G. C. Papanicolaou and S. R. S. Varadhan :
“A limit theorem with strong mixing in Banach space and two applications to stochastic differential equations ,”
Comm. Pure Appl. Math.
26 : 4
(July 1973 ),
pp. 497–524 .
MR
0383530
Zbl
0253.60065
article

Abstract
People
BibTeX

In [Papanicolaou and Hersh 1972] the asymptotic behavior of the expected value of the solution of an abstract stochastic equation was investigated. This work was motivated by the results of Khasminskii [1966] and Stratonovich [1968] for stochastic ordinary differential equations and other works (cf. [Kubo 1963; Lax 1966; Papanicolaou and Keller 1971]) concerned with operator equations. The results obtained in [Papanicolaou and Hersh 1972] were limited by severe restrictions on the allowed from of the stochastic perturbation. Recently, Cogburn and Hersh [Cogburn and Hersh 1973] have generalized the results of [Papanicolaou and Hersh 1972] considerably by allowing a much broader class of stochastic perturbations and requiring only a strong mixing condition. Our aim here is to improve the results of [Cogburn and Hersh 1973] by giving an estimate for the error committed in the asymptotic approximation. If \( \varepsilon \) denotes the small parameter of the problem we show that the error is \( O(\varepsilon) \) as \( \varepsilon\to 0 \) . The results in [Papanicolaou and Hersh 1972; Khasminskii 1966; Stratonovich 1968; Cogburn and Hersh 1973] show only that the error is \( o(1) \) . Our estimate is best possible since it is achieved for the classical central limit theorem which is a special case of our Theorem 2.

@article {key0383530m,
AUTHOR = {Papanicolaou, G. C. and Varadhan, S.
R. S.},
TITLE = {A limit theorem with strong mixing in
{B}anach space and two applications
to stochastic differential equations},
JOURNAL = {Comm. Pure Appl. Math.},
FJOURNAL = {Communications on Pure and Applied Mathematics},
VOLUME = {26},
NUMBER = {4},
MONTH = {July},
YEAR = {1973},
PAGES = {497--524},
DOI = {10.1002/cpa.3160260405},
NOTE = {MR:0383530. Zbl:0253.60065.},
ISSN = {0010-3640},
}
G. C. Papanicolaou, D. Stroock, and S. R. S. Varadhan :
“Martingale approach to some limit theorems ,”
pp. ii+120 pp.
in
Duke turbulence conference
(Durham, NC, April 23–25, 1976 ).
Edited by P. L. Chow .
Duke University Mathematics Series III .
Duke University (Durham, NC ),
1977 .
MR
0461684
incollection

People
BibTeX
@incollection {key0461684m,
AUTHOR = {Papanicolaou, G. C. and Stroock, D.
and Varadhan, S. R. S.},
TITLE = {Martingale approach to some limit theorems},
BOOKTITLE = {Duke turbulence conference},
EDITOR = {Chow, P. L.},
SERIES = {Duke University Mathematics Series},
NUMBER = {III},
PUBLISHER = {Duke University},
ADDRESS = {Durham, NC},
YEAR = {1977},
PAGES = {ii+120 pp.},
NOTE = {(Durham, NC, April 23--25, 1976). MR:0461684.},
}
G. C. Papanicolaou and S. R. S. Varadhan :
“Diffusion in regions with many small holes ,”
pp. 190–206
in
Stochastic differential systems
(Vilnius, Lithuania, 28 August–2 September, 1978 ).
Edited by B. Grigelionis .
Lecture Notes in Control and Information Sciences 25 .
Springer (Berlin ),
1980 .
MR
609184
Zbl
0485.60076
incollection

Abstract
People
BibTeX

Let \( D \) be a bounded open set containing the origin, having \( C^2 \) boundary and with diameter less than or equal to one. For each \( N = 1,2,\dots \) , let \( y_1^{(N)}, y_2^{(N)},\dots,y_N^{(N)} \) be points in \( \mathbb{R}^3 \) and define sets \( D_i^{(N)} \) by
\[ D_i^{(N)} = \bigl\{x\in \mathbb{R}^3\mid N(x-y_i^{(N)})\in D\bigr\}, \]
\( i = 1,2,\ldots,N \) . We shall call the set \( D_i^{(N)} \) the hole centered at \( y_i^{(N)} \) with diameter less than or equal to \( N^{-1} \) . Let \( G^{(N)} \) denote the region
\[ G^{(N)} = \mathbb{R}^3-\bigcup_{i=1}^N D_i^{(N)} \]
which is \( \mathbb{R}^3 \) with holes of diameter \( \leq N^{-1} \) centered at \( y_1^{(N)},\dots,y_N^{(N)} \) . We shall analyze the asymptotic behavior of \( u^{(N)}(x,t) \) as \( N\to\infty \) which is the solution of
\begin{align*}
\frac{\partial}{\partial t} u^{(N)}(x,t) &= \frac{1}{2}\Delta u^{(N)}(x,t), && t > 0,\ x\in G^{(N)},\\
u^{(N)}(x,t) &=0, && t > 0,\ x\in\partial G^{(N)} = \bigcup_{i=1}^N\partial D_i^{(N)},\\
u^{(N)}(x,0) &=f(x), && x\in G^{(N)}, \end{align*}
with \( f(x) \) a given bounded continuous function with compact support in \( \mathbb{R}^3 \) .

@incollection {key609184m,
AUTHOR = {Papanicolaou, G. C. and Varadhan, S.
R. S.},
TITLE = {Diffusion in regions with many small
holes},
BOOKTITLE = {Stochastic differential systems},
EDITOR = {Grigelionis, Bronius},
SERIES = {Lecture Notes in Control and Information
Sciences},
NUMBER = {25},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1980},
PAGES = {190--206},
DOI = {10.1007/BFb0004010},
NOTE = {(Vilnius, Lithuania, 28 August--2 September,
1978). MR:609184. Zbl:0485.60076.},
ISBN = {9783540104988},
}
G. C. Papanicolaou and S. R. S. Varadhan :
“Boundary value problems with rapidly oscillating random coefficients ,”
pp. 835–873
in
Random fields
(Esztergom, Hungary, 1979 ),
vol. II .
Edited by J. Fritz, J. L. Lebowitz, and D. Szász .
Colloquia mathematica Societatis János Bolyai 27 .
North-Holland (Amsterdam ),
1981 .
MR
712714
Zbl
0499.60059
incollection

People
BibTeX
@incollection {key712714m,
AUTHOR = {Papanicolaou, G. C. and Varadhan, S.
R. S.},
TITLE = {Boundary value problems with rapidly
oscillating random coefficients},
BOOKTITLE = {Random fields},
EDITOR = {Fritz, J. and Lebowitz, Joel Louis and
Sz\'asz, D.},
VOLUME = {II},
SERIES = {Colloquia mathematica Societatis J\'anos
Bolyai},
NUMBER = {27},
PUBLISHER = {North-Holland},
ADDRESS = {Amsterdam},
YEAR = {1981},
PAGES = {835--873},
NOTE = {(Esztergom, Hungary, 1979). MR:712714.
Zbl:0499.60059.},
}
G. C. Papanicolaou and S. R. S. Varadhan :
“Diffusions with random coefficients ,”
pp. 547–552
in
Statistics and probability: essays in honor of C. R. Rao .
Edited by G. Kallianpur, P. R. Krishnaiah, and J. K. Ghosh .
North-Holland (Amsterdam ),
1982 .
MR
659505
Zbl
0486.60076
incollection

People
BibTeX
@incollection {key659505m,
AUTHOR = {Papanicolaou, George C. and Varadhan,
S. R. S.},
TITLE = {Diffusions with random coefficients},
BOOKTITLE = {Statistics and probability: essays in
honor of {C}. {R}. {R}ao},
EDITOR = {Kallianpur, G. and Krishnaiah, Paruchuri
R. and Ghosh, J. K.},
PUBLISHER = {North-Holland},
ADDRESS = {Amsterdam},
YEAR = {1982},
PAGES = {547--552},
NOTE = {MR:659505. Zbl:0486.60076.},
ISBN = {9780444861306},
}
G. Papanicolaou and S. R. S. Varadhan :
“Ornstein–Uhlenbeck process in a random potential ,”
Comm. Pure Appl. Math.
38 : 6
(November 1985 ),
pp. 819–834 .
MR
812349
Zbl
0617.60078
article

People
BibTeX
@article {key812349m,
AUTHOR = {Papanicolaou, G. and Varadhan, S. R.
S.},
TITLE = {Ornstein--{U}hlenbeck process in a random
potential},
JOURNAL = {Comm. Pure Appl. Math.},
FJOURNAL = {Communications on Pure and Applied Mathematics},
VOLUME = {38},
NUMBER = {6},
MONTH = {November},
YEAR = {1985},
PAGES = {819--834},
DOI = {10.1002/cpa.3160380611},
NOTE = {MR:812349. Zbl:0617.60078.},
ISSN = {0010-3640},
CODEN = {CPAMA},
}
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},
}
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},
}