Akad. Nauk 371(4), 442–447 (2000) [Russian Acad. Sci. Dokl. Math. 61 (2), 230–234 (2000)].
O. G. Smolyanov, H. v. Weizsäcker, O. Wittich, and N. A. Sidorova, “Surface measures on trajectories in Riemannian manifolds generated by diffusions,” Dokl. Ross. Akad. Nauk 377(4), 441–446 (2001) [Russian Acad. Sci. Dokl. Math. 63 (2), 203–207 (2001)].
O. G. Smolyanov, H. v. Weizsäcker, O. Wittich, and N. A. Sidorova, “Wiener surface measures on trajectories in Riemannian manifolds,” on the trajectories in the Riemannian manifolds,” Dokl. Ross. Akad. Nauk 383(4), 458–463 (2002) [Russian Acad. Sci. Dokl. Math. 65 (2), 239–244 (2002)].
R. P. Chernoff, “Note on product formulas for operator semigroups,” J. Funct. Anal. 2(2), 238–242 (1968).
R. P. Chernoff, Product Formulas, Nonlinear Semigroups, and Addition of Unbounded Operators, in Mem. Amer. Math. Soc. (Amer. Math. Soc., Providence, RI, 1974), Vol. 140.
E. Nelson, “Feynman Integrals and the Schrödinger Equation,” J. Math. Phys. 5(3), 332–343 (1964).
N. A. Sidorova, “The Smolyanov surface measure on trajectories in a Riemannian manifold,” Infin. Dimens. Anal. Quantum Probab. Relat. Top. 7(3), 461–471 (2004).
J. Nash, “C 1 isometric embeddings,” [J] Ann. Math. 60(2), 383–396 (1954).
J. Nash, “The embedding problem for Riemannian manifolds,” Uspekhi Mat. Nauk 26(4), 173–216 (1971).
L. Andersson and B. K. Driver, “Finite-dimensional approximations to Wiener measure and path integral formulas on manifolds,” J. Func. Anal. 165(2), 430–498 (1999).
B. K. Driver, A Primer on Riemannian Geometry and Stochastic Analysis on Path Spaces, http://math.ucsd.edu/:_driver/DRIVER/Preprints/stochastic_geometry.htm.
S. Watanabe and N. Ikeda, Stochastic Differential Equations and Diffusion Processes (North-Holland, Amsterdam-Oxford-New York, 1981; Nauka, Moscow, 1986).
O. G. Smolyanov and A. Truman, “Feynman integrals over trajectories in Riemannian manifolds,” Dokl. Ross. Akad. Nauk 392(2), 174–179 (2003).
O. G. Smolyanov, A. G. Tokarev, and A. Truman, “Hamiltonian Feynman path integrals via Chernoff formula,” J. Math. Phys. 43(10), 5161–5171 (2002).
K. D. Elworthy, Stochastic Differential Equations on Manifolds, in London Math. Soc. Lecture Note Ser. (Cambridge Univ. Press, Cambridge, 1982), Vol. 70.
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), 121–129 (2004).
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Original Russian Text © Y. A. Butko, 2008, published in Matematicheskie Zametki, 2008, Vol. 83, No. 3, pp. 333–349.
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Butko, Y.A. Feynman formulas and functional integrals for diffusion with drift in a domain on a manifold. Math Notes 83, 301–316 (2008). https://doi.org/10.1134/S0001434608030024
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DOI: https://doi.org/10.1134/S0001434608030024