D. W. Stroock and S. R. S. Varadhan :
“Diffusion processes with continuous coefficients. II ,”
Comm. Pure Appl. Math.
22
(July 1969 ),
pp. 479–530 .
MR
0254923
Zbl
0167.43904
article
People
BibTeX
@article {key0254923m,
AUTHOR = {Stroock, Daniel W. and Varadhan, S.
R. S.},
TITLE = {Diffusion processes with continuous
coefficients. {II}},
JOURNAL = {Comm. Pure Appl. Math.},
FJOURNAL = {Communications on Pure and Applied Mathematics},
VOLUME = {22},
MONTH = {July},
YEAR = {1969},
PAGES = {479--530},
DOI = {10.1002/cpa.3160220404},
NOTE = {MR:0254923. Zbl:0167.43904.},
ISSN = {0010-3640},
}
D. W. Stroock and S. R. S. Varadhan :
“Diffusion processes with continuous coefficients. I ,”
Comm. Pure Appl. Math.
22
(May 1969 ),
pp. 345–400 .
MR
0253426
Zbl
0167.43903
article
Abstract
People
BibTeX
The aim of this article is to study diffusion processes on \( \mathbb{R}^d \) corresponding to coefficients \( a=a_{ij}(t,x) \) , \( 1\leq i,j\leq d \) , and \( b=b_i(t,x) \) , \( 1\leq i\leq d \) .
@article {key0253426m,
AUTHOR = {Stroock, Daniel W. and Varadhan, S.
R. S.},
TITLE = {Diffusion processes with continuous
coefficients. {I}},
JOURNAL = {Comm. Pure Appl. Math.},
FJOURNAL = {Communications on Pure and Applied Mathematics},
VOLUME = {22},
MONTH = {May},
YEAR = {1969},
PAGES = {345--400},
DOI = {10.1002/cpa.3160220304},
NOTE = {MR:0253426. Zbl:0167.43903.},
ISSN = {0010-3640},
}
D. W. Stroock and S. R. S. Varadhan :
“Diffusion processes with boundary conditions ,”
Comm. Pure Appl. Math.
24
(March 1971 ),
pp. 147–225 .
MR
0277037
Zbl
0227.76131
article
People
BibTeX
@article {key0277037m,
AUTHOR = {Stroock, Daniel W. and Varadhan, S.
R. S.},
TITLE = {Diffusion processes with boundary conditions},
JOURNAL = {Comm. Pure Appl. Math.},
FJOURNAL = {Communications on Pure and Applied Mathematics},
VOLUME = {24},
MONTH = {March},
YEAR = {1971},
PAGES = {147--225},
DOI = {10.1002/cpa.3160240206},
NOTE = {MR:0277037. Zbl:0227.76131.},
ISSN = {0010-3640},
}
D. W. Stroock and S. R. S. Varadhan :
“Diffusion processes and martingales. II ,”
pp. 67–75
in
Martingales
(Oberwolfach, May 17–30, 1970 ).
Edited by H. Dinges .
Lecture Notes in Mathematics 190 .
Springer (Berlin ),
1971 .
MR
0359025
incollection
People
BibTeX
@incollection {key0359025m,
AUTHOR = {Stroock, Daniel W. and Varadhan, S.
R. S.},
TITLE = {Diffusion processes and martingales.
{II}},
BOOKTITLE = {Martingales},
EDITOR = {Dinges, Hermann},
SERIES = {Lecture Notes in Mathematics},
NUMBER = {190},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1971},
PAGES = {67--75},
DOI = {10.1007/BFb0065894},
NOTE = {(Oberwolfach, May 17--30, 1970). MR:0359025.},
ISBN = {9783540053965},
}
D. W. Stroock and S. R. S. Varadhan :
“Diffusion processes and martingales. I ,”
pp. 60–66
in
Martingales
(Oberwolfach, May 17–30, 1970 ).
Edited by H. Dinges .
Lecture Notes in Mathematics 190 .
Springer (Berlin ),
1971 .
MR
0359024
incollection
Abstract
People
BibTeX
Let \( a:[0,\infty)\times\mathbb{R}^d\to S_d \) and \( b:[0,\infty)\times\mathbb{R}^d\to\mathbb{R}^d \) be bounded measurable functions, and form the elliptic operator
\[ L_t=\frac{1}{2}\sum_{i,j=1}^d a_{ij}(t,x)\frac{\partial^2}{\partial x_i\partial x_j}+\sum_{i=1}^d b_i(t,x)\frac{\partial}{\partial x_i} .\]
Given \( s\geq 0 \) and \( x\in\mathbb{R}^d \) , our aim is to prove the existence and uniqueness of a measure \( P_{s,x} \) which bears the same relation to \( L_t \) as the \( d \) dimensional Wiener \( W_{s,x} \) measure, conditioned to start from \( x \) at time \( s \) , bears to \( \frac{1}{2}\Delta \) (\( \Delta = \sum_{i=1}^d \partial^2/\partial x_i^2 \) ). We have succeeded in this program for the case when \( a \) is continuous and positive-definite valued.
@incollection {key0359024m,
AUTHOR = {Stroock, Daniel W. and Varadhan, S.
R. S.},
TITLE = {Diffusion processes and martingales.
{I}},
BOOKTITLE = {Martingales},
EDITOR = {Dinges, Hermann},
SERIES = {Lecture Notes in Mathematics},
NUMBER = {190},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1971},
PAGES = {60--66},
DOI = {10.1007/BFb0065893},
NOTE = {(Oberwolfach, May 17--30, 1970). MR:0359024.},
}
D. W. Stroock and S. R. S. Varadhan :
“Diffusion processes ,”
pp. 361–368
in
Proceedings of the sixth Berkeley symposium on mathematical statistics and probability
(Berkeley, CA, June 21–July 18, 1970 ),
vol. III: Probability theory .
Edited by L. M. Le Cam, J. Neyman, and E. L. Scott .
Univ. California Press (Berkeley, CA ),
1972 .
MR
0397899
Zbl
0255.60055
inproceedings
People
BibTeX
@inproceedings {key0397899m,
AUTHOR = {Stroock, Daniel W. and Varadhan, S.
R. S.},
TITLE = {Diffusion processes},
BOOKTITLE = {Proceedings of the sixth {B}erkeley
symposium on mathematical statistics
and probability},
EDITOR = {Le Cam, Lucien M. and Neyman, Jerzy
and Scott, Elizabeth L.},
VOLUME = {III: Probability theory},
PUBLISHER = {Univ. California Press},
ADDRESS = {Berkeley, CA},
YEAR = {1972},
PAGES = {361--368},
NOTE = {(Berkeley, CA, June 21--July 18, 1970).
MR:0397899. Zbl:0255.60055.},
ISBN = {9780520021853},
}
D. W. Stroock and S. R. S. Varadhan :
“On the support of diffusion processes with applications to the strong maximum principle ,”
pp. 333–359
in
Proceedings of the sixth Berkeley symposium on mathematical statistics and probability
(Berkeley, CA, June 21–July 18, 1970 ),
vol. III: Probability theory .
Edited by L. M. Le Cam, J. Neyman, and E. L. Scott .
University of California Press (Berkeley, CA ),
1972 .
MR
0400425
Zbl
0255.60056
inproceedings
People
BibTeX
@inproceedings {key0400425m,
AUTHOR = {Stroock, Daniel W. and Varadhan, S.
R. S.},
TITLE = {On the support of diffusion processes
with applications to the strong maximum
principle},
BOOKTITLE = {Proceedings of the sixth {B}erkeley
symposium on mathematical statistics
and probability},
EDITOR = {Le Cam, Lucien M. and Neyman, Jerzy
and Scott, Elizabeth L.},
VOLUME = {III: Probability theory},
PUBLISHER = {University of California Press},
ADDRESS = {Berkeley, CA},
YEAR = {1972},
PAGES = {333--359},
NOTE = {(Berkeley, CA, June 21--July 18, 1970).
MR:0400425. Zbl:0255.60056.},
ISBN = {9780520021853},
}
D. Stroock and S. R. S. Varadhan :
“On degenerate elliptic-parabolic operators of second order and their associated diffusions ,”
Comm. Pure Appl. Math.
25 : 6
(November 1972 ),
pp. 651–713 .
MR
0387812
Zbl
0344.35041
article
Abstract
People
BibTeX
This paper consists of two parts. In the first part we extend our earlier results [Stroock and Varadhan 1972] on the strong maximum principle to a broader class of operators, namely degenerate parabolic operators \( \partial/\partial t + L_t \) , where
\[ L_t = \tfrac{1}{2}\nabla\cdot(a(t,x)\nabla)+b(t,x)\cdot\nabla \]
with \( a \) and \( b \) suitably smooth. This leads to a generalization of the results of M. Bony [1969] that was sought by C. D. Hill [1970]. It is also related to a recent result of M. Redheffer [1971]. The second part of the paper is devoted to the study of the first boundary value problem for degenerate elliptic operators
\[ L = \tfrac{1}{2}\nabla\cdot(a(x)\nabla) + b(x)\cdot\nabla - k(x) \]
in smooth regions.
@article {key0387812m,
AUTHOR = {Stroock, D. and Varadhan, S. R. S.},
TITLE = {On degenerate elliptic-parabolic operators
of second order and their associated
diffusions},
JOURNAL = {Comm. Pure Appl. Math.},
FJOURNAL = {Communications on Pure and Applied Mathematics},
VOLUME = {25},
NUMBER = {6},
MONTH = {November},
YEAR = {1972},
PAGES = {651--713},
DOI = {10.1002/cpa.3160250603},
NOTE = {MR:0387812. Zbl:0344.35041.},
ISSN = {0010-3640},
}
S. R. S. Varadhan :
“Strassen’s version of the law of the iterated logarithm ,”
pp. 15–31
in
Topics in probability theory
(Courant Institute, New York, 1971–1972 ).
Edited by D. W. Stroock and S. R. S. Varadhan .
Courant Institute of Mathematical Sciences (New York ),
1973 .
MR
0413234
Zbl
0269.60025
incollection
People
BibTeX
@incollection {key0413234m,
AUTHOR = {Varadhan, S. R. S.},
TITLE = {Strassen's version of the law of the
iterated logarithm},
BOOKTITLE = {Topics in probability theory},
EDITOR = {Stroock, Daniel W. and Varadhan, S.
R. S.},
PUBLISHER = {Courant Institute of Mathematical Sciences},
ADDRESS = {New York},
YEAR = {1973},
PAGES = {15--31},
NOTE = {(Courant Institute, New York, 1971--1972).
MR:0413234. Zbl:0269.60025.},
}
D. W. Stroock and S. R. S. Varadhan :
“Limit theorems for random walks on Lie groups ,”
Sankhyā Ser. A
35 : 3
(September 1973 ),
pp. 277–294 .
MR
0517406
Zbl
0299.60007
article
Abstract
People
BibTeX
@article {key0517406m,
AUTHOR = {Stroock, Daniel W. and Varadhan, S.
R. S.},
TITLE = {Limit theorems for random walks on {L}ie
groups},
JOURNAL = {Sankhy\=a Ser. A},
FJOURNAL = {Sankhy\=a (Statistics). The Indian Journal
of Statistics. Series A},
VOLUME = {35},
NUMBER = {3},
MONTH = {September},
YEAR = {1973},
PAGES = {277--294},
URL = {http://www.jstor.org/pss/25049879},
NOTE = {MR:0517406. Zbl:0299.60007.},
ISSN = {0581-572X},
}
D. W. Stroock and S. R. S. Varadhan :
“Martingales. I, II, III ,”
pp. 113–161
in
Topics in probability theory
(Courant Institute, New York, 1971–1972 ).
Edited by D. W. Stroock and S. R. S. Varadhan .
Courant Institute of Mathematical Sciences (New York ),
1973 .
MR
0410912
incollection
People
BibTeX
@incollection {key0410912m,
AUTHOR = {Stroock, D. W. and Varadhan, S. R. S.},
TITLE = {Martingales. {I}, {II}, {III}},
BOOKTITLE = {Topics in probability theory},
EDITOR = {Stroock, Daniel W. and Varadhan, S.
R. S.},
PUBLISHER = {Courant Institute of Mathematical Sciences},
ADDRESS = {New York},
YEAR = {1973},
PAGES = {113--161},
NOTE = {(Courant Institute, New York, 1971--1972).
MR:0410912.},
}
D. W. Stroock and S. R. S. Varadhan :
“Probability theory and the strong maximum principle ,”
pp. 215–220
in
Partial differential equations
(Berkeley, CA, 9–27 August, 1971 ).
Edited by D. C. Spencer .
Proceedings of Symposia in Pure Mathematics 23 .
Amererican Mathematical Society (Providence, RI ),
1973 .
MR
0380109
Zbl
0262.35028
incollection
People
BibTeX
@incollection {key0380109m,
AUTHOR = {Stroock, Daniel W. and Varadhan, S.
R. S.},
TITLE = {Probability theory and the strong maximum
principle},
BOOKTITLE = {Partial differential equations},
EDITOR = {Spencer, Donald Clayton},
SERIES = {Proceedings of Symposia in Pure Mathematics},
NUMBER = {23},
PUBLISHER = {Amererican Mathematical Society},
ADDRESS = {Providence, RI},
YEAR = {1973},
PAGES = {215--220},
NOTE = {(Berkeley, CA, 9--27 August, 1971).
MR:0380109. Zbl:0262.35028.},
ISBN = {9780821814239},
}
D. Stroock and S. R. S. Varadhan :
“A probabilistic approach to \( H^{p}(R^{d}) \) ,”
Trans. Amer. Math. Soc.
192
(1974 ),
pp. 245–260 .
MR
0365696
Zbl
0289.60029
article
Abstract
People
BibTeX
The relationship between \( H^p(R^d) \) , \( 1\leq p < \infty \) , and the integrability of certain functionals of Brownian motion is established using the connection between probabilistic and analytic notions of functions with bounded mean oscillation. An application of this relationship is given in the derivation of an interpolation theorem for operators taking \( H^1(R^d) \) to \( L^1(R^d) \) .
@article {key0365696m,
AUTHOR = {Stroock, D. and Varadhan, S. R. S.},
TITLE = {A probabilistic approach to \$H^{p}(R^{d})\$},
JOURNAL = {Trans. Amer. Math. Soc.},
FJOURNAL = {Transactions of the American Mathematical
Society},
VOLUME = {192},
YEAR = {1974},
PAGES = {245--260},
DOI = {10.2307/1996832},
NOTE = {MR:0365696. Zbl:0289.60029.},
ISSN = {0002-9947},
}
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.},
}
D. W. Stroock and S. R. S. Varadhan :
Multidimensional diffusion processes .
Grundlehren der Mathematischen Wissenschaften 233 .
Springer (Berlin ),
1979 .
MR
532498
Zbl
0426.60069
book
People
BibTeX
@book {key532498m,
AUTHOR = {Stroock, Daniel W. and Varadhan, S.
R. Srinivasa},
TITLE = {Multidimensional diffusion processes},
SERIES = {Grundlehren der Mathematischen Wissenschaften},
NUMBER = {233},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1979},
PAGES = {xii+338},
NOTE = {MR:532498. Zbl:0426.60069.},
ISBN = {9780387903538},
}
S. R. S. Varadhan, A. D. Wentzell, E. B. Dynkin, S. A. Molchanov, S. P. Novikov, Ya. G. Sinai, I. M. Sonin, and D. W. Stroock :
“Mark Iosifovich Freidlin ,”
Russian Mathematical Surveys
59 : 3
(2004 ),
pp. 593–597 .
Translation of Russian original from Uspekhi Mat. Nauk 59 :3(357), 185–188. No abstract.
People
BibTeX
@article {key13700106,
AUTHOR = {Varadhan, S. R. S. and Wentzell, A.
D. and Dynkin, E. B. and Molchanov,
S. A. and Novikov, S. P. and Sinai,
Ya. G. and Sonin, I. M. and Stroock,
D. W.},
TITLE = {Mark {I}osifovich {F}reidlin},
JOURNAL = {Russian Mathematical Surveys},
VOLUME = {59},
NUMBER = {3},
YEAR = {2004},
PAGES = {593--597},
NOTE = {Translation of Russian original from
\textit{Uspekhi Mat. Nauk} \textbf{59}:3(357),
185--188. No abstract. Available at
http://dx.doi.org/10.1070/RM2004v059n03ABEH000756.},
ISSN = {0042-1316},
}
S. R. S. Varadhan, A. D. Wentzell, E. B. Dynkin, S. A. Molchanov, S. P. Novikov, Ya. G. Sinai, I. M. Sonin, and D. W. Stroock :
“Mark Iosifovich Freĭdlin ,”
Uspekhi Mat. Nauk
59 : 3(357)
(2004 ),
pp. 185–188 .
In Russian. Translated in Russian Mathematical Surveys 59 :3 (2004), 593–597.
MR
2116554
article
People
BibTeX
@article {key2116554m,
AUTHOR = {Varadhan, S. R. S. and Wentzell, A.
D. and Dynkin, E. B. and Molchanov,
S. A. and Novikov, S. P. and Sinai,
Ya. G. and Sonin, I. M. and Stroock,
D. W.},
TITLE = {Mark {I}osifovich {F}re\u\i dlin},
JOURNAL = {Uspekhi Mat. Nauk},
FJOURNAL = {Rossi\u\i skaya Akademiya Nauk. Moskovskoe
Matematicheskoe Obshchestvo. Uspekhi
Matematicheskikh Nauk},
VOLUME = {59},
NUMBER = {3(357)},
YEAR = {2004},
PAGES = {185--188},
URL = {http://mi.mathnet.ru/eng/umn756},
NOTE = {In Russian. Translated in \textit{Russian
Mathematical Surveys} \textbf{59}:3
(2004), 593--597. MR:2116554.},
ISSN = {0042-1316},
}
D. W. Stroock and S. R. S. Varadhan :
Multidimensional diffusion processes ,
Reprint edition.
Classics in Mathematics 233 .
Springer (Berlin ),
2006 .
MR
2190038
Zbl
1103.60005
book
People
BibTeX
@book {key2190038m,
AUTHOR = {Stroock, Daniel W. and Varadhan, S.
R. Srinivasa},
TITLE = {Multidimensional diffusion processes},
EDITION = {Reprint},
SERIES = {Classics in Mathematics},
NUMBER = {233},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {2006},
PAGES = {xii+338},
NOTE = {MR:2190038. Zbl:1103.60005.},
ISBN = {9783540289982, 3540289984},
}
D. Stroock and S. R. S. Varadhan :
“Theory of diffusion processes ,”
pp. 149–191
in
Stochastic differential equations
(Cortona, Italy, May 29–June 10, 1978 ).
Edited by J. Cecconi .
CIME Summer Schools 77 .
Springer (Heidelberg ),
2010 .
MR
2830392
incollection
People
BibTeX
@incollection {key2830392m,
AUTHOR = {Stroock, D. and Varadhan, S. R. S.},
TITLE = {Theory of diffusion processes},
BOOKTITLE = {Stochastic differential equations},
EDITOR = {Cecconi, Jaures},
SERIES = {CIME Summer Schools},
NUMBER = {77},
PUBLISHER = {Springer},
ADDRESS = {Heidelberg},
YEAR = {2010},
PAGES = {149--191},
NOTE = {(Cortona, Italy, May 29--June 10, 1978).
MR:2830392.},
ISBN = {9783642110771},
}