Celebratio Mathematica

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

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

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

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.

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

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

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

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

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. 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].

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S. R. Srinivasa Varadhan

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