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 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 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},
}
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. 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},
}
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},
}
