Celebratio Mathematica

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

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: “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

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

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

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

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: “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

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

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.

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

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: “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

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

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

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. 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

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

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: “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: “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: “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

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

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

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: 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

year	title	people

			clear

S. R. Srinivasa Varadhan

Large deviations

Filter the Bibliography List