Celebratio Mathematica

S. R. S. Varadhan: “On the behavior of the fundamental solution of the heat equation with variable coefficients,” Comm. Pure Appl. Math. 20 : 2 (May 1967), pp. 431–455. MR 0208191 Zbl 0155.16503 article

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
Daniel W. Stroock	Related

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
Elizabeth Leonard Scott	Related
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.

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

Daniel W. Stroock	Related
Donald Clayton Spencer	Related

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

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.

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

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

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 \( \mathbb{R}^3 \).

George C. Papanicolaou	Related
Bronius Grigelionis	Related

S. R. S. Varadhan: Lectures on diffusion problems and partial differential equations. Tata Institute of Fundamental Research Lectures on Mathematics and Physics 64. Tata Institute of Fundamental Research (Bombay), 1980. MR 607678 Zbl 0489.35002 book

S. R. S. Varadhan and R. J. Williams: “Brownian motion in a wedge with oblique reflection,” Comm. Pure Appl. Math. 38 : 4 (July 1985), pp. 405–443. MR 792398 Zbl 0579.60082 article

S. R. S. Varadhan: “Stochastic differential equations–large deviations,” pp. 625–678 in Phénomènes critiques, systèmes aléatoires, théories de jauge (Les Houches, 1984), Part II. Edited by K. Osterwalder, R. Stora, and D. Brydges. North-Holland (Amsterdam), 1986. MR 880536 incollection

Raymond Stora	Related
Konrad Osterwalder	Related
David Brydges	Related

A.-S. Sznitman and S. R. S. Varadhan: “A multidimensional process involving local time,” Probab. Theory Relat. Fields 71 : 4 (1986), pp. 553–579. MR 833269 Zbl 0613.60050 article

J. Quastel and S. R. S. Varadhan: “Diffusion semigroups and diffusion processes corresponding to degenerate divergence form operators,” Comm. Pure Appl. Math. 50 : 7 (July 1997), pp. 667–706. MR 1447057 Zbl 0907.47040 article

In this article we study existence and uniqueness problems for semigroups and processes determined by time-dependent, second-order linear partial differential operators of the form \[ L_tu = \nabla\cdot a\nabla u + \sigma c\cdot\nabla u .\] For simplicity we will take the space to be the \( d \)-dimensional torus \( \mathbb{T}^d \). The matrix \[ a:[0,T]\times\mathbb{T}^d\to\mathbb{R}^d\otimes\mathbb{R}^d \] is assumed to be symmetric and nonnegative definite. We define \( \sigma \) to be the symmetric, nonnegative definite square root of \( a \), and \( c:[0,T]\times\mathbb{T}^d\to\mathbb{R}^d \) to be a function about which assumptions will be made later. The focus of our study is on cases where the coefficients \( a \) and \( c \) have minimal regularity, and \( a \) is allowed to degenerate.

S. R. S. Varadhan: “Diffusion processes,” pp. 853–872 in Stochastic processes: theory and methods. Edited by D. N. Shanbhag and C. R. Rao. Handbook of Statistics 19. North-Holland (Amsterdam), 2001. MR 1861741 Zbl 0986.60075 incollection

D. N. Shanbhag	Related
Calyampudi Radhakrishna Rao	Related

S. R. S. Varadhan: “Stochastic analysis and applications,” Bull. Amer. Math. Soc. (N.S.) 40 : 1 (2003), pp. 89–97. MR 1943135 Zbl 1012.60004 article

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

Daniel W. Stroock	Related
Jaures Cecconi	Related

year	title	people

			clear

S. R. Srinivasa Varadhan

Diffusion theory

Filter the Bibliography List