Chernoff approximations of Feller semigroups in Riemannian manifolds. (English) Zbl 1529.58014
Sufficient conditions are presented for second order elliptic operators on manifolds of bounded geometry to generate Feller semigroups. Moreover, Chernoff approximations are constructed for these Feller semigroups in terms of shift operators, which in turn to provide approximations of solutions to the associated parabolic equations, and also yields the weak convergence of a sequence of random walks on the manifolds to the generated diffusion processes. In particular, the Brownian motion on parallelizable manifolds is represented as the weak limit of the corresponding random walks.
Reviewer: Feng-Yu Wang (Tianjin)
MSC:
| 58J65 | Diffusion processes and stochastic analysis on manifolds |
| 58A05 | Differentiable manifolds, foundations |
| 35K15 | Initial value problems for second-order parabolic equations |
Keywords:
Chernoff product formula; diffusion processes; evolution equations; Feller semigroups; Feynman formula; Feynman-Kac formula; one-parameter operator semigroupsReferences:
| [1] | L.Andersson and B. K.Driver, Finite dimensional approximations to Wiener measure and path integral formulas on manifolds, J. Funct. Anal.165 (1999), no. 2, 430-498. · Zbl 0943.58024 |
| [2] | P.Baldi, Stochastic calculus. An introduction through theory and exercises, Springer, Cham2017. · Zbl 1382.60001 |
| [3] | C.Bär and F.Pfäffle, Path integrals on manifolds by finite dimensional approximation, J. Reine Angew. Math.625 (2008), 29-57. · Zbl 1165.58015 |
| [4] | B.Baur, F.Conrad, and M.Grothaus, Smooth contractive embeddings and application to Feynman formula for parabolic equations on smooth bounded domains, Comm. Stat. Theory Methods40 (2011), no. 19-20, 3452-3464. · Zbl 1278.47047 |
| [5] | P.Billingsley, Convergence of probability measures. 2nd ed., Wiley, New York, 1999. · Zbl 0172.21201 |
| [6] | V. I.Bogachev and O. G.Smolyanov, Real and functional analysis, vol. 4, Springer Nature, London2020. · Zbl 1466.26002 |
| [7] | L. A.Borisov, Y. N.Orlov, and V. Z.Sakbaev, Feynman averaging of semigroups generated by Schrödinger operators, Infinite Dimens. Anal. Quantum Probab. Relat. Top.21 (2018), no. 02, 1850010. · Zbl 1391.35338 |
| [8] | Y. A.Butko, Function integrals corresponding to a solution of the Cauchy‐Dirichlet problem for the heat equation in a domain of a Riemannian manifold, J. Math. Sci.151 (2008), no. 1, 2629-2638. · Zbl 1151.35375 |
| [9] | Y. A.Butko, Chernoff approximation for semigroups generated by killed Feller processes and Feynman formulae for time‐fractional Fokker-Planck-Kolmogorov equations, Fract. Calc. Appl. Anal.21 (2018), no. 5, 1203-1237. · Zbl 1422.35162 |
| [10] | Y. A.Butko, Chernoff approximation of subordinate semigroups, Stoch. Dyn.18 (2018), no. 03, 1850021. · Zbl 06871300 |
| [11] | Y. A.Butko, The method of Chernoff approximation, Conf. Semigroups Operators: Theory and Applications, Springer, Cham, 2020, pp. 19-46. · Zbl 1501.47063 |
| [12] | Y. A.Butko, R. L.Schilling, and O. G.Smolyanov, Feynman formulae for Feller semigroups, Dokl. Math.82 (2010), no. 2, 697-683. · Zbl 1213.47047 |
| [13] | Y. A.Butko, R. L.Schilling, and O. G.Smolyanov, Lagrangian and Hamiltonian Feynman formulae for some Feller semigroups and their perturbations, Infin. Dimens. Anal. Quantum Probab. Relat. Top.15 (2012), no. 03, 1250015. · Zbl 1273.47070 |
| [14] | P. R.Chernoff, Note on product formulas for operator semigroups, J. Funct. Anal.2 (1968), no. 2, 238-242. · Zbl 0157.21501 |
| [15] | A.Debiard, B.Gaveau, and E.Mazet, Théoremes de comparaison en géométrie riemannienne, Publ. Res. Inst. Math. Sci.12 (1976), no. 2, 391-425. · Zbl 0382.31007 |
| [16] | M.Disconzi, Y.Shao, and G.Simonett, Some remarks on uniformly regular Riemannian manifolds, Math. Nachr.289 (2016), no. 2‐3, 232-242. · Zbl 1342.53055 |
| [17] | M. P.doCarmo, Riemannian Geometry, 1992, Birkhäuser, Boston, M.A., 1999. |
| [18] | B. K.Driver and J. S.Semko, Controlled rough paths on manifolds I, Rev. Mat. Iberoam.33 (2017), no. 3, 885-950. · Zbl 1382.58029 |
| [19] | K. D.Elworthy, Stochastic differential equations on manifolds, vol. 70, Cambridge University Press, Cambridge, 1982. · Zbl 0514.58001 |
| [20] | K.‐J.Engel and R.Nagel, One‐parameter semigroups for linear evolution equations, Springer, Berlin, Heidelberg, 2000. · Zbl 0952.47036 |
| [21] | S. N.Ethier and T. G.Kurtz, Markov processes: characterization and convergence, vol. 282, Wiley, New York, 2009. |
| [22] | R. P.Feynman, Space‐time approach to non‐relativistic quantum mechanics, Rev. Modern Phys.20 (1948), no. 2, 367-387. · Zbl 1371.81126 |
| [23] | R. P.Feynman, An operator calculus having applications in quantum electrodynamics, Phys. Rev.84 (1951), no. 1, 108-128. · Zbl 0044.23304 |
| [24] | M.Gordina and T.Laetsch, A convergence to Brownian motion on sub‐Riemannian manifolds, Trans. Amer. Math. Soc.369 (2017), no. 9, 6263-6278. · Zbl 1372.60119 |
| [25] | B.Güneysu, Covariant schrödinger semigroups on noncompact Riemannian manifolds, Oper. Theory Adv. Appl.264 (2017). · Zbl 1422.47048 |
| [26] | L.Hörmander, The analysis of linear partial differential operators I: distribution theory and Fourier analysis, Springer, Cham, 2015. |
| [27] | E. P.Hsu, Stochastic analysis on manifolds, vol. 38, Amer. Math. Soc., Providence, R.I., 2002. · Zbl 0994.58019 |
| [28] | N.Ikeda and S.Watanabe, Stochastic differential equations and diffusion processes, Elsevier, New York, 2014. |
| [29] | E.Jørgensen, The central limit problem for geodesic random walks, Z. Wahrscheinlichkeitstheorie Verw. Geb.32 (1975), no. 1-2, 1-64. · Zbl 0292.60103 |
| [30] | J.Jost, Riemannian geometry and geometric analysis, vol. 42005, Springer, Berlin, Heidelberg, 2008. · Zbl 1143.53001 |
| [31] | M.Kac, On distributions of certain Wiener functionals, Trans. Amer. Math. Soc.65 (1949), no. 1, 1-13. · Zbl 0032.03501 |
| [32] | O.Kallenberg, Foundations of Modern Probability, 2nd ed., Springer‐Verlag, Berlin, Heidelberg, 2002. · Zbl 0996.60001 |
| [33] | S.Kobayashi and K.Nomizu, Foundations of differential geometry. vol 1, Wiley, New York, 2009. |
| [34] | V. N.Kolokoltsov, Markov processes, semigroups and generators, De Gruyter, Berlin, 2011. · Zbl 1220.60003 |
| [35] | Y. A.Kordyukov, \({L}^p\)-theory of elliptic differential operators on manifolds of bounded geometry, Acta Appl. Math.23 (1991), no. 3, 223-260. · Zbl 0743.58030 |
| [36] | J. M.Lee, Introduction to Riemannian manifolds, Springer, Cham, 2018. · Zbl 1409.53001 |
| [37] | X.‐M.Li, Properties at infinity of diffusion semigroups and stochastic flows via weak uniform covers, Potential Anal.3 (1994), no. 4, 339-357. · Zbl 0864.58061 |
| [38] | J. H.Manton, A primer on stochastic differential geometry for signal processing, IEEE J. Sel. Top. Signal Process.7 (2013), no. 4, 681-699. |
| [39] | S. A.Molchanov, Diffusion processes and Riemannian geometry, Russian Math. Surv.30 (1975), no. 1, 3-59. · Zbl 0315.53026 |
| [40] | V.Moretti, Spectral theory and quantum mechanics: mathematical foundations of quantum theories, symmetries and introduction to the algebraic formulation, vol. 110, Springer, Cham, 2018. |
| [41] | B.O’neill, Semi‐Riemannian geometry with applications to relativity, Academic Press, San Diego, C.A., 1983. · Zbl 0531.53051 |
| [42] | M. A.Pinsky, Isotropic transport process on a Riemannian manifold, Trans. Amer. Math. Soc.218 (1976), 353-360. · Zbl 0341.60041 |
| [43] | M.Reed and B.Simon, Methods of Modern Mathematical Physics II: Fourier Analysis, Self‐Adjointness, Academic Press, San Diego, C.A. 1975. · Zbl 0308.47002 |
| [44] | I. D.Remizov, Quasi‐Feynman formulas-a method of obtaining the evolution operator for the Schrödinger equation, J. Funct. Anal.270 (2016), no. 12, 4540-4557. · Zbl 1337.81053 |
| [45] | I.Remizov, New method for constructing Chernoff functions, Differ. Equations53 (2017), no. 4, 566-570. · Zbl 06763546 |
| [46] | I. D.Remizov, Approximations to the solution of Cauchy problem for a linear evolution equation via the space shift operator (second‐order equation example), Appl. Math. Comput.328 (2018), 243-246. · Zbl 1427.35094 |
| [47] | I. D.Remizov, Solution‐giving formula to Cauchy problem for multidimensional parabolic equation with variable coefficients, J. Math. Phys.60 (2019), no. 7, 071505. · Zbl 1418.35212 |
| [48] | I. D.Remizov, Formulas that represent Cauchy problem solution for momentum and position Schrödinger equation, Potential Anal.52 (2020), no. 3, 339-370. · Zbl 1433.81084 |
| [49] | L.Rogers and D.Williams, Diffusions, Markov Processes and Martingales, Wiley, New York, 1987. · Zbl 0627.60001 |
| [50] | K.Schmüdgen, Unbounded self‐adjoint operators on Hilbert space, vol. 265, Springer Science & Business Media, New York, 2012. · Zbl 1257.47001 |
| [51] | M. A.Shubin, Spectral theory of elliptic operators on noncompact manifolds, Astérisque207 (1992), no. 5, 35-108. · Zbl 0793.58039 |
| [52] | O. G.Smolyanov, Feynman formulae for evolutionary equations, Trends Stochastic Anal.353 (2009), 283-302. · Zbl 1173.58006 |
| [53] | O. G.Smolyanov, Schrödinger type semigroups via Feynman formulae and all that, Proc. Quantum Bio‐Informatics V, Tokyo University of Science, Japan, 7-12, March 2011, World Scientific, Singapore, 2011, pp. 301-313. · Zbl 1354.47030 |
| [54] | O.Smolyanov, A.Tokarev, and A.Truman, Hamiltonian Feynman path integrals via the Chernoff formula, J. Math. Phys.43 (2002), no. 10, 5161-5171. · Zbl 1060.58009 |
| [55] | O.Smolyanov, H.vonWeizsäcker, and O.Wittich, Brownian motion on a manifold as a limit of stepwise conditioned standard Brownian motions, Stochastic Processes, Physics and Geometry: New Interplays II (Proc. Conf. Infinite Dimensional (Stochastic) Analysis and Quantum Physics, Leipzig, Germany, January 18-22, 1999), CMS Conf. Proc., vol. 29, Amer. Math. Soc., pp. 589-602. · Zbl 0978.58015 |
| [56] | O.Smolyanov, O.Wittich, and H. V.Weizsäcker, Diffusion on compact Riemannian manifolds and surface measures, Dokl. Math.61 (2000), no. 2, 230-234. · Zbl 1047.58006 |
| [57] | O. G.Smolyanov, H. v.Weizsäcker, and O.Wittich, Chernoff’s theorem and discrete time approximations of Brownian motion on manifolds, Potential Anal.26 (2007), no. 1, 1-29. · Zbl 1107.58014 |
| [58] | H. F.Trotter, Approximation of semi‐groups of operators, Pacific J. Math.8 (1958), no. 4, 887-919. · Zbl 0099.10302 |
| [59] | A.Vedenin et al. Speed of convergence of Chernoff approximations to solutions of evolution equations, Math. Notes108 (2020), no. 3, 451-456. · Zbl 1507.47095 |
| [60] | F. Y.Wang, Analysis for diffusion processes on Riemannian manifolds, vol. 18, World Scientific, Singapore, 2014. · Zbl 1296.58001 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
