K. L. Chung and M. Rao :
On existence of a dual process .
Technical report 25 ,
Mathematics Institute, Aarhus University ,
1978 .
23 pages.
Zbl
0392.60052
techreport

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BibTeX
@techreport {key0392.60052z,
AUTHOR = {Chung, K. L. and Rao, Murali},
TITLE = {On existence of a dual process},
NUMBER = {25},
INSTITUTION = {Mathematics Institute, Aarhus University},
YEAR = {1978},
URL = {http://books.google.com/books/about/On_Existence_of_a_Dual_Process.html?id=u7ZVHQAACAAJ},
NOTE = {23 pages. Zbl:0392.60052.},
}
K. L. Chung and M. Rao :
Potential theory without duality .
Technical report 22 ,
Mathematics Institute, Aarhus University ,
1978 .
15 pages.
Zbl
0369.31006
techreport

People
BibTeX
@techreport {key0369.31006z,
AUTHOR = {Chung, K. L. and Rao, Murali},
TITLE = {Potential theory without duality},
NUMBER = {22},
INSTITUTION = {Mathematics Institute, Aarhus University},
YEAR = {1978},
URL = {http://books.google.com/books/about/Potential_Theory_Without_Duality.html?id=WGEDGwAACAAJ},
NOTE = {15 pages. Zbl:0369.31006.},
}
K. L. Chung and K. M. Rao :
“Sur la théorie du potentiel avec la fonctionnelle de Feynman–Kac ,”
C. R. Acad. Sci. Paris Sér. A-B
290 : 13
(1980 ),
pp. A629–A631 .
MR
572652
Zbl
0439.31005
article

People
BibTeX
@article {key572652m,
AUTHOR = {Chung, Kai Lai and Rao, K. Murali},
TITLE = {Sur la th\'eorie du potentiel avec la
fonctionnelle de {F}eynman--{K}ac},
JOURNAL = {C. R. Acad. Sci. Paris S\'er. A-B},
FJOURNAL = {Comptes Rendus Hebdomadaires des S\'eances
de l'Acad\'emie des Sciences. S\'eries
A et B},
VOLUME = {290},
NUMBER = {13},
YEAR = {1980},
PAGES = {A629--A631},
NOTE = {MR:572652. Zbl:0439.31005.},
ISSN = {0151-0509},
CODEN = {CHASAP},
}
K. L. Chung and M. Rao :
“Equilibrium and energy ,”
Probab. Math. Statist.
1 : 2
(1980 ),
pp. 99–108 .
MR
626304
Zbl
0502.60060
article

Abstract
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In this paper it is shown that the equilibrium measure \( \nu \) for a compact \( K \) in potential theory can be related with a unique invariant measure \( \pi \) for a discrete time Markov process by the formula
\[ \pi(dy) = \phi(y)\,\nu(dy) .\]
The chain has the transition function \( L(x,A) \) , where \( L \) is the last-exit kernel in [Chung 1973]. For a general non-symmetric potential density \( u \) the modified energy
\[ I(\lambda) = \iint \lambda(dx)\,u(x,y)\,\phi(y)^{-1}(dy) \]
and the Gauss quadratic
\[ G(\lambda) = I(\lambda) - 2\lambda(K) \]
are introduced. Then \( G \) is minimized by \( \pi \) among all signed measures \( \lambda \) on \( K \) of finite modified energy, provided \( I \) is positive. This includes the classical symmetric case of Newtonian and M. Riesz potentials as a special case. The modification corresponds to a time change for the underlying Markov process. The positivity of \( I \) is established for a class of signed measures associated with continuous additive functionals in the sense of Revuz.

@article {key626304m,
AUTHOR = {Chung, Kai Lai and Rao, Murali},
TITLE = {Equilibrium and energy},
JOURNAL = {Probab. Math. Statist.},
FJOURNAL = {Probability and Mathematical Statistics},
VOLUME = {1},
NUMBER = {2},
YEAR = {1980},
PAGES = {99--108},
URL = {http://www.math.uni.wroc.pl/~pms/files/1.2/Abstract/1.2.1.abs.pdf},
NOTE = {MR:626304. Zbl:0502.60060.},
ISSN = {0208-4147},
}
K. L. Chung and K. M. Rao :
“A new setting for potential theory, I ,”
Ann. Inst. Fourier (Grenoble)
30 : 3
(1980 ),
pp. 167–198 .
MR
597022
Zbl
0424.31004
article

Abstract
People
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In Hunt’s theory of Markov processes certain duality assumptions are made to generalize well known classical potential theoretic results such as F. Riesz representation theorem, uniqueness, existence of equilibrium potential, etc. A standard treatment developed by several subsequent authors can be found in [Blumenthal and Getoor 1968]. In a different direction, it was shown in [Chung 1973] under simple analytic conditions that the equilibrium measure is inherently linked to the last exit distribution of the process. It thus appears feasible that this last result, namely on “equilibrium principle” may be made the starting point to which other major results are related.

In this paper we exploit the line of thought in [Chung 1973] to derive some of these major results under the same set of conditions as in [Chung 1973]. It turns out that these sets of conditions automatically imply the existence of a dual. However, this will not be proved here.

@article {key597022m,
AUTHOR = {Chung, K. L. and Rao, K. Murali},
TITLE = {A new setting for potential theory,
{I}},
JOURNAL = {Ann. Inst. Fourier (Grenoble)},
FJOURNAL = {Universit\'e de Grenoble. Annales de
l'Institut Fourier},
VOLUME = {30},
NUMBER = {3},
YEAR = {1980},
PAGES = {167--198},
URL = {http://www.numdam.org/item?id=AIF_1980__30_3_167_0},
NOTE = {MR:597022. Zbl:0424.31004.},
ISSN = {0373-0956},
CODEN = {AIFUA7},
}
K. L. Chung and K. M. Rao :
“Feynman–Kac functional and the Schrödinger equation ,”
pp. 1–29
in
Seminar on stochastic processes, 1981
(Northwestern University, Evanston, IL, April 1981 ).
Edited by E. Çinlar, K. L. Chung, and R. K. Getoor .
Progress in Probability and Statistics 1 .
Birkhäuser (Boston, MA ),
1981 .
MR
647779
Zbl
0492.60073
incollection

People
BibTeX
@incollection {key647779m,
AUTHOR = {Chung, K. L. and Rao, K. M.},
TITLE = {Feynman--{K}ac functional and the {S}chr\"odinger
equation},
BOOKTITLE = {Seminar on stochastic processes, 1981},
EDITOR = {Erhan \c{C}inlar and K. L. Chung and
R. K. Getoor},
SERIES = {Progress in Probability and Statistics},
NUMBER = {1},
PUBLISHER = {Birkh\"auser},
ADDRESS = {Boston, MA},
YEAR = {1981},
PAGES = {1--29},
NOTE = {(Northwestern University, Evanston,
IL, April 1981). MR:647779. Zbl:0492.60073.},
ISBN = {3764330724, 9783764330729},
}
K. L. Chung and K. M. Rao :
“Erratum: ‘A new setting for potential theory, I’ ,”
Ann. Inst. Fourier (Grenoble)
31 : 2
(1981 ).
loose errata.
MR
617248
article

People
BibTeX
@article {key617248m,
AUTHOR = {Chung, K. L. and Rao, K. Murali},
TITLE = {Erratum: ``{A} new setting for potential
theory, {I}''},
JOURNAL = {Ann. Inst. Fourier (Grenoble)},
FJOURNAL = {Universit\'e de Grenoble. Annales de
l'Institut Fourier},
VOLUME = {31},
NUMBER = {2},
YEAR = {1981},
NOTE = {loose errata. MR:617248.},
ISSN = {0373-0956},
CODEN = {AIFUA7},
}
K. L. Chung, M. Liao, and K. M. Rao :
“Duality under a new setting ,”
pp. 23–38
in
Seminar on stochastic processes, 1983
(University of Florida, Gainesville, FL, 1983 ).
Edited by E. Çinlar, K. L. Chung, and R. K. Getoor .
Progress in Probability and Statistics 7 .
Birkhäuser (Boston, MA ),
1984 .
MR
902410
Zbl
0558.60056
incollection

People
BibTeX
@incollection {key902410m,
AUTHOR = {Chung, K. L. and Liao, Ming and Rao,
K. M.},
TITLE = {Duality under a new setting},
BOOKTITLE = {Seminar on stochastic processes, 1983},
EDITOR = {Erhan \c{C}inlar and K. L. Chung and
R. K. Getoor},
SERIES = {Progress in Probability and Statistics},
NUMBER = {7},
PUBLISHER = {Birkh\"auser},
ADDRESS = {Boston, MA},
YEAR = {1984},
PAGES = {23--38},
NOTE = {(University of Florida, Gainesville,
FL, 1983). MR:902410. Zbl:0558.60056.},
ISBN = {081763293X, 9780817632939},
}
K. L. Chung and K. M. Rao :
“General gauge theorem for multiplicative functionals ,”
Trans. Amer. Math. Soc.
306 : 2
(April 1988 ),
pp. 819–836 .
MR
933320
Zbl
0647.60083
article

Abstract
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We generalize our previous work on the gauge theorem and its various consequences and complements, initiated in [Chung and Rao 1981] and somewhat extended by subsequent investigations (see [Chung and Zhao]). The generalization here is two-fold. First, instead of the Brownian motion, the underlying process is now a fairly broad class of Markov processes, not necessarily having continuous paths. Second, instead of the Feynman–Kac functional, the exponential of a general class of additive functionals is treated. The case of Schrödinger operator \( \Delta/2+\nu \) , where \( \nu \) is a suitable measure, is a simple special case. The most general operator, not necessarily a differential one, which may arise from our potential equations is briefly discussed toward the end of the paper. Concrete instances of applications in this case should be of great interest.

@article {key933320m,
AUTHOR = {Chung, K. L. and Rao, K. M.},
TITLE = {General gauge theorem for multiplicative
functionals},
JOURNAL = {Trans. Amer. Math. Soc.},
FJOURNAL = {Transactions of the American Mathematical
Society},
VOLUME = {306},
NUMBER = {2},
MONTH = {April},
YEAR = {1988},
PAGES = {819--836},
DOI = {10.2307/2000825},
NOTE = {MR:933320. Zbl:0647.60083.},
ISSN = {0002-9947},
CODEN = {TAMTAM},
}