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Classes of time-dependent measures, non-homogeneous Markov processes, and Feynman-Kac propagators


Author: Archil Gulisashvili
Journal: Trans. Amer. Math. Soc. 360 (2008), 4063-4098
MSC (2000): Primary 47D08; Secondary 60J35
DOI: https://doi.org/10.1090/S0002-9947-08-04492-9
Published electronically: March 11, 2008
MathSciNet review: 2395164
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Abstract: We study the inheritance of properties of free backward propagators associated with transition probability functions by backward Feynman-Kac propagators corresponding to functions and time-dependent measures from non-autonomous Kato classes. The inheritance of the following properties is discussed: the strong continuity of backward propagators on the space $ L^r$, the $ (L^r-L^q)$-smoothing property of backward propagators, and various generalizations of the Feller property. We also prove that a propagator on a Banach space is strongly continuous if and only if it is separately strongly continuous and locally uniformly bounded.


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  • [1] M. Aizenman and B. Simon, Brownian motion and Harnack's inequality for Schrödinger operators, Comm. Pure Appl. Math. 35 (1982), 209-271. MR 644024 (84a:35062)
  • [2] S. Albeverio, Ph. Blanchard, and Z. M. Ma, Feynman-Kac semigroups in terms of signed smooth measures, in: Proc. Random Partial Differential Equations, Internat. Series of Numerical Math. 102, Birkhäuser, Boston, 1991, 1-31. MR 1185735 (93i:60140)
  • [3] S. Albeverio and Z. M. Ma, Additive functionals, nowhere Radon and Kato class smooth measures associated with Dirichlet forms, Osaka J. Math. 29 (1991), 247-265. MR 1173989 (93m:60158)
  • [4] Ph. Blanchard and Z. M. Ma, Semigroup of Schrödinger operators with potentials given by Radon measures, in: Stochastic Processes, Physics and Geometry (eds. S. Albeverio, et al.), World Scientific, Singapore, 1989. MR 1124210 (93a:35034)
  • [5] Ph. Blanchard and Z. M. Ma, New results on the Schrödinger semigroups with potentials given by signed smooth measures, Proc. Silvri Workshop (eds. Korezlioglu, et al.), Lecture Notes in Math. 1444, Springer-Verlag, Basel, 1990. MR 1078851 (91m:35067)
  • [6] K. L. Chung and Z. Zhao, From Brownian Motion to Schrödinger's Equation, Springer-Verlag, Berlin, 1995. MR 1329992 (96f:60140)
  • [7] M. Demuth and J. A. van Casteren, Stochastic Spectral Theory for Selfadjoint Feller Operators. A functional integration approach, Birkhäuser Verlag, Basel, 2000. MR 1772266 (2002d:47066)
  • [8] E. B. Dynkin, Theory of Markov Processes, Pergamon Press, Oxford-London-Paris, 1961. MR 0131900 (24:A1747)
  • [9] E. B. Dynkin, Superprocesses and partial differential equations, Ann. Probab. 21 (1993), 1185-1262. MR 1235414 (94j:60156)
  • [10] E. B. Dynkin, Diffusions, Superdiffusions, and Partial Differential Equaitons, American Mathematical Society Colloquium Publications 50, Amer. Math. Soc., Providence, RI, 2002. MR 1883198 (2003c:60001)
  • [11] S. D. Eidelman, Parabolic equations, in: Partial Differential Equations VI, Elliptic and Parabolic Operators, Yu. V. Egorov, M. A. Shubin (Eds.), Springer-Verlag, Berlin, 1994, pp. 203-325. MR 1313734 (95i:35002)
  • [12] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice Hall, Englewood Cliffs, NJ, 1964. MR 0181836 (31:6062)
  • [13] M. Fukushima, Dirichlet Forms and Markov Processes, North-Holland, Amsterdam, 1980. MR 569058 (81f:60105)
  • [14] R. K. Getoor, Measure perturbations of Markovian semigroups, Potential Analysis 11 (1999), 101-133. MR 1703827 (2001c:60119)
  • [15] I. I. Gihman and A. V. Skorokhod, The Theory of Stochastic Processes II, Springer-Verlag, Berlin, 1975. MR 0375463 (51:11656)
  • [16] A. Gulisashvili, Classes of time-dependent measures and the behavior of Feynman-Kac propagators, C. R. Acad. Sci. Paris, Ser. I 334 (2002), 1-5. MR 1890631 (2003g:60133)
  • [17] A. Gulisashvili, On the heat equation with a time-dependent singular potential, J. Funct. Anal. 194 (2002), 17-52. MR 1929138 (2003i:35117)
  • [18] A. Gulisashvili, Nonautonomous Kato classes and the behavior of backward Feynman-Kac propagators, in: Analyse stochastique et theorie du potentiel, Saint Priest de Gimel (2002), Association Laplace-Gauss, Paris, 2003.
  • [19] A. Gulisashvili, Free propagators and Feynman-Kac propagators, Seminar of Mathematical Analysis: proceedings, Universities of Málaga and Seville (Spain), September 2003-June 2004, Daniel Girela Álvarez, Genaro López Acedo, Rafael Villa Caro (eds.), Universidad de Sevilla, 2004. MR 2117061
  • [20] A. Gulisashvili, Markov processes and Feynman-Kac propagators, Centre de Recerca Matematica, Preprint No. 573, Bellaterra (Barcelona), Spain, April 2004, 43 pp. MR 2117061
  • [21] A. Gulisashvili, Nonautonomous Kato classes of measures and Feynman-Kac propagators, Trans. Amer. Math. Soc. 357 (2005), 4607-4632. MR 2156723 (2007f:35138)
  • [22] A. Gulisashvili and J. A. van Casteren, Feynman-Kac propagators and viscosity solutions, J. Evol. Equ. 5 (2005), 105-121. MR 2125408 (2007j:47092)
  • [23] A. M. Il'in, A. S. Kalashnikov, and O. A. Oleinik, Linear equations of the second order of parabolic type, Russian Math. Surveys 17 (1962), 1-143. MR 0138888 (25:2328)
  • [24] G. W. Johnson and M. Lapidus, The Feynman Integral and Feynman's Operational Calculus, Clarendon Press, Oxford, 2000. MR 1771173 (2001i:58015)
  • [25] T. Kato, Schrödinger operators with singular potentials, Israel J. Math. 13 (1973), 135-148. MR 0333833 (48:12155)
  • [26] J. L. Kelley, General Topology, Van Nostrand Reinhold Company, New York, 1955. MR 0070144 (16:1136c)
  • [27] A. N. Kolmogorov, Über die Analytischen Methoden in der Wahrscheinlichkeitsrechnung, Math. Ann. 104 (1931), 415-458. English translation: On analytical methods in probability theory, in: Selected Works of A. N. Kolmogorov, Vol. II, Kluwer Academic Publishers, Dordrecht, Netherlands, 1992, pp. 62-108. MR 1512678
  • [28] S. E. Kuznetsov, Nonhomogeneous Markov processes, Sovremennye Problemy Matematiki 20, VINITI, Moscow, (1982), 37-178 (in Russian). English translation in: J. Soviet Math. 25 (1984), 1380-1498. MR 716331 (85f:60104)
  • [29] V. Liskevich and Y. Semenov, Estimates for fundamental solutions of second-order parabolic equations, J. London Math. Soc. 62 (2000), 521-543. MR 1783642 (2002j:35132)
  • [30] V. Liskevich, H. Vogt, and J. Voigt, Gaussian bounds for propagators perturbed by potentials, J. Funct. Anal. 238 (2006), 245-277. MR 2253015 (2007g:35078)
  • [31] M. Nagasawa, Stochastic Processes in Quantum Physics, Birkhäuser, Basel, 2000. MR 1739699 (2001g:60148)
  • [32] E. M. Ouhabaz, P. Stollmann, K.-Th. Sturm, and J. Voigt, The Feller property for absorption semigroups, J. Funct. Anal. 138 (1996), 351-378. MR 1395962 (97j:47061)
  • [33] F. O. Porper and S. D. Eidelman, Two-sided estimates of the fundamental solutions of second-order parabolic equations, and some applications, Uspekhi Mat. Nauk 39 (1984), 107-156 (in Russian). English translation in: Russian Math. Surveys 39 (1984), 119-178. MR 747792 (86b:35078)
  • [34] F. Räbiger, A. Rhandi, R. Schnaubelt, and J. Voigt, Non-autonomous Miyadera perturbation, Differential Integral Equations 13 (2000), 341-368. MR 1811962 (2002b:34095)
  • [35] D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, Springer-Verlag, Berlin, 1999. MR 1725357 (2000h:60050)
  • [36] R. Schnaubelt and J. Voigt, The non-autonomous Kato class, Arch. Math. 72 (1999), 454-460. MR 1687500 (2000h:35058)
  • [37] B. Simon, Schrödinger semigroups, Bull. Amer. Math. Soc. 7 (1982), 445-526. MR 670130 (86b:81001a)
  • [38] K.-Th. Sturm, Harnack's inequality for parabolic operators with singular low order terms, Math. Z. 216 (1994), 593-611. MR 1288047 (95g:35027)
  • [39] J. Voigt, Absorption semigroups, their generators, and Schrödinger semigroups, J. Funct. Anal. 67 (1986), 167-205. MR 845197 (88a:81036)
  • [40] J. Voigt, Absorption semigroups, Feller property, and Kato class, Operator Theory: Advances and Applications, 78, Birkhäuser, 1995, 389-396. MR 1365353 (96j:47039)
  • [41] J. Yeh, Martingales and Stochastic Analysis, World Scientific, Singapore, 1995. MR 1412800 (97j:60002)
  • [42] Qi Zhang, On a parabolic equation with a singular lower order term, Trans. Amer. Math. Soc. 348 (1996), 2811-2844. MR 1360232 (96k:35073)
  • [43] Qi Zhang, On a parabolic equation with a singular lower order term, Part 2: The Gaussian bounds, Indiana Univ. Math. J. 46 (1997), 989-1020. MR 1488344 (98m:35079)
  • [44] Qi Zhang, A sharp comparison result concerning Schrödinger heat kernels, Bull. London Math. Soc., 35 (2003), 461-472. MR 1978999 (2004b:35135)
  • [45] T. S. Zhang, Generalized Feynman-Kac semigroups, associated quadratic forms and asymptotic properties, Potential Analysis, 14 (2001), 387-408. MR 1825693 (2002d:31012)

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Additional Information

Archil Gulisashvili
Affiliation: Department of Mathematics, Ohio University, Athens, Ohio 45701
Email: guli@math.ohiou.edu

DOI: https://doi.org/10.1090/S0002-9947-08-04492-9
Keywords: Propagators and backward propagators, non-homogeneous Markov processes, non-autonomous Kato classes, free propagators, Feynman-Kac propagators, the Feller property, the inheritance problem.
Received by editor(s): March 27, 2006
Published electronically: March 11, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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