C. Kipnis and S. R. S. Varadhan :
“Central limit theorems for additive functionals of reversible Markov chains and applications ,”
pp. 65–70
in
Colloque en l’honneur de Laurent Schwartz, volume 2
(École Polytechnique, Palaiseau, 30 May–3 June, 1983 ).
Astérisque 132 .
1985 .
MR
816760 Zbl
0584.60043 incollection
People
BibTeX
@incollection {key816760m,
AUTHOR = {Kipnis, Claude and Varadhan, S. R. S.},
TITLE = {Central limit theorems for additive
functionals of reversible {M}arkov chains
and applications},
BOOKTITLE = {Colloque en l'honneur de {L}aurent {S}chwartz,
volume 2},
SERIES = {Ast\'erisque},
NUMBER = {132},
YEAR = {1985},
PAGES = {65--70},
NOTE = {(\'Ecole Polytechnique, Palaiseau, 30
May--3 June, 1983). MR:816760. Zbl:0584.60043.},
ISSN = {0303-1179},
}
C. Kipnis and S. R. S. Varadhan :
“Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions Comm. Math. Phys.
104 : 1
(1986 ),
pp. 1–19 .
MR
834478 Zbl
0588.60058 article
Abstract
People
BibTeX
@article {key834478m,
AUTHOR = {Kipnis, C. and Varadhan, S. R. S.},
TITLE = {Central limit theorem for additive functionals
of reversible {M}arkov processes and
applications to simple exclusions},
JOURNAL = {Comm. Math. Phys.},
FJOURNAL = {Communications in Mathematical Physics},
VOLUME = {104},
NUMBER = {1},
YEAR = {1986},
PAGES = {1--19},
URL = {http://projecteuclid.org/euclid.cmp/1104114929},
NOTE = {MR:834478. Zbl:0588.60058.},
ISSN = {0010-3616},
CODEN = {CMPHAY},
}
M. Z. Guo, G. C. Papanicolaou, and S. R. S. Varadhan :
“Nonlinear diffusion limit for a system with nearest neighbor interactions Comm. Math. Phys.
118 : 1
(1988 ),
pp. 31–59 .
MR
954674 Zbl
0652.60107 article
Abstract
People
BibTeX
We consider a system of interacting diffusions. The variables are to be thought of as charges at sites indexed by a periodic one-dimensional lattice. The diffusion preserves the total charge and the interaction is of nearest neighbor type. With the appropriate scaling of lattice spacing and time, a nonlinear diffusion equation is derived for the time evolution of the macroscopic charge density.
@article {key954674m,
AUTHOR = {Guo, M. Z. and Papanicolaou, G. C. and
Varadhan, S. R. S.},
TITLE = {Nonlinear diffusion limit for a system
with nearest neighbor interactions},
JOURNAL = {Comm. Math. Phys.},
FJOURNAL = {Communications in Mathematical Physics},
VOLUME = {118},
NUMBER = {1},
YEAR = {1988},
PAGES = {31--59},
URL = {http://projecteuclid.org/euclid.cmp/1104161907},
NOTE = {MR:954674. Zbl:0652.60107.},
ISSN = {0010-3616},
CODEN = {CMPHAY},
}
C. Kipnis, S. Olla, and S. R. S. Varadhan :
“Hydrodynamics and large deviation for simple exclusion processes Comm. Pure Appl. Math.
42 : 2
(1989 ),
pp. 115–137 .
MR
978701 Zbl
0644.76001 article
Abstract
People
BibTeX
@article {key978701m,
AUTHOR = {Kipnis, C. and Olla, S. and Varadhan,
S. R. S.},
TITLE = {Hydrodynamics and large deviation for
simple exclusion processes},
JOURNAL = {Comm. Pure Appl. Math.},
FJOURNAL = {Communications on Pure and Applied Mathematics},
VOLUME = {42},
NUMBER = {2},
YEAR = {1989},
PAGES = {115--137},
DOI = {10.1002/cpa.3160420202},
NOTE = {MR:978701. Zbl:0644.76001.},
ISSN = {0010-3640},
CODEN = {CPAMA},
}
M. D. Donsker and S. R. S. Varadhan :
“Large deviations from a hydrodynamic scaling limit Comm. Pure Appl. Math.
42 : 3
(April 1989 ),
pp. 243–270 .
MR
982350 Zbl
0780.60027 article
Abstract
People
BibTeX
The problem of describing how a large system evolves towards its equilibrium in terms of the evolution of certain macroscopic quantities can be formulated and studied in widely different contexts. Recently in [Guo, Papanicolaou and Varadhan 1988] one such model has been studied using estimates based on entropy and its rate of change. The main result in [Guo, Papanicolaou and Varadhan 1988] is a law of large numbers type result which asserts that the macroscopic functions evolve according to a specific deterministic motion (to be more specific a nonlinear diffusion equation) as the size gets large and the fluctuations disappear in the scaling. The present article is a complement to [Guo, Papanicolaou and Varadhan 1988] and we are interested here in estimating precisely the probabilities of significant (large) deviations from the deterministic limit. A similar result in a different context has been derived in [Kipnis, Olla and Varadhan 1989].
@article {key982350m,
AUTHOR = {Donsker, M. D. and Varadhan, S. R. S.},
TITLE = {Large deviations from a hydrodynamic
scaling limit},
JOURNAL = {Comm. Pure Appl. Math.},
FJOURNAL = {Communications on Pure and Applied Mathematics},
VOLUME = {42},
NUMBER = {3},
MONTH = {April},
YEAR = {1989},
PAGES = {243--270},
DOI = {10.1002/cpa.3160420303},
NOTE = {MR:982350. Zbl:0780.60027.},
ISSN = {0010-3640},
CODEN = {CPAMA},
}
S. R. S. Varadhan :
“On the derivation of conservation laws for stochastic dynamics ,”
pp. 677–694
in
Analysis, et cetera .
Edited by P. H. Rabinowitz and E. Zehnder .
Academic Press (Boston, MA ),
1990 .
MR
1039368 Zbl
0699.60097 incollection
People
BibTeX
@incollection {key1039368m,
AUTHOR = {Varadhan, S. R. S.},
TITLE = {On the derivation of conservation laws
for stochastic dynamics},
BOOKTITLE = {Analysis, et cetera},
EDITOR = {Rabinowitz, Paul H. and Zehnder, Eduard},
PUBLISHER = {Academic Press},
ADDRESS = {Boston, MA},
YEAR = {1990},
PAGES = {677--694},
NOTE = {MR:1039368. Zbl:0699.60097.},
ISBN = {9780125742498},
}
J. Fritz :
“On the diffusive nature of entropy flow in infinite systems: remarks to a paper: ‘Nonlinear diffusion limit for a system with nearest neighbor interactions’ by M. Z. Guo, G. C. Papanicolau and S. R. S. Varadhan Comm. Math. Phys.
133 : 2
(1990 ),
pp. 331–352 .
MR
1090429 article
Abstract
People
BibTeX
The hydrodynamic behaviour of interacting diffusion processes is investigated by means of entropy (free energy) arguments. The methods of [Guo, Papanicolau and Varadhan 1988] are simplified and extended to infinite systems including a case of anharmonic oscillators in a degenerate thermal noise. Following [Holley 1971; Holley and Stroock 1977] and [Fritz 1982a; 1982b; 1986], we derive a priori bounds for the rate of entropy production in finite volumes as the size of the whole system is infinitely extended. The flow of entropy through the boundary is controlled in much the same way as energy flow in diffusive systems [Fritz 1982b]
@article {key1090429m,
AUTHOR = {Fritz, J.},
TITLE = {On the diffusive nature of entropy flow
in infinite systems: remarks to a paper:
``{N}onlinear diffusion limit for a
system with nearest neighbor interactions''
by {M}. {Z}. {G}uo, {G}. {C}. {P}apanicolau
and {S}. {R}. {S}. {V}aradhan},
JOURNAL = {Comm. Math. Phys.},
FJOURNAL = {Communications in Mathematical Physics},
VOLUME = {133},
NUMBER = {2},
YEAR = {1990},
PAGES = {331--352},
URL = {http://projecteuclid.org/euclid.cmp/1104201401},
NOTE = {MR:1090429.},
ISSN = {0010-3616},
CODEN = {CMPHAY},
}
S. Olla and S. R. S. Varadhan :
“Scaling limit for interacting Ornstein–Uhlenbeck processes Comm. Math. Phys.
135 : 2
(1991 ),
pp. 355–378 .
MR
1087388 Zbl
0725.60086 article
Abstract
People
BibTeX
@article {key1087388m,
AUTHOR = {Olla, Stefano and Varadhan, S. R. S.},
TITLE = {Scaling limit for interacting {O}rnstein--{U}hlenbeck
processes},
JOURNAL = {Comm. Math. Phys.},
FJOURNAL = {Communications in Mathematical Physics},
VOLUME = {135},
NUMBER = {2},
YEAR = {1991},
PAGES = {355--378},
URL = {http://projecteuclid.org/euclid.cmp/1104202030},
NOTE = {MR:1087388. Zbl:0725.60086.},
ISSN = {0010-3616},
CODEN = {CMPHAY},
}
S. R. S. Varadhan :
“Scaling limits for interacting diffusions Comm. Math. Phys.
135 : 2
(1991 ),
pp. 313–353 .
MR
1087387 Zbl
0725.60085 article
Abstract
BibTeX
@article {key1087387m,
AUTHOR = {Varadhan, S. R. S.},
TITLE = {Scaling limits for interacting diffusions},
JOURNAL = {Comm. Math. Phys.},
FJOURNAL = {Communications in Mathematical Physics},
VOLUME = {135},
NUMBER = {2},
YEAR = {1991},
PAGES = {313--353},
URL = {http://projecteuclid.org/euclid.cmp/1104202029},
NOTE = {MR:1087387. Zbl:0725.60085.},
ISSN = {0010-3616},
CODEN = {CMPHAY},
}
S. R. S. Varadhan :
“Entropy methods in hydrodynamical scaling ,”
pp. 103–112
in
Mathematical physics X
(Leipzig, 30 July–9 August, 1991 ).
Edited by K. Schmüdgen .
Springer (Berlin ),
1992 .
MR
1386400 Zbl
0947.82510 incollection
People
BibTeX
@incollection {key1386400m,
AUTHOR = {Varadhan, S. R. S.},
TITLE = {Entropy methods in hydrodynamical scaling},
BOOKTITLE = {Mathematical physics {X}},
EDITOR = {Schm\"udgen, Konrad},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1992},
PAGES = {103--112},
NOTE = {(Leipzig, 30 July--9 August, 1991).
MR:1386400. Zbl:0947.82510.},
ISBN = {9780387551661},
}
S. R. S. Varadhan :
“Entropy methods in hydrodynamic scaling 112–145
in
Nonequilibrium problems in many-particle systems
(Montecatini, Italy, June 15–27, 1992 ).
Edited by L. Arkeryd .
Lecture Notes in Mathematics 1551 .
Springer (Berlin ),
1993 .
MR
1296260 Zbl
0791.60098 incollection
Abstract
People
BibTeX
@incollection {key1296260m,
AUTHOR = {Varadhan, S. R. S.},
TITLE = {Entropy methods in hydrodynamic scaling},
BOOKTITLE = {Nonequilibrium problems in many-particle
systems},
EDITOR = {Arkeryd, Leif},
SERIES = {Lecture Notes in Mathematics},
NUMBER = {1551},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1993},
PAGES = {112--145},
DOI = {10.1007/BFb0090931},
NOTE = {(Montecatini, Italy, June 15--27, 1992).
MR:1296260. Zbl:0791.60098.},
ISBN = {9783540569459},
}
S. R. S. Varadhan :
“Relative entropy and hydrodynamic limits ,”
pp. 329–336
in
Stochastic processes .
Edited by S. Cambanis .
Springer (New York ),
1993 .
MR
1427330 Zbl
0790.60094 incollection
People
BibTeX
@incollection {key1427330m,
AUTHOR = {Varadhan, S. R. S.},
TITLE = {Relative entropy and hydrodynamic limits},
BOOKTITLE = {Stochastic processes},
EDITOR = {Cambanis, Stamatis},
PUBLISHER = {Springer},
ADDRESS = {New York},
YEAR = {1993},
PAGES = {329--336},
NOTE = {MR:1427330. Zbl:0790.60094.},
ISBN = {9780387979212},
}
S. Olla, S. R. S. Varadhan, and H.-T. Yau :
“Hydrodynamical limit for a Hamiltonian system with weak noise Comm. Math. Phys.
155 : 3
(1993 ),
pp. 523–560 .
MR
1231642 Zbl
0781.60101 article
Abstract
People
BibTeX
Starting from a general Hamiltonian system with superstable pairwise
potential, we construct a stochastic dynamics by adding a noise term which exchanges
the momenta of nearby particles. We prove that, in the scaling limit, the time conserved
quantities, energy, momenta and density, satisfy the Euler equation of conservation
laws up to a fixed time \( t \) provided that the Euler equation has a smooth solution with
a given initial data up to time \( t \) . The strength of the noise term is chosen to be very
small (but nonvanishing) so that it disappears in the scaling limit.
@article {key1231642m,
AUTHOR = {Olla, S. and Varadhan, S. R. S. and
Yau, H.-T.},
TITLE = {Hydrodynamical limit for a {H}amiltonian
system with weak noise},
JOURNAL = {Comm. Math. Phys.},
FJOURNAL = {Communications in Mathematical Physics},
VOLUME = {155},
NUMBER = {3},
YEAR = {1993},
PAGES = {523--560},
URL = {http://projecteuclid.org/euclid.cmp/1104253396},
NOTE = {MR:1231642. Zbl:0781.60101.},
ISSN = {0010-3616},
CODEN = {CMPHAY},
}
S. R. S. Varadhan :
“Nonlinear diffusion limit for a system with nearest neighbor interactions. II ,”
pp. 75–128
in
Asymptotic problems in probability theory: stochastic models and diffusions on fractals
(Sanda/Kyoto, 1990 ),
vol. 1 .
Edited by K. D. Elworthy and N. Ikeda .
Pitman Research Notes in Mathematics 283 .
Longman Scientific & Technical (Harlow ),
1993 .
MR
1354152 Zbl
0793.60105 incollection
People
BibTeX
@incollection {key1354152m,
AUTHOR = {Varadhan, S. R. S.},
TITLE = {Nonlinear diffusion limit for a system
with nearest neighbor interactions.
{II}},
BOOKTITLE = {Asymptotic problems in probability theory:
stochastic models and diffusions on
fractals},
EDITOR = {Elworthy, K. D. and Ikeda, Nobuyuki},
VOLUME = {1},
SERIES = {Pitman Research Notes in Mathematics},
NUMBER = {283},
PUBLISHER = {Longman Scientific \& Technical},
ADDRESS = {Harlow},
YEAR = {1993},
PAGES = {75--128},
NOTE = {(Sanda/Kyoto, 1990). MR:1354152. Zbl:0793.60105.},
ISBN = {9780582086566},
}
S. R. S. Varadhan :
“Regularity of self-diffusion coefficient ,”
pp. 387–397
in
The Dynkin Festschrift: Markov processes and their applications .
Edited by M. I. Freĭdlin .
Progress in Probability 34 .
Birkhäuser (Boston, MA ),
1994 .
MR
1311731 Zbl
0822.60089 incollection
People
BibTeX
@incollection {key1311731m,
AUTHOR = {Varadhan, S. R. S.},
TITLE = {Regularity of self-diffusion coefficient},
BOOKTITLE = {The {D}ynkin {F}estschrift: {M}arkov
processes and their applications},
EDITOR = {Fre\u\i dlin, Mark Iosifovich},
SERIES = {Progress in Probability},
NUMBER = {34},
PUBLISHER = {Birkh\"auser},
ADDRESS = {Boston, MA},
YEAR = {1994},
PAGES = {387--397},
NOTE = {MR:1311731. Zbl:0822.60089.},
ISBN = {9780817636968},
}
S. R. S. Varadhan :
“Entropy methods in hydrodynamic scaling 196–208
in
Proceedings of the International Congress of Mathematicians
(Zürich, August 3–11, 1994 ),
vol. I .
Edited by S. D. Chatterji .
Birkhäuser (Basel ),
1995 .
MR
1403922 Zbl
0872.60081 inproceedings
People
BibTeX
@inproceedings {key1403922m,
AUTHOR = {Varadhan, S. R. S.},
TITLE = {Entropy methods in hydrodynamic scaling},
BOOKTITLE = {Proceedings of the {I}nternational {C}ongress
of {M}athematicians},
EDITOR = {Chatterji, S. D.},
VOLUME = {I},
PUBLISHER = {Birkh\"auser},
ADDRESS = {Basel},
YEAR = {1995},
PAGES = {196--208},
URL = {http://www.mathunion.org/ICM/ICM1994.1/Main/icm1994.1.0196.0208.ocr.pdf},
NOTE = {(Z\"urich, August 3--11, 1994). MR:1403922.
Zbl:0872.60081.},
ISBN = {9783764351533},
}
S. R. S. Varadhan :
“Self-diffusion of a tagged particle in equilibrium for asymmetric mean zero random walk with simple exclusion Ann. Inst. H. Poincaré Probab. Statist.
31 : 1
(1995 ),
pp. 273–285 .
MR
1340041 Zbl
0816.60093 article
Abstract
BibTeX
We consider a tagged particle in a simple exclusion model where the probability distribution of the jump sizes has zero mean, but is not necessarily symmetric. We establish for the tagged particle, a central limit theorem under the usual scaling.
@article {key1340041m,
AUTHOR = {Varadhan, S. R. S.},
TITLE = {Self-diffusion of a tagged particle
in equilibrium for asymmetric mean zero
random walk with simple exclusion},
JOURNAL = {Ann. Inst. H. Poincar\'e Probab. Statist.},
FJOURNAL = {Annales de l'Institut Henri Poincar\'e.
Probabilit\'es et Statistiques},
VOLUME = {31},
NUMBER = {1},
YEAR = {1995},
PAGES = {273--285},
URL = {http://www.numdam.org/item?id=AIHPB_1995__31_1_273_0},
NOTE = {MR:1340041. Zbl:0816.60093.},
ISSN = {0246-0203},
CODEN = {AHPBAR},
}
S. R. S. Varadhan :
“The complex story of simple exclusion ,”
pp. 385–400
in
Itô’s stochastic calculus and probability theory .
Edited by N. Ikeda, S. Watanabe, M. Fukushima, and H. Kunita .
Springer (Tokyo ),
1996 .
MR
1439538 Zbl
0866.60092 incollection
People
BibTeX
@incollection {key1439538m,
AUTHOR = {Varadhan, S. R. S.},
TITLE = {The complex story of simple exclusion},
BOOKTITLE = {It\^o's stochastic calculus and probability
theory},
EDITOR = {Ikeda, N. and Watanabe, S. and Fukushima,
M. and Kunita, H.},
PUBLISHER = {Springer},
ADDRESS = {Tokyo},
YEAR = {1996},
PAGES = {385--400},
NOTE = {MR:1439538. Zbl:0866.60092.},
ISBN = {9784431701866},
}
C. Landim, S. Sethuraman, and S. Varadhan :
“Spectral gap for zero-range dynamics Ann. Probab.
24 : 4
(1996 ),
pp. 1871–1902 .
MR
1415232 Zbl
0870.60095 article
Abstract
People
BibTeX
We give a lower bound on the spectral gap for symmetric zero-range processes. Under some conditions on the rate function, we show that the gap shrinks as \( n^{-2} \) , independent of the density, for the dynamics localized on a cube of size \( n^d \) . We follow the method outlined by Lu and Yau, where a similar spectral gap is proved for Kawasaki dynamics.
@article {key1415232m,
AUTHOR = {Landim, C. and Sethuraman, S. and Varadhan,
S.},
TITLE = {Spectral gap for zero-range dynamics},
JOURNAL = {Ann. Probab.},
FJOURNAL = {The Annals of Probability},
VOLUME = {24},
NUMBER = {4},
YEAR = {1996},
PAGES = {1871--1902},
DOI = {10.1214/aop/1041903209},
NOTE = {MR:1415232. Zbl:0870.60095.},
ISSN = {0091-1798},
CODEN = {APBYAE},
}
S. R. S. Varadhan :
“Nongradient models in hydrodynamic scaling ,”
pp. 397–416
in
Analysis, geometry and probability .
Edited by R. Bhatia .
Texts and Readings in Mathematics 10 .
Hindustan Book Agency (Delhi ),
1996 .
MR
1477705 Zbl
0977.60099 incollection
People
BibTeX
@incollection {key1477705m,
AUTHOR = {Varadhan, S. R. S.},
TITLE = {Nongradient models in hydrodynamic scaling},
BOOKTITLE = {Analysis, geometry and probability},
EDITOR = {Bhatia, Rajendra},
SERIES = {Texts and Readings in Mathematics},
NUMBER = {10},
PUBLISHER = {Hindustan Book Agency},
ADDRESS = {Delhi},
YEAR = {1996},
PAGES = {397--416},
NOTE = {MR:1477705. Zbl:0977.60099.},
ISBN = {9788185931128},
}
S. R. S. Varadhan and H. T. Yau :
“Scaling limits for lattice gas models ,”
pp. 39–43
in
Stochastic processes and functional analysis
(Riverside, CA, November 18–20, 1994 ).
Edited by J. A. Goldstein, N. E. Gretsky, and J. J. Uhl .
Lecture Notes in Pure and Applied Mathematics 186 .
Dekker (New York ),
1997 .
MR
1440413 Zbl
0884.60090 incollection
People
BibTeX
@incollection {key1440413m,
AUTHOR = {Varadhan, S. R. S. and Yau, H. T.},
TITLE = {Scaling limits for lattice gas models},
BOOKTITLE = {Stochastic processes and functional
analysis},
EDITOR = {Goldstein, Jerome A. and Gretsky, Neil
E. and Uhl, J. Jerry},
SERIES = {Lecture Notes in Pure and Applied Mathematics},
NUMBER = {186},
PUBLISHER = {Dekker},
ADDRESS = {New York},
YEAR = {1997},
PAGES = {39--43},
NOTE = {(Riverside, CA, November 18--20, 1994).
MR:1440413. Zbl:0884.60090.},
ISBN = {9780824798017},
}
S. R. S. Varadhan and H.-T. Yau :
“Diffusive limit of lattice gas with mixing conditions Asian J. Math.
1 : 4
(December 1997 ),
pp. 623–678 .
MR
1621569 Zbl
0947.60089 article
Abstract
People
BibTeX
We prove, under certain mixing conditions, that the hydrodynamical limit of a stochastic lattice gas on the cubic lattice \( \mathbb{Z}^d \) is governed by a nonlinear diffusion equation. Following [Varadhan 1994], we characterize the diffusion coefficient by a variational formula, which is equivalent to the Green–Kubo formula. The fluctuation-dissipation equation is established rigorously as an important step of the proof. Our mixing conditions are implied by the Dobrushin–Shlosman mixing conditions which are always valid at high temperatures.
@article {key1621569m,
AUTHOR = {Varadhan, S. R. S. and Yau, Horng-Tzer},
TITLE = {Diffusive limit of lattice gas with
mixing conditions},
JOURNAL = {Asian J. Math.},
FJOURNAL = {The Asian Journal of Mathematics},
VOLUME = {1},
NUMBER = {4},
MONTH = {December},
YEAR = {1997},
PAGES = {623--678},
URL = {http://intlpress.com/AJM/p/1997/1_4/AJM-1-4-623-678.pdf},
NOTE = {MR:1621569. Zbl:0947.60089.},
ISSN = {1093-6106},
}
J. Quastel, F. Rezakhanlou, and S. R. S. Varadhan :
“Large deviations for the symmetric simple exclusion process in dimensions \( d\geq 3 \) Probab. Theory Related Fields
113 : 1
(1999 ),
pp. 1–84 .
MR
1670733 Zbl
0928.60087 article
Abstract
People
BibTeX
We consider symmetric simple exclusion processes with \( L=\overline{\rho}N^d \) particles in a periodic \( d \) -dimensional lattice of width \( N \) . We perform the diffusive hydrodynamic scaling of space and time. The initial
condition is arbitrary and is typically far away form equilibrium. It specifies in the scaling limit a density profile on the \( d \) -dimensional torus. We are interested in the large deviations of the empirical process,
\[ N^{-d}\Bigl[\sum_1^L\delta_{x_i(\,\cdot\,)}\Bigr] \]
as random variables taking values in the space of measures on \( D[0,1] \) . We prove a large deviation principle, with a rate function that is more or less universal, involving explicitly besides the initial profile, only such canonical objects as bulk and self diffusion coefficients.
@article {key1670733m,
AUTHOR = {Quastel, J. and Rezakhanlou, F. and
Varadhan, S. R. S.},
TITLE = {Large deviations for the symmetric simple
exclusion process in dimensions \$d\geq
3\$},
JOURNAL = {Probab. Theory Related Fields},
FJOURNAL = {Probability Theory and Related Fields},
VOLUME = {113},
NUMBER = {1},
YEAR = {1999},
PAGES = {1--84},
DOI = {10.1007/s004400050202},
NOTE = {MR:1670733. Zbl:0928.60087.},
ISSN = {0178-8051},
CODEN = {PTRFEU},
}
S. R. S. Varadhan :
“Large deviations for interacting particle systems ,”
pp. 373–383
in
Perplexing problems in probability .
Edited by M. Bramson and R. Durrett .
Progress in Probability 44 .
Birkhäuser (Boston, MA ),
1999 .
MR
1703141 Zbl
0941.60096 incollection
People
BibTeX
@incollection {key1703141m,
AUTHOR = {Varadhan, S. R. S.},
TITLE = {Large deviations for interacting particle
systems},
BOOKTITLE = {Perplexing problems in probability},
EDITOR = {Bramson, Maury and Durrett, Rick},
SERIES = {Progress in Probability},
NUMBER = {44},
PUBLISHER = {Birkh\"auser},
ADDRESS = {Boston, MA},
YEAR = {1999},
PAGES = {373--383},
NOTE = {MR:1703141. Zbl:0941.60096.},
ISBN = {9780817640934},
}
S. R. S. Varadhan :
“Infinite particle systems and their scaling limits 306–315
in
Mathematical physics 2000 .
Edited by A. Fokas, A. Grigoryan, T. Kibble, and B. Zegarlinski .
Imperial College Press (London ),
2000 .
MR
1773051 Zbl
1028.82516 incollection
Abstract
People
BibTeX
The subject of nonequlibrium statistical mechanics deals with the question of how, a complex system with one or more conserved quantities, approaches its equilibrium. Although initially investigated in the context of classical mechanical systems that conserve mass, momenta and energy, its paradigm is relevent in the study of stochastic paticle systems with conserved quantities and mutiple equilibria. In fact the inherent noise in the stochastic systems provides a degree of stability that is absent in classical systems. During last twenty years several such models have been studied, with varying amounts of noise, and a body of rigorous results has been obtained.
@incollection {key1773051m,
AUTHOR = {Varadhan, S. R. S.},
TITLE = {Infinite particle systems and their
scaling limits},
BOOKTITLE = {Mathematical physics 2000},
EDITOR = {Fokas, A. and Grigoryan, A. and Kibble,
T. and Zegarlinski, B.},
PUBLISHER = {Imperial College Press},
ADDRESS = {London},
YEAR = {2000},
PAGES = {306--315},
DOI = {10.1142/9781848160224_0016},
NOTE = {MR:1773051. Zbl:1028.82516.},
ISBN = {9781848160224},
}
S. Sethuraman, S. R. S. Varadhan, and H.-T. Yau :
“Diffusive limit of a tagged particle in asymmetric simple exclusion processes Comm. Pure Appl. Math.
53 : 8
(August 2000 ),
pp. 972–1006 .
MR
1755948 Zbl
1029.60084 article
Abstract
People
BibTeX
@article {key1755948m,
AUTHOR = {Sethuraman, Sunder and Varadhan, S.
R. S. and Yau, Horng-Tzer},
TITLE = {Diffusive limit of a tagged particle
in asymmetric simple exclusion processes},
JOURNAL = {Comm. Pure Appl. Math.},
FJOURNAL = {Communications on Pure and Applied Mathematics},
VOLUME = {53},
NUMBER = {8},
MONTH = {August},
YEAR = {2000},
PAGES = {972--1006},
DOI = {10.1002/1097-0312(200008)53:8%3C972::AID-CPA2%3E3.0.CO;2-%23},
NOTE = {MR:1755948. Zbl:1029.60084.},
ISSN = {0010-3640},
CODEN = {CPAMA},
}
S. R. S. Varadhan :
“Lectures on hydrodynamic scaling ,”
pp. 3–40
in
Hydrodynamic limits and related topics
(Toronto, October 7–10, 1998 ).
Edited by S. Feng, A. T. Lawniczak, and S. R. S. Varadhan .
Fields Institute Communications 27 .
American Mathematical Society (Providence, RI ),
2000 .
MR
1798641 Zbl
1060.82514 incollection
Abstract
People
BibTeX
@incollection {key1798641m,
AUTHOR = {Varadhan, S. R. S.},
TITLE = {Lectures on hydrodynamic scaling},
BOOKTITLE = {Hydrodynamic limits and related topics},
EDITOR = {Feng, Shui and Lawniczak, Anna T. and
Varadhan, S. R. S.},
SERIES = {Fields Institute Communications},
NUMBER = {27},
PUBLISHER = {American Mathematical Society},
ADDRESS = {Providence, RI},
YEAR = {2000},
PAGES = {3--40},
NOTE = {(Toronto, October 7--10, 1998). MR:1798641.
Zbl:1060.82514.},
ISBN = {9780821819937},
}
Hydrodynamic limits and related topics
(Toronto, October 7–10, 1998 ).
Edited by S. Feng, A. T. Lawniczak, and S. R. S. Varadhan .
Fields Institute Communications 27 .
American Mathematical Society (Providence, RI ),
2000 .
MR
1798640 book
People
BibTeX
@book {key1798640m,
TITLE = {Hydrodynamic limits and related topics},
EDITOR = {Feng, Shui and Lawniczak, Anna T. and
Varadhan, S. R. S.},
SERIES = {Fields Institute Communications},
NUMBER = {27},
PUBLISHER = {American Mathematical Society},
ADDRESS = {Providence, RI},
YEAR = {2000},
PAGES = {x+141},
NOTE = {(Toronto, October 7--10, 1998). MR:1798640.},
ISBN = {9780821819937},
}
C. Landim, S. Olla, and S. R. S. Varadhan :
“Asymptotic behavior of a tagged particle in simple exclusion processes Bol. Soc. Brasil. Mat. (N.S.)
31 : 3
(2000 ),
pp. 241–275 .
MR
1817088 Zbl
0983.60100 article
Abstract
People
BibTeX
We review in this article central limit theorems for a tagged particle in the
simple exclusion process. In the first two sections we present a general method to prove
central limit theorems for additive functional of Markov processes. These results are
then applied to the case of a tagged particle in the exclusion process. Related questions,
such as smoothness of the diffusion coefficient and finite dimensional approximations,
are considered in the last section.
@article {key1817088m,
AUTHOR = {Landim, C. and Olla, S. and Varadhan,
S. R. S.},
TITLE = {Asymptotic behavior of a tagged particle
in simple exclusion processes},
JOURNAL = {Bol. Soc. Brasil. Mat. (N.S.)},
FJOURNAL = {Boletim da Sociedade Brasileira de Matem\'atica.
Nova S\'erie},
VOLUME = {31},
NUMBER = {3},
YEAR = {2000},
PAGES = {241--275},
DOI = {10.1007/BF01241629},
NOTE = {MR:1817088. Zbl:0983.60100.},
ISSN = {0100-3569},
}
S. R. S. Varadhan :
“Scaling limits of large interacting systems ,”
pp. 144–158
in
The mathematical sciences after the year 2000
(Beirut, 11–15 January, 1999 ).
Edited by K. Bitar, A. Chamseddine, and W. Sabra .
World Scientific (River Edge, NJ ),
2000 .
MR
1799447 Zbl
1112.82320 incollection
People
BibTeX
@incollection {key1799447m,
AUTHOR = {Varadhan, S. R. S.},
TITLE = {Scaling limits of large interacting
systems},
BOOKTITLE = {The mathematical sciences after the
year 2000},
EDITOR = {Bitar, Khalil and Chamseddine, Ali and
Sabra, Wafic},
PUBLISHER = {World Scientific},
ADDRESS = {River Edge, NJ},
YEAR = {2000},
PAGES = {144--158},
NOTE = {(Beirut, 11--15 January, 1999). MR:1799447.
Zbl:1112.82320.},
ISBN = {9789810242237},
}
C. Landim, S. Olla, and S. R. S. Varadhan :
“Symmetric simple exclusion process: regularity of the self-diffusion coefficient 307–321
in
Dedicated to Joel L. Lebowitz on his seventieth birthday ,
published as Comm. Math. Phys.
224 : 1
(2001 ).
MR
1869001 Zbl
0994.60093 incollection
Abstract
People
BibTeX
We prove that the self-diffusion coefficient of a tagged particle in the symmetric exclusion process in \( Z^d \) , which is in equilibrium at density \( \alpha \) , is of class \( C^{\infty} \) as a function of \( \alpha \) in the closed interval \( [0,1] \) . The proof provides also a recursive method to compute the Taylor expansion at the boundaries.
@article {key1869001m,
AUTHOR = {Landim, C. and Olla, S. and Varadhan,
S. R. S.},
TITLE = {Symmetric simple exclusion process:
regularity of the self-diffusion coefficient},
JOURNAL = {Comm. Math. Phys.},
FJOURNAL = {Communications in Mathematical Physics},
VOLUME = {224},
NUMBER = {1},
YEAR = {2001},
PAGES = {307--321},
DOI = {10.1007/s002200100513},
NOTE = {\textit{Dedicated to Joel L. Lebowitz
on his seventieth birthday}. MR:1869001.
Zbl:0994.60093.},
ISSN = {0010-3616},
CODEN = {CMPHAY},
}
S. R. S. Varadhan :
“Large deviation and hydrodynamic scaling ,”
pp. 265–286
in
Taniguchi conference on mathematics Nara ’98
(Nara, Japan, December 15–20, 1998 ).
Edited by M. Maruyama and T. Sunada .
Advanced Studies in Pure Mathematics 31 .
Mathematical Society of Japan (Tokyo ),
2001 .
MR
1865096 Zbl
1006.60019 incollection
People
BibTeX
@incollection {key1865096m,
AUTHOR = {Varadhan, Srinivasa R. S.},
TITLE = {Large deviation and hydrodynamic scaling},
BOOKTITLE = {Taniguchi conference on mathematics
{N}ara '98},
EDITOR = {Maruyama, Masaki and Sunada, Toshikazu},
SERIES = {Advanced Studies in Pure Mathematics},
NUMBER = {31},
PUBLISHER = {Mathematical Society of Japan},
ADDRESS = {Tokyo},
YEAR = {2001},
PAGES = {265--286},
NOTE = {(Nara, Japan, December 15--20, 1998).
MR:1865096. Zbl:1006.60019.},
ISBN = {9784931469136},
}
C. Landim, S. Olla, and S. R. S. Varadhan :
“Finite-dimensional approximation of the self-diffusion coefficient for the exclusion process Ann. Probab.
30 : 2
(2002 ),
pp. 483–508 .
MR
1905849 Zbl
1018.60097 article
Abstract
People
BibTeX
@article {key1905849m,
AUTHOR = {Landim, C. and Olla, S. and Varadhan,
S. R. S.},
TITLE = {Finite-dimensional approximation of
the self-diffusion coefficient for the
exclusion process},
JOURNAL = {Ann. Probab.},
FJOURNAL = {The Annals of Probability},
VOLUME = {30},
NUMBER = {2},
YEAR = {2002},
PAGES = {483--508},
DOI = {10.1214/aop/1023481000},
NOTE = {MR:1905849. Zbl:1018.60097.},
ISSN = {0091-1798},
CODEN = {APBYAE},
}
S. R. S. Varadhan :
“Particle systems and partial differential equations ,”
pp. 217–222
in
Lectures on partial differential equations: Proceedings in honor of Louis Nirenberg’s 75th birthday .
Edited by S.-Y. A. Chang, C.-S. Lin, and H.-T. Yau .
New Studies in Advanced Mathematics 2 .
International Press (Somerville, MA ),
2003 .
MR
2055850 Zbl
1167.82351 incollection
People
BibTeX
@incollection {key2055850m,
AUTHOR = {Varadhan, S. R. S.},
TITLE = {Particle systems and partial differential
equations},
BOOKTITLE = {Lectures on partial differential equations:
{P}roceedings in honor of {L}ouis {N}irenberg's
75th birthday},
EDITOR = {Chang, Sun-Yung A. and Lin, Chang-Shou
and Yau, Horng-Tzer},
SERIES = {New Studies in Advanced Mathematics},
NUMBER = {2},
PUBLISHER = {International Press},
ADDRESS = {Somerville, MA},
YEAR = {2003},
PAGES = {217--222},
NOTE = {MR:2055850. Zbl:1167.82351.},
ISBN = {9781571461117},
}
S. R. S. Varadhan :
“Large deviations for the asymmetric simple exclusion process ,”
pp. 1–27
in
Stochastic analysis on large scale interacting systems .
Edited by T. Funaki and H. Osada .
Advanced Studies in Pure Mathematics 39 .
Math. Soc. Japan (Tokyo ),
2004 .
MR
2073328 Zbl
1114.60026 incollection
People
BibTeX
@incollection {key2073328m,
AUTHOR = {Varadhan, Srinivasa R. S.},
TITLE = {Large deviations for the asymmetric
simple exclusion process},
BOOKTITLE = {Stochastic analysis on large scale interacting
systems},
EDITOR = {Funaki, Tadahisa and Osada, Hirofumi},
SERIES = {Advanced Studies in Pure Mathematics},
NUMBER = {39},
PUBLISHER = {Math. Soc. Japan},
ADDRESS = {Tokyo},
YEAR = {2004},
PAGES = {1--27},
NOTE = {MR:2073328. Zbl:1114.60026.},
ISBN = {9784931469242},
}
C. Landim, S. Olla, and S. R. S. Varadhan :
“On viscosity and fluctuation-dissipation in exclusion processes J. Statist. Phys.
115 : 1–2
(2004 ),
pp. 323–363 .
MR
2070098 Zbl
1157.82355 article
Abstract
People
BibTeX
@article {key2070098m,
AUTHOR = {Landim, C. and Olla, S. and Varadhan,
S. R. S.},
TITLE = {On viscosity and fluctuation-dissipation
in exclusion processes},
JOURNAL = {J. Statist. Phys.},
FJOURNAL = {Journal of Statistical Physics},
VOLUME = {115},
NUMBER = {1-2},
YEAR = {2004},
PAGES = {323--363},
DOI = {10.1023/B:JOSS.0000019814.73545.28},
NOTE = {\textit{In honor of {G}ianni {J}ona-{L}asinio's
70th birthday}. MR:2070098. Zbl:1157.82355.},
ISSN = {0022-4715},
CODEN = {JSTPSB},
}
C. Landim, S. Olla, and S. R. S. Varadhan :
“Diffusive behaviour of the equilibrium fluctuations in the asymmetric exclusion processes 307–324
in
Stochastic analysis on large scale interacting systems .
Edited by T. Funaki and H. Osada .
Advanced Studies in Pure Mathematics 39 .
Mathemnatical Society of Japan (Tokyo ),
2004 .
MR
2073338 Zbl
1080.60094 incollection
Abstract
People
BibTeX
@incollection {key2073338m,
AUTHOR = {Landim, Claudio and Olla, Stefano and
Varadhan, Srinivasa R. S.},
TITLE = {Diffusive behaviour of the equilibrium
fluctuations in the asymmetric exclusion
processes},
BOOKTITLE = {Stochastic analysis on large scale interacting
systems},
EDITOR = {Funaki, Tadahisa and Osada, Hirofumi},
SERIES = {Advanced Studies in Pure Mathematics},
NUMBER = {39},
PUBLISHER = {Mathemnatical Society of Japan},
ADDRESS = {Tokyo},
YEAR = {2004},
PAGES = {307--324},
URL = {http://basepub.dauphine.fr/xmlui/handle/123456789/},
NOTE = {MR:2073338. Zbl:1080.60094.},
ISBN = {9784931469242},
}
S. R. S. Varadhan :
“Large deviations and scaling limit Lett. Math. Phys.
88 : 1–3
(2009 ),
pp. 175–185 .
MR
2512145 Zbl
1185.60024 article
Abstract
BibTeX
@article {key2512145m,
AUTHOR = {Varadhan, Srinivasa R. S.},
TITLE = {Large deviations and scaling limit},
JOURNAL = {Lett. Math. Phys.},
FJOURNAL = {Letters in Mathematical Physics},
VOLUME = {88},
NUMBER = {1-3},
YEAR = {2009},
PAGES = {175--185},
DOI = {10.1007/s11005-009-0303-x},
NOTE = {MR:2512145. Zbl:1185.60024.},
ISSN = {0377-9017},
CODEN = {LMPHDY},
}
S. R. S. Varadhan :
“Scaling limits 247–262
in
Perspectives in mathematical sciences ,
vol. I: Probability and statistics .
Edited by N. S. N. Sastry, T. S. S. R. K. Rao, M. Delampady, and B. Rajeev .
Statistical Science and Interdisciplinary Research 7 .
World Scientific (Hackensack, NJ ),
2009 .
MR
2581747 incollection
People
BibTeX
@incollection {key2581747m,
AUTHOR = {Varadhan, S. R. S.},
TITLE = {Scaling limits},
BOOKTITLE = {Perspectives in mathematical sciences},
EDITOR = {Sastry, N. S. Narasimha and Rao, T.
S. S. R. K. and Delampady, Mohan and
Rajeev, B.},
VOLUME = {I: Probability and statistics},
SERIES = {Statistical Science and Interdisciplinary
Research},
NUMBER = {7},
PUBLISHER = {World Scientific},
ADDRESS = {Hackensack, NJ},
YEAR = {2009},
PAGES = {247--262},
DOI = {10.1142/9789814273633_0011},
NOTE = {MR:2581747.},
ISBN = {9789814273626},
}
S. Sethuraman and S. R. S. Varadhan :
Large deviations for the current and tagged particle in 1D nearest-neighbor symmetric simple exclusion .
Preprint ,
January 2011 .
ArXiv
1101.1479 techreport
Abstract
People
BibTeX
Laws of large numbers, starting from certain nonequilibrium measures, have been shown for the integrated current across a bond, and a tagged particle in one dimensional symmetric nearest-neighbor simple exclusion ([Jara and Landim 2006]). In this article, we prove corresponding large deviation principles, and evaluate the rate functions, showing different growth behaviors near and far from their zeroes which connect with results in [Derrida and Gerschenfeld 2009].
@techreport {key1101.1479a,
AUTHOR = {Sethuraman, Sunder and Varadhan, S.
R. S.},
TITLE = {Large deviations for the current and
tagged particle in 1{D} nearest-neighbor
symmetric simple exclusion},
TYPE = {Preprint},
MONTH = {January},
YEAR = {2011},
PAGES = {42},
NOTE = {ArXiv:1101.1479.},
}
