Celebratio Mathematica

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

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

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

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

Hermann Dinges	Related
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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

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.

Hermann Dinges	Related
Daniel W. Stroock	Related

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

Daniel W. Stroock	Related
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Jerzy Neyman	Related
Lucien Marie Le Cam	Related

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

Daniel W. Stroock	Related
Elizabeth Leonard Scott	Related
Jerzy Neyman	Related
Lucien Marie Le Cam	Related

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

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.

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

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

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

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

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Topics in probability theory (Courant Institute of Mathematical Sciences, New York, 1971–1972). Edited by D. W. Stroock and S. R. S. Varadhan. Courant Institute of Mathematical Sciences (New York), 1973. MR 0353393 Zbl 0258.00010 book

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

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

Daniel W. Stroock	Related
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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

K. Itō: Selected papers. Edited by S. R. S. Varadhan and D. W. Stroock. Springer (New York), 1987. With an introduction by the editors. MR 868862 Zbl 0621.01019 book

Daniel W. Stroock	Related
Kiyoshi Itō	Related

“1996 Steele Prizes,” Notices Amer. Math. Soc. 43 : 11 (1996), pp. 1340–1347. MR 1412777 article

Daniel W. Stroock	Related
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Bruce C. Berndt	Related

V. S. Varadarajan: “Some mathematical reminiscences,” pp. v–xviii in Special issue dedicated to Daniel W. Stroock and Srinivasa S. R. Varadhan on the occasion of their 60th birthday, published as Methods Appl. Anal. 9 : 3 (2002). MR 2023128 Zbl 1071.01523 incollection

Special issue dedicated to Daniel W. Stroock and Srinivasa S. R. Varadhan on the occasion of their 60th birthday, published as Methods Appl. Anal. 9 : 3. International Press (Somerville, MA), 2002. MR 2023127 book

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.

Evgeniĭ Borisovich Dynkin	Related
Sergey Petrovich Novikov	Related
Mark Iosifovich Freĭdlin	Related
Daniel W. Stroock	Related
Ya. G. Sinai	Related
I. M. Sonin	Related
S. A. Molchanov	Related
A. D. Wentzell	Related

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

Evgeniĭ Borisovich Dynkin	Related
Sergey Petrovich Novikov	Related
Mark Iosifovich Freĭdlin	Related
Daniel W. Stroock	Related
Ya. G. Sinai	Related
I. M. Sonin	Related
S. A. Molchanov	Related
A. D. Wentzell	Related

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

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

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