K. L. Chung, P. Erdős, and T. Sirao :
“On the Lipschitz’s condition for Brownian motion J. Math. Soc. Japan
11 : 4
(1959 ),
pp. 263–274 .
MR
0121873 Zbl
0091.13301 article
People
BibTeX
@article {key0121873m,
AUTHOR = {Chung, K. L. and Erd{\H{o}}s, P. and
Sirao, T.},
TITLE = {On the {L}ipschitz's condition for {B}rownian
motion},
JOURNAL = {J. Math. Soc. Japan},
FJOURNAL = {Journal of the Mathematical Society
of Japan},
VOLUME = {11},
NUMBER = {4},
YEAR = {1959},
PAGES = {263--274},
DOI = {10.2969/jmsj/01140263},
NOTE = {MR:0121873. Zbl:0091.13301.},
ISSN = {0025-5645},
}
K. L. Chung :
“On last exit times Illinois J. Math.
4 : 4
(1960 ),
pp. 629–639 .
MR
0130715 Zbl
0094.32102 article
BibTeX
@article {key0130715m,
AUTHOR = {Chung, K. L.},
TITLE = {On last exit times},
JOURNAL = {Illinois J. Math.},
FJOURNAL = {Illinois Journal of Mathematics},
VOLUME = {4},
NUMBER = {4},
YEAR = {1960},
PAGES = {629--639},
URL = {http://projecteuclid.org/euclid.ijm/1255456281},
NOTE = {MR:0130715. Zbl:0094.32102.},
ISSN = {0019-2082},
}
K. L. Chung :
“Probabilistic approach in potential theory to the equilibrium problem Ann. Inst. Fourier (Grenoble)
23 : 3
(1973 ),
pp. 313–322 .
MR
0391277 Zbl
0258.31012 article
Abstract
BibTeX
@article {key0391277m,
AUTHOR = {Chung, Kai Lai},
TITLE = {Probabilistic approach in potential
theory to the equilibrium problem},
JOURNAL = {Ann. Inst. Fourier (Grenoble)},
FJOURNAL = {Universit\'e de Grenoble. Annales de
l'Institut Fourier},
VOLUME = {23},
NUMBER = {3},
YEAR = {1973},
PAGES = {313--322},
URL = {http://www.numdam.org/item?id=AIF_1973__23_3_313_0},
NOTE = {MR:0391277. Zbl:0258.31012.},
ISSN = {0373-0956},
}
K. L. Chung :
“Note on last exit decomposition 20–21
in
Séminaire de Probabilités VIII
(Université de Strasbourg, année universitaire 1972–1973 ).
Edited by C. Dellarcherie, P. Meyer, and M. Weil .
Lecture Notes in Mathematics 381 .
Springer (Berlin ),
1974 .
MR
0378117 Zbl
0295.60056 incollection
Abstract
People
BibTeX
@incollection {key0378117m,
AUTHOR = {Chung, K. L.},
TITLE = {Note on last exit decomposition},
BOOKTITLE = {S\'eminaire de {P}robabilit\'es {VIII}},
EDITOR = {C. Dellarcherie and P.A. Meyer and M.
Weil},
SERIES = {Lecture Notes in Mathematics},
NUMBER = {381},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1974},
PAGES = {20--21},
DOI = {10.1007/BFb0057252},
NOTE = {(Universit\'e de Strasbourg, ann\'ee
universitaire 1972--1973). MR:0378117.
Zbl:0295.60056.},
ISSN = {0075-8438, 0720-8766},
ISBN = {3540067833, 9783540067832},
}
K. L. Chung :
“An expression for canonical entrance laws ,”
pp. 311–314
in
Stochastic analysis: A tribute to the memory of Rollo Davidson .
Edited by D. G. Kendall and E. F. Harding .
Wiley Series in Probability and Statistics .
Wiley (London ),
1974 .
MR
0368189 incollection
People
BibTeX
@incollection {key0368189m,
AUTHOR = {Chung, K. L.},
TITLE = {An expression for canonical entrance
laws},
BOOKTITLE = {Stochastic analysis: {A} tribute to
the memory of {R}ollo {D}avidson},
EDITOR = {D. G. Kendall and E. F. Harding},
SERIES = {Wiley Series in Probability and Statistics},
PUBLISHER = {Wiley},
ADDRESS = {London},
YEAR = {1974},
PAGES = {311--314},
NOTE = {MR:0368189.},
ISBN = {0471351415, 9780471351412},
}
K. L. Chung :
“Remarks on equilibrium potential and energy xiv, 131–138
in
Collection of articles dedicated to Marcel Brelot on the occasion of his 70th birthday ,
published as Ann. Inst. Fourier (Grenoble)
25 : 3–4
(1975 ).
MR
0405601 Zbl
0338.31001 incollection
Abstract
People
BibTeX
@article {key0405601m,
AUTHOR = {Chung, Kai Lai},
TITLE = {Remarks on equilibrium potential and
energy},
JOURNAL = {Ann. Inst. Fourier (Grenoble)},
FJOURNAL = {Universit\'e de Grenoble. Annales de
l'Institut Fourier},
VOLUME = {25},
NUMBER = {3--4},
YEAR = {1975},
PAGES = {xiv, 131--138},
URL = {http://www.numdam.org/item?id=AIF_1975__25_3-4_131_0},
NOTE = {\textit{Collection of articles dedicated
to {M}arcel {B}relot on the occasion
of his 70th birthday}. MR:0405601. Zbl:0338.31001.},
ISSN = {0373-0956},
}
K. L. Chung :
“Maxima in Brownian excursions Bull. Amer. Math. Soc.
81 : 4
(July 1975 ),
pp. 742–745 .
MR
0373035 Zbl
0325.60077 article
BibTeX
@article {key0373035m,
AUTHOR = {Chung, Kai Lai},
TITLE = {Maxima in {B}rownian excursions},
JOURNAL = {Bull. Amer. Math. Soc.},
FJOURNAL = {Bulletin of the American Mathematical
Society},
VOLUME = {81},
NUMBER = {4},
MONTH = {July},
YEAR = {1975},
PAGES = {742--745},
DOI = {10.1090/S0002-9904-1975-13852-3},
NOTE = {MR:0373035. Zbl:0325.60077.},
ISSN = {0002-9904},
}
K. L. Chung :
“Approximation of a continuous parameter Markov chain by discrete skeletons Bull. London Math. Soc.
7 : 1
(1975 ),
pp. 71–76 .
MR
0388540 Zbl
0301.60046 article
Abstract
BibTeX
The approximation of a given continuous parameter Markov chain by its discrete skeletons raises many interesting questions. Some beginnings of this study were given in §11.13 of [Chung 1967], but little progress seems to have been made since. It is possible that the apparent impenetrability of the “embedding problem” (when is a discrete parameter chain a skeleton?) may have unwarrantedly hampered reasonable and useful investigations. The present note deals with a relatively easy aspect of the approximation theory: the global faithfulness of the skeletal approach as reflected in the moments of the entrance time distributions. Even so our method leans heavily on recurrence. When this hypothesis is dropped these distributions may be improper, but moments can still be defined by excluding the mass at infinity as done in §1.11 of [Chung 1967] in the discrete parameter case. Solidarity results such as Theorem 1 there can be generalized to the continuous parameter case without the intervention of skeletons (see the remarks concerning the last part of the theorem below); but approximation seems an open problem.
@article {key0388540m,
AUTHOR = {Chung, Kai Lai},
TITLE = {Approximation of a continuous parameter
{M}arkov chain by discrete skeletons},
JOURNAL = {Bull. London Math. Soc.},
FJOURNAL = {The Bulletin of the London Mathematical
Society},
VOLUME = {7},
NUMBER = {1},
YEAR = {1975},
PAGES = {71--76},
DOI = {10.1112/blms/7.1.71},
NOTE = {MR:0388540. Zbl:0301.60046.},
ISSN = {0024-6093},
}
K. L. Chung :
“Excursions in Brownian motion Ark. Mat.
14 : 1–2
(1976 ),
pp. 155–177 .
Dedicated to the Memory of Paul Lévy.
MR
0467948 Zbl
0356.60033 article
People
BibTeX
@article {key0467948m,
AUTHOR = {Chung, Kai Lai},
TITLE = {Excursions in {B}rownian motion},
JOURNAL = {Ark. Mat.},
FJOURNAL = {Arkiv f\"or Matematik},
VOLUME = {14},
NUMBER = {1--2},
YEAR = {1976},
PAGES = {155--177},
DOI = {10.1007/BF02385832},
NOTE = {Dedicated to the Memory of Paul L\'evy.
MR:0467948. Zbl:0356.60033.},
ISSN = {0004-2080},
}
K. L. Chung :
“Pedagogic notes on the barrier theorem 27–33
in
Séminaire de probabilités XI
(Université de Strasbourg, année universitaire 1975–1976 ).
Edited by C. Dellacherie, P. A. Meyer, and M. Weil .
Lecture Notes in Mathematics 581 .
Springer (Berlin ),
1977 .
MR
0445620 Zbl
0376.60080 incollection
People
BibTeX
@incollection {key0445620m,
AUTHOR = {Chung, Kai Lai},
TITLE = {Pedagogic notes on the barrier theorem},
BOOKTITLE = {S\'eminaire de probabilit\'es {XI}},
EDITOR = {C. Dellacherie and P. A. Meyer and M.
Weil},
SERIES = {Lecture Notes in Mathematics},
NUMBER = {581},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1977},
PAGES = {27--33},
DOI = {10.1007/BFb0087184},
NOTE = {(Universit\'e de Strasbourg, ann\'ee
universitaire 1975--1976). MR:0445620.
Zbl:0376.60080.},
ISSN = {0075-8438, 0720-8766},
ISBN = {3540081453, 9783540081456},
}
K. L. Chung and R. K. Getoor :
“The condenser problem Ann. Probability
5 : 1
(February 1977 ),
pp. 82–86 .
MR
0428480 Zbl
0381.60064 article
Abstract
People
BibTeX
Read PDF
@article {key0428480m,
AUTHOR = {Chung, K. L. and Getoor, R. K.},
TITLE = {The condenser problem},
JOURNAL = {Ann. Probability},
VOLUME = {5},
NUMBER = {1},
MONTH = {February},
YEAR = {1977},
PAGES = {82--86},
URL = {http://www.jstor.org/stable/2242804},
NOTE = {MR:0428480. Zbl:0381.60064.},
}
K. L. Chung :
“Correction to: ‘Pedagogic notes on the barrier theorem’ 739
in
Séminaire de probabilités XII
(Université de Strasbourg, année universitaire 1976–1977 ).
Lecture Notes in Mathematics 649 .
Springer (Berlin ),
1978 .
MR
520034 incollection
BibTeX
@incollection {key520034m,
AUTHOR = {Chung, K. L.},
TITLE = {Correction to: ``{P}edagogic notes on
the barrier theorem''},
BOOKTITLE = {S\'eminaire de probabilit\'es {XII}},
SERIES = {Lecture Notes in Mathematics},
NUMBER = {649},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1978},
PAGES = {739},
URL = {http://archive.numdam.org/ARCHIVE/SPS/SPS_1978__12_/SPS_1978__12__739_0/SPS_1978__12__739_0.pdf},
NOTE = {(Universit\'e de Strasbourg, ann\'ee
universitaire 1976--1977). MR:520034.},
}
K. L. Chung and S. R. S. Varadhan :
“Kac functional and Schrödinger equation ,”
Studia Math.
68 : 3
(1980 ),
pp. 249–260 .
MR
599148 Zbl
0448.60054 article
People
BibTeX
@article {key599148m,
AUTHOR = {Chung, Kai Lai and Varadhan, S. R. S.},
TITLE = {Kac functional and {S}chr\"odinger equation},
JOURNAL = {Studia Math.},
FJOURNAL = {Polska Akademia Nauk. Institut Matematyczny.
Studia Mathematica},
VOLUME = {68},
NUMBER = {3},
YEAR = {1980},
PAGES = {249--260},
NOTE = {MR:599148. Zbl:0448.60054.},
ISSN = {0039-3223},
CODEN = {SMATAZ},
}
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
People
BibTeX
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
BibTeX
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 :
“On stopped Feynman–Kac functionals ,”
pp. 347–356
in
Séminaire de probabilités XIV
(Paris, 1978–1979 ).
Edited by J. Azéma and M. Yor .
Lecture Notes in Mathematics 784 .
Springer (Berlin ),
1980 .
MR
580141 Zbl
0444.60061 incollection
People
BibTeX
@incollection {key580141m,
AUTHOR = {Chung, Kai Lai},
TITLE = {On stopped {F}eynman--{K}ac functionals},
BOOKTITLE = {S\'eminaire de probabilit\'es {XIV}},
EDITOR = {Az\'ema, J. and Yor, M.},
SERIES = {Lecture Notes in Mathematics},
NUMBER = {784},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1980},
PAGES = {347--356},
NOTE = {(Paris, 1978--1979). MR:580141. Zbl:0444.60061.},
ISSN = {0075-8438, 0720-8766},
ISBN = {3540097600, 9783540097600},
}
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 :
“Brownian motion on the line, I ,”
J. Math. Res. Exposition
1 : 1
(1981 ),
pp. 74–82 .
MR
617355 Zbl
0583.60074 article
BibTeX
@article {key617355m,
AUTHOR = {Chung, K. L.},
TITLE = {Brownian motion on the line, {I}},
JOURNAL = {J. Math. Res. Exposition},
FJOURNAL = {Journal of Mathematical Research \&
Exposition. Shuxue Yanjiu yu Pinglun},
VOLUME = {1},
NUMBER = {1},
YEAR = {1981},
PAGES = {74--82},
NOTE = {MR:617355. Zbl:0583.60074.},
ISSN = {1000-341X},
}
K. L. Chung :
“A cluster of great formulas Acta Math. Acad. Sci. Hungar.
39 : 1–3
(1982 ),
pp. 65–67 .
MR
653674 Zbl
0495.60020 article
BibTeX
@article {key653674m,
AUTHOR = {Chung, K. L.},
TITLE = {A cluster of great formulas},
JOURNAL = {Acta Math. Acad. Sci. Hungar.},
FJOURNAL = {Acta Mathematica Academiae Scientiarum
Hungaricae},
VOLUME = {39},
NUMBER = {1--3},
YEAR = {1982},
PAGES = {65--67},
DOI = {10.1007/BF01895215},
NOTE = {MR:653674. Zbl:0495.60020.},
ISSN = {0001-5954},
CODEN = {ACMTAV},
}
K. L. Chung :
“Brownian motion on the line, II ,”
J. Math. Res. Exposition
2 : 3
(1982 ),
pp. 87–98 .
MR
672218 Zbl
0583.60075 article
BibTeX
@article {key672218m,
AUTHOR = {Chung, K. L.},
TITLE = {Brownian motion on the line, {II}},
JOURNAL = {J. Math. Res. Exposition},
FJOURNAL = {Journal of Mathematical Research \&
Exposition. Shuxue Yanjiu yu Pinglun},
VOLUME = {2},
NUMBER = {3},
YEAR = {1982},
PAGES = {87--98},
NOTE = {MR:672218. Zbl:0583.60075.},
ISSN = {1000-341X},
}
K. L. Chung, R. Durrett, and Z. X. Zhao :
“Extension of domains with finite gauge Math. Ann.
264 : 1
(1983 ),
pp. 73–79 .
MR
709862 Zbl
0519.60080 article
People
BibTeX
@article {key709862m,
AUTHOR = {Chung, K. L. and Durrett, R. and Zhao,
Zhong Xin},
TITLE = {Extension of domains with finite gauge},
JOURNAL = {Math. Ann.},
FJOURNAL = {Mathematische Annalen},
VOLUME = {264},
NUMBER = {1},
YEAR = {1983},
PAGES = {73--79},
DOI = {10.1007/BF01458051},
NOTE = {MR:709862. Zbl:0519.60080.},
ISSN = {0025-5831},
CODEN = {MAANA},
}
K. L. Chung :
“An inequality for boundary value problems ,”
pp. 111–122
in
Seminar on stochastic processes, 1982
(Northwestern University, Evanston, IL, March 1982 ).
Edited by E. Çinlar, K. L. Chung, and R. K. Getoor .
Progress in Probability and Statistics 5 .
Birkhäuser (Boston, MA ),
1983 .
MR
733667 Zbl
0528.60071 incollection
People
BibTeX
@incollection {key733667m,
AUTHOR = {Chung, K. L.},
TITLE = {An inequality for boundary value problems},
BOOKTITLE = {Seminar on stochastic processes, 1982},
EDITOR = {Erhan \c{C}inlar and K. L. Chung and
R. K. Getoor},
SERIES = {Progress in Probability and Statistics},
NUMBER = {5},
PUBLISHER = {Birkh\"auser},
ADDRESS = {Boston, MA},
YEAR = {1983},
PAGES = {111--122},
NOTE = {(Northwestern University, Evanston,
IL, March 1982). MR:733667. Zbl:0528.60071.},
ISBN = {0817631313, 9780817631314},
}
K. L. Chung :
“Properties of finite gauge with an application to local time ,”
pp. 16–24
in
Probability and mathematical statistics: essays in honour of Carl-Gustav Esseen
(Uppsala, Sweden, 1983 ).
Edited by A. Gut and L. Holst .
Uppsala University ,
1983 .
MR
727123 Zbl
0519.60081 incollection
People
BibTeX
@incollection {key727123m,
AUTHOR = {Chung, K. L.},
TITLE = {Properties of finite gauge with an application
to local time},
BOOKTITLE = {Probability and mathematical statistics:
essays in honour of {C}arl-{G}ustav
{E}sseen},
EDITOR = {A. Gut and L. Holst},
PUBLISHER = {Uppsala University},
YEAR = {1983},
PAGES = {16--24},
NOTE = {(Uppsala, Sweden, 1983). MR:727123.
Zbl:0519.60081.},
}
K. L. Chung and Z. X. Zhao :
“Some probabilistic results on the Sturm–Liouville equation J. Systems Sci. Math. Sci.
4 : 4
(1984 ),
pp. 320–324 .
MR
820144 Zbl
0591.60075 article
People
BibTeX
@article {key820144m,
AUTHOR = {Chung, K. L. and Zhao, Zhong Xin},
TITLE = {Some probabilistic results on the {S}turm--{L}iouville
equation},
JOURNAL = {J. Systems Sci. Math. Sci.},
FJOURNAL = {Journal of Systems Science and Mathematical
Sciences. Xitong Kexue yu Shuxue},
VOLUME = {4},
NUMBER = {4},
YEAR = {1984},
PAGES = {320--324},
URL = {http://www.sysmath.com/CN/abstract/abstract8901.shtml},
NOTE = {MR:820144. Zbl:0591.60075.},
ISSN = {1000-0577},
CODEN = {XKSHEW},
}
K. L. Chung :
“The lifetime of conditional Brownian motion in the plane Ann. Inst. H. Poincaré Probab. Statist.
20 : 4
(1984 ),
pp. 349–351 .
MR
771894 Zbl
0573.60070 article
Abstract
BibTeX
@article {key771894m,
AUTHOR = {Chung, Kai Lai},
TITLE = {The lifetime of conditional {B}rownian
motion in the plane},
JOURNAL = {Ann. Inst. H. Poincar\'e Probab. Statist.},
FJOURNAL = {Annales de l'Institut Henri Poincar\'e.
Probabilit\'es et Statistique},
VOLUME = {20},
NUMBER = {4},
YEAR = {1984},
PAGES = {349--351},
URL = {http://www.numdam.org/item?id=AIHPB_1984__20_4_349_0},
NOTE = {MR:771894. Zbl:0573.60070.},
ISSN = {0246-0203},
CODEN = {AHPBAR},
}
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 :
“Conditional gauges ,”
pp. 17–22
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
902409 Zbl
0552.60073 incollection
People
BibTeX
@incollection {key902409m,
AUTHOR = {Chung, K. L.},
TITLE = {Conditional gauges},
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 = {17--22},
NOTE = {(University of Florida, Gainesville,
FL, 1983). MR:902409. Zbl:0552.60073.},
ISBN = {081763293X, 9780817632939},
}
K. L. Chung :
“The gauge and conditional gauge theorem 496–503
in
Séminaire de probabilités XIX
(Université de Strasbourg, année universitaire, 1983–84 ).
Edited by J. Azéma .
Lecture Notes in Mathematics 1123 .
Springer (Berlin ),
1985 .
MR
889497 Zbl
0561.60084 incollection
People
BibTeX
@incollection {key889497m,
AUTHOR = {Chung, K. L.},
TITLE = {The gauge and conditional gauge theorem},
BOOKTITLE = {S\'eminaire de probabilit\'es {XIX}},
EDITOR = {Az\'ema, J.},
SERIES = {Lecture Notes in Mathematics},
NUMBER = {1123},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1985},
PAGES = {496--503},
DOI = {10.1007/BFb0075868},
NOTE = {(Universit\'e de Strasbourg, ann\'ee
universitaire, 1983--84). MR:889497.
Zbl:0561.60084.},
ISBN = {038715230X, 9780387152301},
}
K. L. Chung and Z. X. Zhao :
“A simple proof of a new theorem on conditional Brownian motion ,”
Kexue Tongbao (Chinese)
30 : 5
(1985 ),
pp. 329–332 .
MR
803587 article
People
BibTeX
@article {key803587m,
AUTHOR = {Chung, Kai Lai and Zhao, Zhong Xin},
TITLE = {A simple proof of a new theorem on conditional
{B}rownian motion},
JOURNAL = {Kexue Tongbao (Chinese)},
FJOURNAL = {Kexue Tongbao. Chinese Science Bulletin},
VOLUME = {30},
NUMBER = {5},
YEAR = {1985},
PAGES = {329--332},
NOTE = {MR:803587.},
ISSN = {0023-074X},
}
K. L. Chung :
“Probabilistic approach to boundary value problems for Schrödinger’s equation Exposition. Math.
3 : 2
(1985 ),
pp. 175–178 .
MR
816403 Zbl
0577.60066 article
BibTeX
@article {key816403m,
AUTHOR = {Chung, Kai Lai},
TITLE = {Probabilistic approach to boundary value
problems for {S}chr\"odinger's equation},
JOURNAL = {Exposition. Math.},
FJOURNAL = {Expositiones Mathematicae. International
Journal for Pure and Applied Mathematics},
VOLUME = {3},
NUMBER = {2},
YEAR = {1985},
PAGES = {175--178},
URL = {http://ebooks.worldscinet.com/ISBN/9789812567130/preserved-docs/9789812567130_0019.pdf},
NOTE = {MR:816403. Zbl:0577.60066.},
ISSN = {0723-0869},
}
K. L. Chung and P. Li :
“Comparison of probability and eigenvalue methods for the Schrödinger equation ,”
pp. 25–34
in
Probability, statistical mechanics, and number theory .
Edited by G.-C. Rota .
Advances in Mathematics Supplementary Studies 9 .
Academic Press (Orlando, FL ),
1986 .
Volume dedicated to Mark Kac.
MR
875443 Zbl
0608.60061 incollection
People
BibTeX
@incollection {key875443m,
AUTHOR = {Chung, K. L. and Li, Peter},
TITLE = {Comparison of probability and eigenvalue
methods for the {S}chr\"odinger equation},
BOOKTITLE = {Probability, statistical mechanics,
and number theory},
EDITOR = {Rota, Gian-Carlo},
SERIES = {Advances in Mathematics Supplementary
Studies},
NUMBER = {9},
PUBLISHER = {Academic Press},
ADDRESS = {Orlando, FL},
YEAR = {1986},
PAGES = {25--34},
NOTE = {Volume dedicated to Mark Kac. MR:875443.
Zbl:0608.60061.},
ISSN = {0001-8708},
ISBN = {0125985436, 9780125985437},
}
K. L. Chung and P. Hsu :
“Gauge theorem for the Neumann problem ,”
pp. 63–70
in
Seminar on stochastic processes, 1984
(Northwestern University, Evanston, IL, 1984 ).
Edited by E. Çinlar, K. L. Chung, and R. K. Getoor .
Progress in Probability and Statistics 9 .
Birkhäuser (Boston, MA ),
1986 .
MR
896722 Zbl
0585.60064 incollection
People
BibTeX
@incollection {key896722m,
AUTHOR = {Chung, K. L. and Hsu, Pei},
TITLE = {Gauge theorem for the {N}eumann problem},
BOOKTITLE = {Seminar on stochastic processes, 1984},
EDITOR = {\c{C}inlar, E. and Chung, K. L. and
Getoor, R. K.},
SERIES = {Progress in Probability and Statistics},
NUMBER = {9},
PUBLISHER = {Birkh\"auser},
ADDRESS = {Boston, MA},
YEAR = {1986},
PAGES = {63--70},
NOTE = {(Northwestern University, Evanston,
IL, 1984). MR:896722. Zbl:0585.60064.},
ISBN = {0817633278, 9780817633271},
}
K. L. Chung, P. Li, and R. J. Williams :
“Comparison of probability and classical methods for the Schrödinger equation ,”
Exposition. Math.
4 : 3
(1986 ),
pp. 271–278 .
MR
880127 Zbl
0623.60097 article
People
BibTeX
@article {key880127m,
AUTHOR = {Chung, K. L. and Li, P. and Williams,
R. J.},
TITLE = {Comparison of probability and classical
methods for the {S}chr\"odinger equation},
JOURNAL = {Exposition. Math.},
FJOURNAL = {Expositiones Mathematicae. International
Journal for Pure and Applied Mathematics},
VOLUME = {4},
NUMBER = {3},
YEAR = {1986},
PAGES = {271--278},
NOTE = {MR:880127. Zbl:0623.60097.},
ISSN = {0723-0869},
}
K. L. Chung :
“Notes on the inhomogeneous Schrödinger equation ,”
pp. 55–62
in
Seminar on stochastic processes, 1984
(Northwestern University, Evanston, IL, 1984 ).
Edited by E. Çinlar, K. L. Chung, and R. K. Getoor .
Progress in Probability and Statistics 9 .
Birkhäuser (Boston, MA ),
1986 .
MR
896721 Zbl
0585.60063 incollection
People
BibTeX
@incollection {key896721m,
AUTHOR = {Chung, K. L.},
TITLE = {Notes on the inhomogeneous {S}chr\"odinger
equation},
BOOKTITLE = {Seminar on stochastic processes, 1984},
EDITOR = {\c{C}inlar, E. and Chung, K. L. and
Getoor, R. K.},
SERIES = {Progress in Probability and Statistics},
NUMBER = {9},
PUBLISHER = {Birkh\"auser},
ADDRESS = {Boston, MA},
YEAR = {1986},
PAGES = {55--62},
NOTE = {(Northwestern University, Evanston,
IL, 1984). MR:896721. Zbl:0585.60063.},
ISBN = {0817633278, 9780817633271},
}
K. L. Chung :
“Remark on the conditional gauge theorem 423–425
in
Séminaire de Probabilités XX
(Université de Strasbourg, année universitaire 1984/85 ).
Edited by J. Azéma and M. Yor .
Lecture Notes in Mathematics 1204 .
Springer (Berlin ),
1986 .
MR
942036 incollection
People
BibTeX
@incollection {key942036m,
AUTHOR = {Chung, K. L.},
TITLE = {Remark on the conditional gauge theorem},
BOOKTITLE = {S\'eminaire de {P}robabilit\'es {XX}},
EDITOR = {Az\'ema, J. and Yor, M.},
SERIES = {Lecture Notes in Mathematics},
NUMBER = {1204},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1986},
PAGES = {423--425},
DOI = {10.1007/BFb0075733},
NOTE = {(Universit\'e de Strasbourg, ann\'ee
universitaire 1984/85). MR:942036.},
}
K. L. Chung :
“Green’s function for a ball 1–13
in
Seminar on stochastic processes, 1986
(University of Virginia, Charlottesville, VA, March, 1986 ).
Edited by E. Çinlar, K. L. Chung, R. K. Getoor, and J. Glover .
Progress in Probability and Statistics 13 .
Birkhäuser (Boston, MA ),
1987 .
MR
902422 Zbl
0643.31007 incollection
Abstract
People
BibTeX
@incollection {key902422m,
AUTHOR = {Chung, K. L.},
TITLE = {Green's function for a ball},
BOOKTITLE = {Seminar on stochastic processes, 1986},
EDITOR = {E. \c{C}inlar and K. L. Chung and R.
K. Getoor and J. Glover},
SERIES = {Progress in Probability and Statistics},
NUMBER = {13},
PUBLISHER = {Birkh\"auser},
ADDRESS = {Boston, MA},
YEAR = {1987},
PAGES = {1--13},
URL = {http://en.scientificcommons.org/19809614},
NOTE = {(University of Virginia, Charlottesville,
VA, March, 1986). MR:902422. Zbl:0643.31007.},
ISBN = {0817633537, 9780817633530},
}
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
People
BibTeX
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},
}
K. L. Chung :
“Probability methods in potential theory 42–54
in
Potential theory — surveys and problems
(Prague, July 19–24, 1987 ).
Edited by J. Král, J. Lukeš, I. Netuka, and J. Veselý .
Lecture Notes in Mathematics 1344 .
Springer (Berlin ),
1988 .
MR
973880 Zbl
0677.60085 incollection
People
BibTeX
@incollection {key973880m,
AUTHOR = {Chung, Kai Lai},
TITLE = {Probability methods in potential theory},
BOOKTITLE = {Potential theory---surveys and problems},
EDITOR = {J. Kr\'al and J. Luke\v{s} and I. Netuka
and J. Vesel\'y},
SERIES = {Lecture Notes in Mathematics},
NUMBER = {1344},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1988},
PAGES = {42--54},
DOI = {10.1007/BFb0103343},
NOTE = {(Prague, July 19--24, 1987). MR:973880.
Zbl:0677.60085.},
ISBN = {3540502106, 9783540502104},
}
K. L. Chung :
“Gauge theorem for unbounded domains ,”
pp. 87–98
in
Seminar on stochastic processes, 1988
(University of Florida, Gainesville, FL, March 3–5, 1988 ).
Edited by E. Çinlar, K. L. Chung, R. K. Getoor, and J. Glover .
Progress in Probability 17 .
Birkhäuser (Boston, MA ),
1989 .
MR
990475 Zbl
0689.60076 incollection
People
BibTeX
@incollection {key990475m,
AUTHOR = {Chung, Kai Lai},
TITLE = {Gauge theorem for unbounded domains},
BOOKTITLE = {Seminar on stochastic processes, 1988},
EDITOR = {E. \c{C}inlar and K. L. Chung and R.
K. Getoor and J. Glover},
SERIES = {Progress in Probability},
NUMBER = {17},
PUBLISHER = {Birkh\"auser},
ADDRESS = {Boston, MA},
YEAR = {1989},
PAGES = {87--98},
NOTE = {(University of Florida, Gainesville,
FL, March 3--5, 1988). MR:990475. Zbl:0689.60076.},
ISSN = {0892-063X},
ISBN = {0817634223, 9780817634223},
}
K. L. Chung :
“Greenian bounds for Markov processes Potential Anal.
1 : 1
(1992 ),
pp. 83–92 .
MR
1245226 Zbl
1081.60543 article
Abstract
BibTeX
In Section 1, a temporal bound is estimated by a spatial bound, for a Markov process whose transition density satisfies a simple condition. This includes the Brownian motion, for which comparison with a more special method is made. In Section 2, the result is related to the Green operator and examples are given. In Section 3, the result is applied to an old problem of eigenvalues of the Laplacian. In Section 4, recent extensions from the Laplacian to the Schrödinger circle-of-ideas are briefly described. In this case, time is measured by an exponential functional of the process, commonly known under the names Feynman–Kac.
This article was written as script for two talks, one general and one technical, given in Taiwan in January 1991. In the spirit of the occasion, the style of exposition is deliberately paced and discursive.
@article {key1245226m,
AUTHOR = {Chung, Kai Lai},
TITLE = {Greenian bounds for {M}arkov processes},
JOURNAL = {Potential Anal.},
FJOURNAL = {Potential Analysis. An International
Journal Devoted to the Interactions
between Potential Theory, Probability
Theory, Geometry and Functional Analysis},
VOLUME = {1},
NUMBER = {1},
YEAR = {1992},
PAGES = {83--92},
DOI = {10.1007/BF00249787},
NOTE = {MR:1245226. Zbl:1081.60543.},
ISSN = {0926-2601},
CODEN = {POANE2},
}
