Celebratio Mathematica

S. R. S. Varadhan: “Limit theorems for sums of independent random variables with values in a Hilbert space,” Sankhyā Ser. A 24 : 3 (August 1962), pp. 213–238. MR 0171305 Zbl 0113.34101 article

K. R. Parthasarathy, R. Ranga Rao, and S. R. S. Varadhan: “On the category of indecomposable distributions on topological groups,” Trans. Amer. Math. Soc. 102 (1962), pp. 200–217. MR 0153041 Zbl 0104.36205 article

According to a theorem of A. I. Khinchine [1937] any distribution on the real line can be written as the convolution of two distributions one of which is the convolution of a finite or a countable number of indecomposable distributions and the other is an infinitely divisible distribution without indecomposable factors. Further, any distribution which is not infinitely divisible has an indecomposable component. This result gives an indication of the existence of a large collection of indecomposable distributions. It is, however, not clear from this result alone that there exists a non-atomic or absolutely continuous indecomposable distribution. This question was raised by H. Cramér [1948] and an answer in the affirmative was given by P. Lévy [1952]. However, what is available is only a meagre supply of examples [Lévy 1952; Dugué 1951; Dugué and Fisher 1948] even in the case of the real line. In this connection there arises naturally the question of the “size” of the class \( \mathcal{M}_1 \),of indecomposable distributions. Or more precisely, what is the category of \( \mathcal{M}_1 \)? The object of this paper is to answer questions of this type.

R. Ranga Rao	Related
K. R. Parthasarathy	Related

S. R. S. Varadhan: Convolution properties of distributions on topological groups. Ph.D. thesis, Indian Statistical Institute (Baranagar, India), 1963. Advised by C. R. Rao.

K. R. Parthasarathy, R. Ranga Rao, and S. R. S. Varadhan: “Probability distributions on locally compact abelian groups,” Illinois J. Math. 7 : 2 (1963), pp. 337–369. MR 0190968 Zbl 0129.10902 article

For probability distributions on the real line there are three main theorems on which the entire study of limit theorems for sums of independent random variables is based. These are

the Lévy–Khinchin representation of an infinitely divisible distribution,
the criteria for weak convergence of such distributions, and
Khinchin’s theorem on sums of infinitesimal summands stating that these converge weakly if and only if certain associated infinitely divisible laws converge.

For a precise statement of these results we refer to Kolmogorov and Gnedenko [Kolmogorov and Gnedenko 1954]. During the last two decades or so these results have been extended by many authors to varying degrees of generality. We mention in particular the works of Lévy [1939], Kawada and Itô [1940], Takano [1956], Bochner [1955; 1958], Hunt [1956], Urbanik [1958; 1960], Kloss [1961].

In this paper we study probability distributions on a locally compact abelian (separable) group and obtain definitive extensions of all the three main results mentioned above.

R. Ranga Rao	Related
K. R. Parthasarathy	Related

K. R. Parthasarathy and S. R. S. Varadhan: “Extension of stationary stochastic processes,” Theory Probab. Appl. 9 : 1 (1964), pp. 65–71. English translation of Teor. Verojatnost. i Primenen. 9 (1964) 72–78.

S. R. S. Varadhan: “Limit theorems in probability,” Math. Student 32 (1964), pp. 17–21. MR 0182050 Zbl 0168.17004 article

K. R. Parthasarathy and S. R. S. Varadhan: “Extension of stationary stochastic processes,” Teor. Verojatnost. i Primenen. 9 (1964), pp. 72–78. In Russian. English translation appeared as Theory Probab. Appl. 9:1 (1964) 65–71. MR 0164364 Zbl 0138.11001 article

S. R. S. Varadhan: “Asymptotic probabilities and differential equations,” Comm. Pure Appl. Math. 19 (1966), pp. 261–286. MR 0203230 Zbl 0147.15503 article

S. R. S. Varadhan: “Diffusion processes in a small time interval,” Comm. Pure Appl. Math. 20 : 4 (November 1967), pp. 659–685. MR 0217881 Zbl 0278.60051 article

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

S. R. S. Varadhan: Stochastic processes. Courant Lecture Notes 16. Courant Institute of Mathematical Sciences (New York), 1968. Notes based on a course given at New York University during the year 1967/68. MR 0260028 book

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.

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

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.

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

S. Varadhan: Mathematical statistics. Courant Institute of Mathematical Sciences (New York), 1974. Lectures given during the academic year 1973–1974. MR 0362573 Zbl 0293.62001 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

M. D. Donsker and S. R. S. Varadhan: “Asymptotic evaluation of certain Markov process expectations for large time. I,” Comm. Pure Appl. Math. 28 : 1 (January 1975), pp. 1–47. MR 0386024 article

M. D. Donsker and S. R. S. Varadhan: “Erratum: ‘Asymptotics for the Wiener sausage’,” Comm. Pure Appl. Math. 28 : 5 (September 1975), pp. 677. MR 0397902 article

M. D. Donsker and S. R. S. Varadhan: “Asymptotics for the Wiener sausage,” Comm. Pure Appl. Math. 28 : 4 (July 1975), pp. 525–565. MR 0397901 article

M. D. Donsker and S. R. S. Varadhan: “Asymptotic evaluation of certain Markov process expectations for large time. II,” Comm. Pure Appl. Math. 28 : 2 (March 1975), pp. 279–301. MR 52 \#6883

M. D. Donsker and S. R. S. Varadhan: “Large deviations for Markov processes and the asymptotic evaluation of certain Markov process expectations for large times,” pp. 82–88 in Probabilistic methods in differential equations (Victoria, BC, August 19–20, 1974). Edited by M. A. Pinsky. Lecture Notes in Mathematics 451. Springer (Berlin), 1975. MR 0410942 incollection

Mark A. Pinsky	Related
Monroe D. Donsker	Related

M. D. Donsker and S. R. S. Varadhan: “On a variational formula for the principal eigenvalue for operators with maximum principle,” Proc. Nat. Acad. Sci. U.S.A. 72 (1975), pp. 780–783. MR 0361998 Zbl 0353.49039 article

M. D. Donsker and S. R. S. Varadhan: “On some problems of large deviations for Markov processes,” pp. 409–416, 417–418 in Proceedings of the 40th session of the International Statistical Institute (Warsaw, 1975), published as Bulletin of the International Statistical Institute 46 : 1. Héritiers Botta, 1975. MR 0488298 Zbl 0351.60036 inproceedings

Ergodic theory (Courant Institute of Mathematical Sciences, New York, 1973–1974). Edited by J. Moser, E. Phillips, and S. R. S. Varadhan. Courant Institute of Mathematical Sciences (New York), 1975. MR 0486431 Zbl 0334.28019 book

Jürgen K. Moser	Related
Edward Phillips	Related

M. D. Donsker and S. R. S. Varadhan: “Asymptotic evaluation of certain Wiener integrals for large time,” pp. 15–33 in Functional integration and its applications (London, April 1974). Edited by A. M. Arthurs. Clarendon Press (Oxford), 1975. MR 0486395 Zbl 0333.60078 incollection

Monroe D. Donsker	Related
A. B. Arthurs	Related

M. D. Donsker and S. R. S. Varadhan: “Asymptotic evaluation of certain Markov process expectations for large time. III,” Comm. Pure Appl. Math. 29 : 4 (July 1976), pp. 389–461. MR 0428471 Zbl 0348.60032 article

M. D. Donsker and S. R. S. Varadhan: “On the principal eigenvalue of second-order elliptic differential operators,” Comm. Pure Appl. Math. 29 : 6 (November 1976), pp. 595–621. MR 0425380 Zbl 0356.35065 article

M. D. Donsker and S. R. S. Varadhan: “Some problems of large deviations,” pp. 313–318 (INDAM, Rome, 1975). Symposia Mathematica 21. Academic Press (London), 1977. MR 0517541 Zbl 0372.60036 incollection

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
George C. Papanicolaou	Related
P. L. Chow	Related

M. D. Donsker and S. R. S. Varadhan: “On laws of the iterated logarithm for local times,” Comm. Pure Appl. Math. 30 : 6 (November 1977), pp. 707–753. MR 0461682 Zbl 0356.60029 article

M. D. Donsker and S. R. S. Varadhan: “On the principal eigenvalue of elliptic second order differential operators,” pp. 41–47 in Proceedings of the international symposium on stochastic differential equations (Kyoto, 1976). Edited by K. Itō. Wiley (New York), 1978. MR 536002 Zbl 0447.35030 inproceedings

Monroe D. Donsker	Related
Kiyoshi Itō	Related

M. D. Donsker and S. R. S. Varadhan: “On the number of distinct sites visited by a random walk,” pp. 57–62 in Stochastic analysis (Evanston, IL, April 10–14, 1978). Edited by A. Friedman and M. A. Pinsky. Academic Press (New York), 1978. MR 517233 Zbl 0442.60068 incollection

Avner Friedman	Related
Monroe D. Donsker	Related
Mark A. Pinsky	Related

J. Saint-Raymond: “Quelques remarques sur un article de M. D. Donsker et S. R. S. Varadhan: ‘Asymptotic evaluation of certain Markov process expectations for large time, III’,” pp. 468–481 in Séminaire de probabilités, XII (Strasbourg, 1976/1977). Edited by C. Dellacherie, P. A. Meyer, and M. Weil. Lecture Notes in Mathematics 649. Springer (Berlin), 1978. MR 520021 incollection

Monroe D. Donsker	Related
Jean Saint-Raymond	Related
Paul-André Meyer	Related
M. Weil	Related
Claude Dellacherie	Related

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

M. D. Donsker and S. R. S. Varadhan: “On the number of distinct sites visited by a random walk,” Comm. Pure Appl. Math. 32 : 6 (1979), pp. 721–747. MR 539157 Zbl 0418.60074 article

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

K. L. Chung and S. R. S. Varadhan: “Kac functional and Schrödinger equation,” Studia Math. 68 : 3 (1980), pp. 249–260. MR 599148 Zbl 0448.60054 article

M. D. Donsker and S. R. S. Varadhan: “A law of the iterated logarithm for total occupation times of transient Brownian motion,” Comm. Pure Appl. Math. 33 : 3 (1980), pp. 365–393. MR 562740 Zbl 0504.60037 article

Let \( \{\beta(s), 0\leq s < \infty\} \) be Brownian motion in \( R_d \), starting from the origin with \( d\geq 3 \), and let \( T_d(\lambda,\omega) \) be the total time that a particular path \( \omega = \beta(\,\cdot\,) \) occupies the sphere with center at the origin of radius \( \lambda \). In [1962] Ciesielski and Taylor showed that, for almost all Brownian paths, \begin{equation*}\tag{1}\overline{\lim_{\lambda\downarrow 0}}\frac{T_d(\lambda,\omega)}{\lambda^2\log\log(1/\lambda)} = \frac{2}{p_d^2},\end{equation*} where \( p_d \) is the first positive zero of \( J\nu(x) \) with \( \nu = \frac{1}{2}d-2 \). In this paper, motivated by (1), the authors prove a Strassen type law of the iterated logarithm for total Brownian occupation times in three or more dimensions. These theorems involve the \( I \)-function introduced by the authors in their recent work and which we now describe in the present context.

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: “Some problems of large deviations,” pp. 755–762 in Proceedings of the International Congress of Mathematicians (Helsinki, 1978), vol. 2. Edited by O. Lehto. Acad. Sci. Fennica (Helsinki), 1980. MR 562683 Zbl 0421.60025 inproceedings

M. D. Donsker and S. R. S. Varadhan: “The polaron problem and large deviations,” Phys. Rep. 77 : 3 (November 1981), pp. 235–237. MR 639028 article

Let \( \{x_t(s),0\leq s\leq t\} \) be three dimensional Brownian motion tied down at the ends of the time interval, i.e., \( x_t(0) = x_t(t) = 0 \). Let \( a > 0 \) and consider the following function space integral: \[ A(t,a) = E\Bigl\{\exp\Bigl[\alpha\int_0^t\int_0^t\frac{e^{-|\sigma-s|}}{\|x_t(\sigma)-x_t(s)\|}ds\,d\sigma\Bigr]\Bigr\} \] The “Polaron Problem” which arises in statistical mechanics [Feynman and Hibbs 1965] is to evaluate \[ G(\alpha) = \lim_{t\to\infty}\frac{1}{t}\log A(t,\alpha) .\] Now, \( G(\alpha \)) is complicated to evaluate and even to estimate, but a conjecture by Pekar is that \( \lim_{\alpha\to\infty}G(\alpha)/\alpha^2=c \) where \[ c=\sup_{\varphi\in L^2(R^3),\,\|\varphi\|_2=1}\Bigl[2\iint\frac{\varphi^2(x)\,\varphi^2(y)}{\|x-y\|}dx\,dy\,-\frac{1}{2}\int|\nabla\varphi|^2dx\Bigr] \] Using methods developed by us in [Donsker and Varadhan, 1975-76], [1983], we succeeded in finding a “fairly explicit” expression for \( G(\alpha) \) and then used this expression to prove rigorously the conjecture of Pekar. The details of that argument will be found in [Donsker and Varadhan 1981]. In this note we briefly describe some of the ideas lying behind our work on asymptotics.

M. D. Donsker and S. R. S. Varadhan: “Some problems of large deviations,” pp. 41–46 in Stochastic differential systems (Visegrád, Hungary, September 15–20, 1980). Edited by M. Arató, D. Vermes, and A. V. Balakrishnan. Lecture Notes in Control and Information Sciences 36. Springer (Berlin), 1981. MR 653644 Zbl 0472.60028 incollection

Let \( E_t \) refer to the expectation with respect to a three dimensional Brownian path \( \beta(\,\cdot\,) \) tied down at both ends with \( \beta(0) = \beta(t) = 0 \). Let \[ G(\alpha,t)=E_t\Bigl\{\exp\Bigl[\alpha\int_0^t\int_0^t\frac{e^{-|\sigma-s|}}{|\beta(\sigma)-\beta(s)|}d\sigma\,ds\Bigr]\Bigr\} \] show that, if \[ \lim_{t\to\infty}\frac{1}{t}\log G(\alpha,t)=g(\alpha) \] exists, \[ \lim_{\alpha\to\infty} \frac{g(\alpha)}{\alpha^2}=g_0 \] exists with \[ g_0 =\!\!\!\sup_{\varphi\in L_2(R^3), \,\|\varphi\|_2=1} \Bigl[2\iint\frac{\varphi^2(x)\,\varphi^2(y)}{|x-y|}\,dx\,dy-\frac{1}{2}\int|\nabla\varphi|^2\,dx\Bigr] \] The problem comes up in statistical mechanics. See for instance the book by Feynman [1972]. The formula for \( g_0 \) has been conjectured by Pekar [1949]. We shall outline a theory that allows us to prove these formulae.

Monroe D. Donsker	Related
Mátyás Arató	Related
A. V. Balakrishnan	Related
D. Vermes	Related

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

J. Fritz	Related
D. Szász	Related
Joel Louis Lebowitz	Related
George C. Papanicolaou	Related

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

Gopinath Kallianpur	Related
Paruchuri R. Krishnaiah	Related
Calyampudi Radhakrishna Rao	Related
Jayanta Kumar Ghosh	Related
George C. Papanicolaou	Related

S. R. S. Varadhan: “Large deviations,” pp. 382–392 in Advances in filtering and optimal stochastic control (Cocoyoc, Mexico, February 1–6, 1982). Edited by W. H. Fleming and L. G. Gorostiza. Lecture Notes in Control and Information Sciences 42. Springer (Berlin), 1982. MR 794532 Zbl 0496.60024 incollection

Luis G. Gorostiza	Related
Wendell H. Fleming	Related

S. R. S. Varadhan: “Random walks among random scatterers,” pp. 282–283 in Theory and application of random fields (Bangalore, January 1982). Edited by G. Kallianpur. Lecture Notes in Control and Information Sciences 49. Springer (Berlin), 1983. MR 799951 Zbl 0505.60028 incollection

M. D. Donsker and S. R. S. Varadhan: “Asymptotics for the polaron,” Comm. Pure Appl. Math. 36 : 4 (1983), pp. 505–528. MR 709647 Zbl 0538.60081 article

Let \( P_t \) and \( E^{P_t}\{\ \} \) denote, respectively, the probability measure and expectation with respect to three-dimensional Brownian motion \( x(\,\cdot\,) \) tied down at both ends, i.e., with \( x(0) = x(t) = 0 \). For \( a > 0 \), let \[ G(\alpha,t)=E^{P_t}\Bigl\{\alpha\int_0^t\int_0^t\frac{e^{-|\sigma-s|}}{|x(\sigma)-x(s)|}d\sigma\, ds\Bigr\} .\] A long standing problem in statistical mechanics (cf. [Feynman 1972]), the “polaron problem,” has been to show that \begin{equation*}\tag{1} \lim_{t\to\infty}\frac{1}{t}\log G(\alpha,t)=g(\alpha) \end{equation*} exists, and, moreover, according to a conjecture of Pekar [1949], that \begin{equation*}\tag{2} \lim_{\alpha\to\infty}\frac{g(\alpha)}{\alpha^2} = g_0 \end{equation*} exists, with \begin{equation*}\tag{3} g_0 = \!\!\!\sup_{\varphi\in L_2(R^3),\,\|\varphi\|=1}\Bigl[2\iint\frac{\varphi^2(x)\,\varphi^2(y)}{|x-y|}\,dx\,dy -\frac{1}{2}\int|\nabla\varphi|^2\,dx\Bigr]. \end{equation*} In this paper we prove (1) obtaining an expression for \( g(\alpha) \) which is explicit enough to allow us to prove also the conjecture (2) of Pekar, \( g_0 \) being indeed given by (3). We make use of large deviation results obtained in our earlier papers [Donsker and Varadhan, 1975-6] and, in particular, [Donsker and Varadhan 1983]. That the polaron problem depends on sharp large deviation theorems is natural, since in determining the asymptotic behavior of \( G(\alpha,t) \) for large \( t \) it is clear that the three-dimensional Brownian motion paths which contribute the most are those which make \( |x(\sigma)- x(s)| \) small. However \( \sigma \) and \( s \) must not be so far apart that the contribution is killed by \( e^{-|\sigma-s|} \). Thus, the influential paths are those which tend to stay awhile near where they have just been. Since this is not the way “typical” Brownian motion paths behave, we are dealing with probabilities of large deviations.

M. D. Donsker and S. R. S. Varadhan: “Asymptotic evaluation of certain Markov process expectations for large time. IV,” Comm. Pure Appl. Math. 36 : 2 (1983), pp. 183–212. MR 690656 Zbl 0512.60068 article

S. R. S. Varadhan: Large deviations and applications. CBMS-NSF Regional Conference Series in Applied Mathematics 46. Society for Industrial and Applied Mathematics (Philadelphia, PA), 1984. MR 758258 Zbl 0549.60023 book

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

S. R. S. Varadhan: “Large deviations and applications,” Exposition. Math. 3 : 3 (1985), pp. 251–272. MR 861018 Zbl 0567.60030 article

M. D. Donsker and S. R. S. Varadhan: “Large deviations for stationary Gaussian processes,” pp. 108–112 in Stochastic differential systems (Marseille-Luminy, 1984). Edited by M. Métivier and É. Pardoux. Lecture Notes in Control and Information Sciences 69. Springer (Berlin), 1985. MR 798313 Zbl 0657.60036 incollection

Étienne Pardoux	Related
Monroe D. Donsker	Related
Michel E. Métivier	Related

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

M. D. Donsker and S. R. S. Varadhan: “Large deviations for stationary Gaussian processes,” Comm. Math. Phys. 97 : 1–2 (1985), pp. 187–210. MR 782966 Zbl 0646.60030 article

C. Kipnis and S. R. S. Varadhan: “Central limit theorems for additive functionals of reversible Markov chains and applications,” pp. 65–70 in Colloque en l’honneur de Laurent Schwartz, volume 2 (École Polytechnique, Palaiseau, 30 May–3 June, 1983). Astérisque 132. 1985. MR 816760 Zbl 0584.60043 incollection

Laurent Schwartz	Related
Claude Kipnis	Related

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

C. Kipnis and S. R. S. Varadhan: “Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions,” Comm. Math. Phys. 104 : 1 (1986), pp. 1–19. MR 834478 Zbl 0588.60058 article

V. F. Kolchin: Random mappings. Translation Series in Mathematics and Engineering. Optimization Software Publications Division (New York), 1986. With a foreword by S. R. S. Varadhan. MR 865130 Zbl 0605.60010 book

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

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

M. D. Donsker and S. R. S. Varadhan: “Large deviations for noninteracting infinite-particle systems,” J. Statist. Phys. 46 : 5–6 (1987), pp. 1195–1232. MR 893138 Zbl 0682.60020 article

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

M. Z. Guo	Related
George C. Papanicolaou	Related

S. R. S. Varadhan: “Large deviations and applications,” pp. 1–49 in École d’été de probabilités de Saint-Flour XV–XVII, 1985–87 (Saint-Flour, France, 1985–87). Edited by P.-L. Hennequin. Lecture Notes in Mathematics 1362. Springer (Berlin), 1988. MR 983371 Zbl 0661.60040 incollection

Colloque Paul Lévy sur les processus stochastiques (École Polytechnique, Palaiseau, June 22–26, 1987). Edited by M. Métivier, J. Neveu, and S. R. S. Varadhan. Astérisque 157–158. Société Mathématique de France (Paris), 1988. MR 976209 book

Michel E. Métivier	Related
Paul Lévy	Related
Jacques E. Neveu	Related

C. Kipnis, S. Olla, and S. R. S. Varadhan: “Hydrodynamics and large deviation for simple exclusion processes,” Comm. Pure Appl. Math. 42 : 2 (1989), pp. 115–137. MR 978701 Zbl 0644.76001 article

Stefano Olla	Related
Claude Kipnis	Related

M. D. Donsker and S. R. S. Varadhan: “Large deviations from a hydrodynamic scaling limit,” Comm. Pure Appl. Math. 42 : 3 (April 1989), pp. 243–270. MR 982350 Zbl 0780.60027 article

The problem of describing how a large system evolves towards its equilibrium in terms of the evolution of certain macroscopic quantities can be formulated and studied in widely different contexts. Recently in [Guo, Papanicolaou and Varadhan 1988] one such model has been studied using estimates based on entropy and its rate of change. The main result in [Guo, Papanicolaou and Varadhan 1988] is a law of large numbers type result which asserts that the macroscopic functions evolve according to a specific deterministic motion (to be more specific a nonlinear diffusion equation) as the size gets large and the fluctuations disappear in the scaling. The present article is a complement to [Guo, Papanicolaou and Varadhan 1988] and we are interested here in estimating precisely the probabilities of significant (large) deviations from the deterministic limit. A similar result in a different context has been derived in [Kipnis, Olla and Varadhan 1989].

J. A. Gubner, B. Gopinath, and S. R. S. Varadhan: “Bounding functions of Markov processes and the shortest queue problem,” Adv. in Appl. Probab. 21 : 4 (1989), pp. 842–860. MR 1039631 Zbl 0707.60086 article

We prove a theorem which can be used to show the expectation of a non-negative function of the state of a time-homogeneous Markov process is uniformly bounded in time. This is reminiscent of the classical theory of non-negative supermartingales, except that our analog of the supermartingale inequality need not hold almost surely. Consequently, the theorem is suitable for establishing the stability of systems that evolve in a stabilizing mode in most states, though from certain states they may jump to a less stable state. We use this theorem to show that ‘joinging the shortest queue’ can bound the expected sum of the squares of the differences between all pairs among \( N \) queues, even under arbitrarily heavy traffic.

B. Gopinath	Related
John A. Gubner	Related

S. R. S. Varadhan: “On the derivation of conservation laws for stochastic dynamics,” pp. 677–694 in Analysis, et cetera. Edited by P. H. Rabinowitz and E. Zehnder. Academic Press (Boston, MA), 1990. MR 1039368 Zbl 0699.60097 incollection

Paul H. Rabinowitz	Related
Eduard J. Zehnder	Related

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

M. Z. Guo	Related
J. Fritz	Related
George C. Papanicolaou	Related

S. Olla and S. R. S. Varadhan: “Scaling limit for interacting Ornstein–Uhlenbeck processes,” Comm. Math. Phys. 135 : 2 (1991), pp. 355–378. MR 1087388 Zbl 0725.60086 article

The problem of describing the bulk behavior of an interacting system consisting of a large number of particles comes up in different contexts. See for example [Deuschel and Stroock 1989] for a recent exposition. In [Varadhan 1991] one of the authors considered the case of interacting diffusions on a circle and proved that the density of particles evolves according to a nonlinear diffusion equation. The interacting particles evolved according to a generator that was symmetric in equilibrium. In this article we consider interacting Ornstein–Uhlenbeck processes. Here the diffusion generator is not symmetric relative to the equilibrium and the earlier methods have to be modified considerably. We use some ideas that were employed in [Papanicolaou and Varadhan 1985] to extend the central limit theorem from the symmetric to nonsymmetric cases.

S. R. S. Varadhan: “Scaling limits for interacting diffusions,” Comm. Math. Phys. 135 : 2 (1991), pp. 313–353. MR 1087387 Zbl 0725.60085 article

J. R. Baxter, N. C. Jain, and S. R. S. Varadhan: “Some familiar examples for which the large deviation principle does not hold,” Comm. Pure Appl. Math. 44 : 8–9 (1991), pp. 911–923. MR 1127039 Zbl 0749.60025 article

Naresh Chandra Jain	Related
J. R. Baxter	Related

S. R. S. Varadhan: “Entropy methods in hydrodynamical scaling,” pp. 103–112 in Mathematical physics X (Leipzig, 30 July–9 August, 1991). Edited by K. Schmüdgen. Springer (Berlin), 1992. MR 1386400 Zbl 0947.82510 incollection

S. R. S. Varadhan: “Entropy methods in hydrodynamic scaling,” pp. 112–145 in Nonequilibrium problems in many-particle systems (Montecatini, Italy, June 15–27, 1992). Edited by L. Arkeryd. Lecture Notes in Mathematics 1551. Springer (Berlin), 1993. MR 1296260 Zbl 0791.60098 incollection

The problem of transition from a microscopic description of a system with a large number of components to the description of a suitably scaled version of the same system by a few macroscopic parameters arises often and is physically natural. However the mathematical justification of such a simplified description is considerably more difficult. The purpose of these lectures is to consider several such models and develop some techniques that can be used for a rigorous study of these models. Our main tool will be entropy and its production. Since entropy production is associated with the existence of stochastic noise in the system we will consider mainly systems with noisy evolution.

S. R. S. Varadhan: “Relative entropy and hydrodynamic limits,” pp. 329–336 in Stochastic processes. Edited by S. Cambanis. Springer (New York), 1993. MR 1427330 Zbl 0790.60094 incollection

S. Olla, S. R. S. Varadhan, and H.-T. Yau: “Hydrodynamical limit for a Hamiltonian system with weak noise,” Comm. Math. Phys. 155 : 3 (1993), pp. 523–560. MR 1231642 Zbl 0781.60101 article

Stefano Olla	Related
Horng-Tzer Yau	Related

H. Berestycki, L. Nirenberg, and S. Varadhan: “État fondamental et principe du maximum pour les opérateurs elliptiques du second ordre dans des domaines généraux,” C. R. Acad. Sci. Paris Sér. I Math. 317 : 1 (1993), pp. 51–56. MR 1228964 Zbl 0798.35038 article

Louis Nirenberg	Related
Henri Berestycki	Related

S. R. S. Varadhan: “Nonlinear diffusion limit for a system with nearest neighbor interactions. II,” pp. 75–128 in Asymptotic problems in probability theory: stochastic models and diffusions on fractals (Sanda/Kyoto, 1990), vol. 1. Edited by K. D. Elworthy and N. Ikeda. Pitman Research Notes in Mathematics 283. Longman Scientific & Technical (Harlow), 1993. MR 1354152 Zbl 0793.60105 incollection

Nobuyuki Ikeda	Related
K. David Elworthy	Related

S. R. S. Varadhan: “Regularity of self-diffusion coefficient,” pp. 387–397 in The Dynkin Festschrift: Markov processes and their applications. Edited by M. I. Freĭdlin. Progress in Probability 34. Birkhäuser (Boston, MA), 1994. MR 1311731 Zbl 0822.60089 incollection

Evgeniĭ Borisovich Dynkin	Related
Mark Iosifovich Freĭdlin	Related

H. Berestycki, L. Nirenberg, and S. R. S. Varadhan: “The principal eigenvalue and maximum principle for second-order elliptic operators in general domains,” Comm. Pure Appl. Math. 47 : 1 (1994), pp. 47–92. MR 1258192 Zbl 0806.35129 article

Louis Nirenberg	Related
Henri Berestycki	Related

T. Digernes, V. S. Varadarajan, and S. R. S. Varadhan: “Finite approximations to quantum systems,” Rev. Math. Phys. 6 : 4 (1994), pp. 621–648. MR 1290691 Zbl 0855.47046 article

S. R. S. Varadhan: “Entropy methods in hydrodynamic scaling,” pp. 196–208 in Proceedings of the International Congress of Mathematicians (Zürich, August 3–11, 1994), vol. I. Edited by S. D. Chatterji. Birkhäuser (Basel), 1995. MR 1403922 Zbl 0872.60081 inproceedings

S. R. S. Varadhan: “Self-diffusion of a tagged particle in equilibrium for asymmetric mean zero random walk with simple exclusion,” Ann. Inst. H. Poincaré Probab. Statist. 31 : 1 (1995), pp. 273–285. MR 1340041 Zbl 0816.60093 article

S. R. S. Varadhan: “The work of Pierre-Louis Lions,” pp. 6–10 in Proceedings of the International Congress of Mathematicians (Zürich, August 3–11, 1994), vol. I. Edited by S. D. Chatterji. Birkhäuser (Basel), 1995. MR 1403909 Zbl 0838.01025 inproceedings

Srishti Dhar Chatterji	Related
Pierre-Louis Lions	Related

S. R. S. Varadhan: “The complex story of simple exclusion,” pp. 385–400 in Itô’s stochastic calculus and probability theory. Edited by N. Ikeda, S. Watanabe, M. Fukushima, and H. Kunita. Springer (Tokyo), 1996. MR 1439538 Zbl 0866.60092 incollection

Hiroshi Kunita	Related
Masatoshi Fukushima	Related
Shinzo Watanabe	Related
Nobuyuki Ikeda	Related

C. Landim, S. Sethuraman, and S. Varadhan: “Spectral gap for zero-range dynamics,” Ann. Probab. 24 : 4 (1996), pp. 1871–1902. MR 1415232 Zbl 0870.60095 article

Claudio Landim	Related
Sunder Sethuraman	Related

S. R. S. Varadhan: “Nongradient models in hydrodynamic scaling,” pp. 397–416 in Analysis, geometry and probability. Edited by R. Bhatia. Texts and Readings in Mathematics 10. Hindustan Book Agency (Delhi), 1996. MR 1477705 Zbl 0977.60099 incollection

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

Daniel W. Stroock	Related
William Fulton	Related
Goro Shimura	Related
Bruce C. Berndt	Related

A. F. Ramirez and S. R. S. Varadhan: “Relative entropy and mixing properties of interacting particle systems,” J. Math. Kyoto Univ. 36 : 4 (1996), pp. 869–875. MR 1443753 Zbl 0884.60094 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 and H. T. Yau: “Scaling limits for lattice gas models,” pp. 39–43 in Stochastic processes and functional analysis (Riverside, CA, November 18–20, 1994). Edited by J. A. Goldstein, N. E. Gretsky, and J. J. Uhl. Lecture Notes in Pure and Applied Mathematics 186. Dekker (New York), 1997. MR 1440413 Zbl 0884.60090 incollection

Horng-Tzer Yau	Related
Jerome A. Goldstein	Related
Neil E. Gretsky	Related
J. Jerry Uhl	Related

S. R. S. Varadhan: “The work of Pierre-Louis Lions,” pp. 555–559 in Fields Medallists’ lectures. Edited by M. F. Atiyah and D. Iagolnitzer. World Scientific Series in 20th Century Mathematics 5. World Scientific (River Edge, NJ), 1997. MR 1622925 incollection

M. F. Atiyah	Related	Volume
Daniel Iagolnitzer	Related
Pierre-Louis Lions	Related

S. R. S. Varadhan and H.-T. Yau: “Diffusive limit of lattice gas with mixing conditions,” Asian J. Math. 1 : 4 (December 1997), pp. 623–678. MR 1621569 Zbl 0947.60089 article

Differential equations: La Pietra 1996 (Florence, July 3–7, 1996). Edited by M. Giaquinta, J. M. Shatah, and S. R. S. Varadhan. Proceedings of Symposia in Pure Mathematics 65. American Mathematical Society (Providence, RI), 1999. MR 1662745 book

Jalal M. Shatah	Related
Mariano Giaquinta	Related

J. Quastel, F. Rezakhanlou, and S. R. S. Varadhan: “Large deviations for the symmetric simple exclusion process in dimensions \( d\geq 3 \),” Probab. Theory Related Fields 113 : 1 (1999), pp. 1–84. MR 1670733 Zbl 0928.60087 article

We consider symmetric simple exclusion processes with \( L=\overline{\rho}N^d \) particles in a periodic \( d \)-dimensional lattice of width \( N \). We perform the diffusive hydrodynamic scaling of space and time. The initial condition is arbitrary and is typically far away form equilibrium. It specifies in the scaling limit a density profile on the \( d \)-dimensional torus. We are interested in the large deviations of the empirical process, \[ N^{-d}\Bigl[\sum_1^L\delta_{x_i(\,\cdot\,)}\Bigr] \] as random variables taking values in the space of measures on \( D[0,1] \). We prove a large deviation principle, with a rate function that is more or less universal, involving explicitly besides the initial profile, only such canonical objects as bulk and self diffusion coefficients.

Fraydoun Rezakhanlou	Related
Jeremy Quastel	Related

S. R. S. Varadhan: “Large deviations for interacting particle systems,” pp. 373–383 in Perplexing problems in probability. Edited by M. Bramson and R. Durrett. Progress in Probability 44. Birkhäuser (Boston, MA), 1999. MR 1703141 Zbl 0941.60096 incollection

Richard Timothy Durrett	Related
Maury D. Bramson	Related

Probability theory and applications (Princeton, NJ, June 23–July 13, 1996). Edited by E. P. Hsu and S. R. S. Varadhan. IAS/Park City Mathematics Series 6. American Mathematical Society (Providence, RI), 1999. MR 1678306 Zbl 0932.60005 book

L. H. Jensen and S. R. S. Varadhan: Large deviations of the asymmetric exclusion process in one dimension. Preprint, 2000.

S. R. S. Varadhan: “Infinite particle systems and their scaling limits,” pp. 306–315 in Mathematical physics 2000. Edited by A. Fokas, A. Grigoryan, T. Kibble, and B. Zegarlinski. Imperial College Press (London), 2000. MR 1773051 Zbl 1028.82516 incollection

The subject of nonequlibrium statistical mechanics deals with the question of how, a complex system with one or more conserved quantities, approaches its equilibrium. Although initially investigated in the context of classical mechanical systems that conserve mass, momenta and energy, its paradigm is relevent in the study of stochastic paticle systems with conserved quantities and mutiple equilibria. In fact the inherent noise in the stochastic systems provides a degree of stability that is absent in classical systems. During last twenty years several such models have been studied, with varying amounts of noise, and a body of rigorous results has been obtained.

Boguslaw Zegarlinski	Related
Athanassios S. Fokas	Related
T. Kibble	Related
Aleksandr Asaturovich Grigoryan	Related

S. Sethuraman, S. R. S. Varadhan, and H.-T. Yau: “Diffusive limit of a tagged particle in asymmetric simple exclusion processes,” Comm. Pure Appl. Math. 53 : 8 (August 2000), pp. 972–1006. MR 1755948 Zbl 1029.60084 article

Sunder Sethuraman	Related
Horng-Tzer Yau	Related

S. R. S. Varadhan: “Lectures on hydrodynamic scaling,” pp. 3–40 in Hydrodynamic limits and related topics (Toronto, October 7–10, 1998). Edited by S. Feng, A. T. Lawniczak, and S. R. S. Varadhan. Fields Institute Communications 27. American Mathematical Society (Providence, RI), 2000. MR 1798641 Zbl 1060.82514 incollection

Anna T. Lawniczak	Related
Shui Feng	Related

Hydrodynamic limits and related topics (Toronto, October 7–10, 1998). Edited by S. Feng, A. T. Lawniczak, and S. R. S. Varadhan. Fields Institute Communications 27. American Mathematical Society (Providence, RI), 2000. MR 1798640 book

Anna T. Lawniczak	Related
Shui Feng	Related

C. Landim, S. Olla, and S. R. S. Varadhan: “Asymptotic behavior of a tagged particle in simple exclusion processes,” Bol. Soc. Brasil. Mat. (N.S.) 31 : 3 (2000), pp. 241–275. MR 1817088 Zbl 0983.60100 article

Claudio Landim	Related
Stefano Olla	Related

S. R. S. Varadhan: “Scaling limits of large interacting systems,” pp. 144–158 in The mathematical sciences after the year 2000 (Beirut, 11–15 January, 1999). Edited by K. Bitar, A. Chamseddine, and W. Sabra. World Scientific (River Edge, NJ), 2000. MR 1799447 Zbl 1112.82320 incollection

Ali H. Chamseddine	Related
Wafic Sabra	Related
Khalil Bitar	Related

K. B. Athreya: “Professor Srinivasa R. S. Varadhan,” Current Sci. 78 : 9 (2000), pp. 1151–1152. MR 1762110 article

S. Sethuraman: “On Professor S. R. S. Varadhan’s contributions to hydrodynamic limits,” Current Sci. 78 : 9 (May 2000), pp. 1153–1154. MR 1762111 article

S. R. S. Varadhan: Probability theory. Courant Lecture Notes in Mathematics 7. Courant Institute of Mathematical Sciences (New York), 2001. MR 1852999 Zbl 0980.60002 book

C. Landim, S. Olla, and S. R. S. Varadhan: “Symmetric simple exclusion process: regularity of the self-diffusion coefficient,” pp. 307–321 in Dedicated to Joel L. Lebowitz on his seventieth birthday, published as Comm. Math. Phys. 224 : 1 (2001). MR 1869001 Zbl 0994.60093 incollection

Claudio Landim	Related
Stefano Olla	Related
Joel Louis Lebowitz	Related

S. R. S. Varadhan: “Large deviation and hydrodynamic scaling,” pp. 265–286 in Taniguchi conference on mathematics Nara ’98 (Nara, Japan, December 15–20, 1998). Edited by M. Maruyama and T. Sunada. Advanced Studies in Pure Mathematics 31. Mathematical Society of Japan (Tokyo), 2001. MR 1865096 Zbl 1006.60019 incollection

Masaki Maruyama	Related
Toshikazu Sunada	Related

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

C. Landim, S. Olla, and S. R. S. Varadhan: “Finite-dimensional approximation of the self-diffusion coefficient for the exclusion process,” Ann. Probab. 30 : 2 (2002), pp. 483–508. MR 1905849 Zbl 1018.60097 article

Claudio Landim	Related
Stefano Olla	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: “Rare events, large deviations,” pp. 85–92 in Mathematical finance — Bachelier Congress 2000 (Paris, June 29–July 1, 2000). Edited by H. Geman, D. Madan, S. R. Pliska, and T. Vorst. Springer Finance. Springer (Berlin), 2002. MR 1960560 incollection

Stanley R. Pliska	Related
Hélyette Geman	Related
Dilip B. Madan	Related
Ton Vorst	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

S. R. S. Varadhan: “Large deviations and entropy,” pp. 199–214 in Entropy. Edited by A. Greven, G. Keller, and G. Warnecke. Princeton Series in Applied Mathematics. Princeton University Press, 2003. MR 2035822 Zbl 1163.60312 incollection

Gerald Warnecke	Related
Gerhard Keller	Related
Andreas Greven	Related

S. R. S. Varadhan: “Particle systems and partial differential equations,” pp. 217–222 in Lectures on partial differential equations: Proceedings in honor of Louis Nirenberg’s 75th birthday. Edited by S.-Y. A. Chang, C.-S. Lin, and H.-T. Yau. New Studies in Advanced Mathematics 2. International Press (Somerville, MA), 2003. MR 2055850 Zbl 1167.82351 incollection

Chang-Shou Lin	Related
Horng-Tzer Yau	Related
Louis Nirenberg	Related
Sun-Yung A. Chang	Related

S. R. S. Varadhan: “Large deviations for random walks in a random environment,” Comm. Pure Appl. Math. 56 : 8 (August 2003), pp. 1222–1245. Dedicated to the memory of Jürgen K. Moser. MR 1989232 Zbl 1042.60071 article

S. R. S. Varadhan: “Large deviations for the asymmetric simple exclusion process,” pp. 1–27 in Stochastic analysis on large scale interacting systems. Edited by T. Funaki and H. Osada. Advanced Studies in Pure Mathematics 39. Math. Soc. Japan (Tokyo), 2004. MR 2073328 Zbl 1114.60026 incollection

Hirofumi Osada	Related
Tadahisa Funaki	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 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

C. Landim, S. Olla, and S. R. S. Varadhan: “On viscosity and fluctuation-dissipation in exclusion processes,” J. Statist. Phys. 115 : 1–2 (2004), pp. 323–363. MR 2070098 Zbl 1157.82355 article

Claudio Landim	Related
Stefano Olla	Related
Gianni Jona-Lasinio	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

S. R. S. Varadhan: “Random walks in a random environment,” Proc. Indian Acad. Sci. Math. Sci. 114 : 4 (2004), pp. 309–318. MR 2067696 Zbl 1077.60078 ArXiv math/0503089 article

C. Landim, S. Olla, and S. R. S. Varadhan: “Diffusive behaviour of the equilibrium fluctuations in the asymmetric exclusion processes,” pp. 307–324 in Stochastic analysis on large scale interacting systems. Edited by T. Funaki and H. Osada. Advanced Studies in Pure Mathematics 39. Mathemnatical Society of Japan (Tokyo), 2004. MR 2073338 Zbl 1080.60094 incollection

Claudio Landim	Related
Tadahisa Funaki	Related
Stefano Olla	Related
Hirofumi Osada	Related

S. Sethuraman and S. R. S. Varadhan: “A martingale proof of Dobrushin’s theorem for non-homogeneous Markov chains,” Electron. J. Probab. 10 (2005), pp. 1221–1235. MR 2164043 Zbl 1111.60057 ArXiv math/0404231 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

E. Kosygina, F. Rezakhanlou, and S. R. S. Varadhan: “Stochastic homogenization of Hamilton–Jacobi–Bellman equations,” Comm. Pure Appl. Math. 59 : 10 (2006), pp. 1489–1521. MR 2248897 Zbl 1111.60055 article

We study the homogenization of some Hamilton–Jacobi–Bellman equations with a vanishing second-order term in a stationary ergodic random medium under the hyperbolic scaling of time and space. Imposing certain convexity, growth, and regularity assumptions on the Hamiltonian, we show the locally uniform convergence of solutions of such equations to the solution of a deterministic “effective” first-order Hamilton–Jacobi equation. The effective Hamiltonian is obtained from the original stochastic Hamiltonian by a minimax formula. Our homogenization results have a large-deviations interpretation for a diffusion in a random environment.

Fraydoun Rezakhanlou	Related
Elena Kosygina	Related

“On the award of the Abel Prize to S. R. S. Varadhan,” Teor. Veroyatn. Primen. 52 : 3 (2007), pp. 417–418. In Russian. English translation in Theory Probab. Appl. 52:3 (2008) 371.

Norwegian Academy of Science and Letters: “Varadhan receives 2007 Abel Prize,” Notices Amer. Math. Soc. 54 : 6 (2007), pp. 738–739. MR 2327974 article

E. Bolthausen and A.-S. Sznitman: “Zur Verleihung des Abel-Preises 2007 an S. R. Srinisava Varadhan,” Mitt. Dtsch. Math.-Ver. 15 : 3 (2007), pp. 173–175. MR 2385129 article

Erwin Bolthausen	Related
Alain-Sol Sznitman	Related

S. R. S. Varadhan: Stochastic processes. Courant Lecture Notes in Mathematics 16. Courant Institute of Mathematical Sciences (New York), 2007. MR 2354349 Zbl 1133.60004 book

S. R. S. Varadhan: “Homogenization,” Math. Student 76 : 1–4 (2007), pp. 129–136. MR 2522935 Zbl 1182.35023 article

S. R. S. Varadhan: “Large deviations,” Ann. Probab. 36 : 2 (2008), pp. 397–419. MR 2393987 Zbl 1146.60003 ArXiv 0804.2330 article

S. R. S. Varadhan: “Homogenization of random Hamilton–Jacobi–Bellman equations,” pp. 397–403 in Probability, geometry and integrable systems. Edited by M. Pinsky and B. Birnir. Mathematical Sciencies Research Institute Publications 55. Cambridge University Press, 2008. MR 2407606 Zbl 1160.35334 incollection

We consider nonlinear parabolic equations of Hamilton–Jacobi–Bellman type. The Lagrangian is assumed to be convex, but with a spatial dependence which is stationary and random. Rescaling in space and time produces a similar equation with a rapidly varying spatial dependence and a small viscosity term. Motivated by corresponding results for the linear elliptic equation with small viscosity, we seek to find the limiting behaviour of the solution of the Cauchy (final value) problem in terms of a homogenized problem, described by a convex function of the gradient of the solution. The main idea is to use the principle of dynamic programming to write a variational formula for the solution in terms of solutions of linear problems. We then show that asymptotically it is enough to restrict the optimization to a subclass, one for which the asymptotic behavior can be fully analyzed. The paper outlines these steps and refers to the recently published work of Kosygina, Rezakhanlou and the author for full details.

Björn Birnir	Related
Mark A. Pinsky	Related

K. A. Borovkov: “The Abel Prize,” Theory Probab. Appl. 52 : 3 (2008), pp. 371. English translation of Russian original from Teor. Veroyatn. Primen. 52:3 (2007) 417–418.

M. Raussen and C. Skau: “Interview with Srinivasa Varadhan,” Notices Amer. Math. Soc. 55 : 2 (2008), pp. 238–246. MR 2375025 Zbl 1149.01311 article

E. Kosygina and S. R. S. Varadhan: “Homogenization of Hamilton–Jacobi–Bellman equations with respect to time-space shifts in a stationary ergodic medium,” Comm. Pure Appl. Math. 61 : 6 (2008), pp. 816–847. MR 2400607 Zbl 1144.35008 article

We consider a family \( \{u_{\varepsilon}(t,x,\omega)\} \), \( \varepsilon < 0 \), of solutions to the equation \[ \frac{\partial u_{\varepsilon}}{\partial t} + \frac{\varepsilon}{2}\,\Delta u_{\varepsilon} + H\Bigl(\frac{t}{\varepsilon}, \frac{x}{\varepsilon}, \nabla u_{\varepsilon},\,\omega\Bigr) = 0 \] with the terminal data \( u_{\varepsilon}(T,x,\omega) = U(x) \). Assuming that the dependence of the Hamiltonian \( H(t,x,p,\omega) \) on time and space is realized through shifts in a stationary ergodic random medium, and that \( H \) is convex in \( p \) and satisfies certain growth and regularity conditions, we show the almost sure locally uniform convergence, in time and space, of \( u_{\varepsilon}(t,x,\omega) \) as \( \varepsilon\to 0 \) to the solution \( u(t,x) \) of a deterministic averaged equation \[ \frac{\partial u}{\partial t} + \overline{H}(\nabla u) = 0, \qquad u(T,x) = U(x) .\] The “effective” Hamiltonian \( \overline{H} \) is given by a variational formula.

S. R. S. Varadhan and N. Zygouras: “Behavior of the solution of a random semilinear heat equation,” Comm. Pure Appl. Math. 61 : 9 (September 2008), pp. 1298–1329. MR 2431704 Zbl 1152.60077 article

R. Bhatia: “A conversation with S. R. S. Varadhan,” Math. Intelligencer 30 : 2 (2008), pp. 24–42. MR 2410123 Zbl 1179.01036 article

F. den Hollander: “Srinivasa Varadhan,” Nieuw Arch. Wiskd. (5) 9 : 3 (2008), pp. 192–196. MR 2527805 article

S. R. S. Varadhan: “Large deviations and scaling limit,” Lett. Math. Phys. 88 : 1–3 (2009), pp. 175–185. MR 2512145 Zbl 1185.60024 article

S. R. S. Varadhan: “The role of weak convergence in probability theory,” pp. 3–10 in Symmetry in mathematics and physics: Conference proceedings in honor of V. S. Varadarajan (UCLA, January 18–20, 2008). Edited by D. Babbitt, V. Chari, and R. Fioresi. Contemporary Mathematics 490. American Mathematical Society (Providence, RI), 2009. MR 2555966 Zbl 1186.60019 incollection

Vyjayanthi Chari	Related
Donald Babbitt	Related
Rita Fioresi	Related

Proceedings of the international symposium on probability theory and stochastic processes (Cochin University of Science and Technology, Kochi, India, February 6–9, 2009), published as Bull. Kerala Math. Assoc. Special Issue. Issue edited by S. R. S. Varadhan. 2009. MR 2590248 book

S. R. S. Varadhan: “Workshop on large deviations: Lecture notes,” pp. 1–14 in Proceedings of the international symposium on probability theory and stochastic processes (Cochin University of Science and Technology, Kochi, India, February 6–9, 2009), published as Bull. Kerala Math. Assoc. Special Issue. Issue edited by S. R. S. Varadhan. 2009. MR 2590249 incollection

S. R. S. Varadhan: “Scaling limits,” pp. 247–262 in Perspectives in mathematical sciences, vol. I: Probability and statistics. Edited by N. S. N. Sastry, T. S. S. R. K. Rao, M. Delampady, and B. Rajeev. Statistical Science and Interdisciplinary Research 7. World Scientific (Hackensack, NJ), 2009. MR 2581747 incollection

B. Rajeev	Related
T. S. S. R. K. Rao	Related
N. S. Narasimha Sastry	Related
Mohan Delampady	Related

The Abel Prize, 2003–2007: The first five years. Edited by H. Holden and R. Piene. Springer (Berlin), 2010. MR 2605604 book

Ragni Piene	Related
M. F. Atiyah	Related	Volume
Jean-Pierre Serre	Related
Peter D. Lax	Related
Lennart Axel Edvard Carleson	Related
Isadore M. Singer	Related	Volume
Helge Holden	Related

Y. Kifer and S. R. S. Varadhan: Nonconventional limit theorems in discrete and continuous time via martingales. Preprint, December 2010. ArXiv 1012.2223 techreport

S. R. S. Varadhan: “Large deviations,” pp. 622–639 in Proceedings of the International Congress of Mathematicians (Hyderabad, India, August 19–27, 2010), vol. I: Plenary lectures and ceremonies. Edited by R. Bhatia. Hindustan Book Agency (New Delhi), 2010. MR 2827907 Zbl 1228.60037 inproceedings

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

S. Chatterjee and S. R. S. Varadhan: “The large deviation principle for the Erdős–Rényi random graph,” European J. Combin. 32 : 7 (2011), pp. 1000–1017. MR 2825532 ArXiv 1008.1946 article

S. Chatterjee and S. R. S. Varadhan: Large deviations for random matrices. Preprint, June 2011. ArXiv 1106.4366 techreport

S. Sethuraman and S. R. S. Varadhan: Large deviations for the current and tagged particle in 1D nearest-neighbor symmetric simple exclusion. Preprint, January 2011. ArXiv 1101.1479 techreport

S. R. Srinivasa Varadhan

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