Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Nonautonomous Kato classes of measures and Feynman-Kac propagators


Author: Archil Gulisashvili
Journal: Trans. Amer. Math. Soc. 357 (2005), 4607-4632
MSC (2000): Primary 35K15; Secondary 60H30
DOI: https://doi.org/10.1090/S0002-9947-04-03603-7
Published electronically: December 9, 2004
MathSciNet review: 2156723
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The behavior of the Feynman-Kac propagator corresponding to a time-dependent measure on $R^n$ is studied. We prove the boundedness of the propagator in various function spaces on $R^n$, and obtain a uniqueness theorem for an exponentially bounded distributional solution to a nonautonomous heat equation.


References [Enhancements On Off] (What's this?)

  • [AS] M. Aizenman and B. Simon, Brownian motion and Harnack's inequality for Schrödinger operators, Comm. Pure Appl. Math. 35 (1982), 209-271. MR 84a:35062
  • [AM] S. Albeverio and Zhi-Ming Ma, Perturbation of Dirichlet forms-lower semiboundedness, closability and form cores, J. Functional Analysis 99 (1991), 332-356. MR 92i:47039
  • [A] D. G. Aronson, Non-negative solutions of linear parabolic equations, Ann. Scuola Norm. Sup. Pisa 22 (1968), 607-694. MR 55:8553
  • [BM1] P. Blanchard and Zhi-Ming Ma, Semigroups of Schrödinger operators with potentials given by Radon measures, In: Stochastic Processes-Physics and Geometry (eds. S. Albeverio, et al.), World Scientific, Singapore, 1989.
  • [BM2] P. Blanchard and Zhi-Ming Ma, New results on the Schrödinger semigroups with potentials given by signed smooth measures, In: Proc. Silvri Workshop (eds. Korezlioglu, et al.), Lecture Notes in Math. 1444, Springer Verlag, Berlin and New York, 1990. MR 91m:35067
  • [DvC] M. Demuth and J. A. van Casteren, Stochastic Spectral Theory for Selfadjoint Feller Operators. A functional integration approach, Birkhäuser Verlag, Basel, 2000. MR 2002d:47066
  • [FLP] D. Feyel and A. de La Pradelle, Étude de l'équation $1/2\Delta-u\mu=0$$\mu$ est une mesure positive, Ann. Inst. Fourier, Grenoble 38 (1988), 199-218. MR 90j:35074
  • [F] M. Fukushima, Dirichlet Forms and Markov Processes, North-Holland, Amsterdam, 1980. MR 81f:60105
  • [Ge] R. K. Getoor, Measure perturbations of Markovian semigroups, Potential Analysis 11 (1999), 101-133. MR 2001c:60119
  • [Gu1] A. Gulisashvili, Sharp estimates in smoothing theorems for Schrödinger semigroups, J. Functional Analysis 170 (2000), 161-187. MR 2001i:47069
  • [Gu2] 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 2003g:60133
  • [Gu3] A. Gulisashvili, On the heat equation with a time-dependent singular potential, J. Funct. Anal. 194 (2002), no. 1, 17-52. MR 2003i:35117
  • [GK] A. Gulisashvili and M. A. Kon, Exact smoothing properties of Schrödinger semigroups, Amer. J. Math. 118 (1996), 1215-1248. MR 97m:47053
  • [H] W. Hansen, Harnack inequalities for Schrödinger operators, Ann. Scuola Norm. Sup. Pisa XVIII (1999), 413-470. MR 2001b:31008
  • [JL] G. W. Johnson and M. Lapidus, The Feynman Integral and Feynman's Operational Calculus, Clarendon Press, Oxford, 2000. MR 2001i:58015
  • [LP] A. de La Pradelle, Sur les perturbations d'espaces harmonique, Bull. Classe Sci. 6-9 (1990), 201-212. MR 93d:31010
  • [LS] V. Liskevich and Y. Semenov, Estimates for fundamental solutions of second-order parabolic equations, J. London Math. Soc. 62 (2000), 521-543. MR 2002j:35132
  • [N] M. Nagasawa, Stochastic Processes in Quantum Physics, Birkhäuser, Basel, 2000. MR 2001g:60148
  • [QZ1] Qi Zhang, On a parabolic equation with a singular lower order term, Transactions Amer. Math. Soc. 348 (1996), 2811-2844. MR 96k:35073
  • [QZ2] 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 98m:35079
  • [RRSV] F. Räbiger, A. Rhandi, R. Schnaubelt, and J. Voigt, Non-autonomous Miyadera perturbation, Differential Integral Equations 13 (2000), 341-368. MR 2002b:34095
  • [SV] R. Schnaubelt and J. Voigt, The non-autonomous Kato class, Arch. Math. 72 (1999), 454-460. MR 2000h:35058
  • [S] B. Simon, Schrödinger semigroups, Bull. Amer. Math. Soc. 7 (1982), 445-526. MR 86b:81001a
  • [T] A. N. Tikhonov, Théorèmes d'unicité pour l'equation de la chaleur, Mat. Sbornik 42 (1935), 199-216.
  • [Ti] E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals, Oxford University Press, London, 1937.
  • [V1] J. Voigt, Absorption semigroups, their generators, and Schrödinger semigroups, J. Functional Analysis 67 (1986), 167-205. MR 88a:81036
  • [V2] J. Voigt, Absorption semigroups, Feller property, and Kato class, In: Operator Theory: Advances and Applications, 78, Birkhäuser, 1995, 389-396. MR 96j:47039
  • [Z] T. S. Zhang, Generalized Feynman-Kac semigroups, associated quadratic forms and asymptotic properties, Potential Analysis 14 (2001), 387-408. MR 2002d:31012

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 35K15, 60H30

Retrieve articles in all journals with MSC (2000): 35K15, 60H30


Additional Information

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

DOI: https://doi.org/10.1090/S0002-9947-04-03603-7
Keywords: Nonautonomous heat equation, classes of time-dependent measures, Feynman-Kac propagators, time-dependent additive functionals
Received by editor(s): October 10, 2003
Received by editor(s) in revised form: December 22, 2003
Published electronically: December 9, 2004
Article copyright: © Copyright 2004 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society