# S.R.S. Varadhan: profile of the National Medal of Science recipient

**Alain-Sol Sznitman, ETH Zurich, profiles former IMS president S.R.S. Varadhan, who, as reported in the last issue, has been named a recipient of the US National Medal of Science.
**

The National Medal of Science is the highest honor bestowed by the United States government on scientists and engineers for outstanding contributions. Varadhan is recognized for his work on probability, in particular for his work on large deviations. The text below provides a glimpse into Varadhan’s accomplishments.

Raghu Varadhan has had a deep influence on the development of probability theory. He has introduced concepts, which have solved hard problems and shaped the way modern theory conceptualizes such problems. His work is intimately attached to the present understanding of very diverse subjects such as diffusion processes, large deviations, random media, and systems of many particles.

At the end of the sixties and beginning of the seventies Varadhan developed with Dan Stroock a new approach to diffusion processes, the so-called martingale problem. This was a far-reaching generalization of Paul Lévy’s famous characterization of Brownian motion in terms of two martingales. The martingale problem approach gave a way to tackle weak solutions of stochastic differential equations, which—unlike Itô’s approach—was ideal to treat questions of approximation, when diffusion processes show up as limit objects. Martingale problems offered a powerful and more transparent way to obtain such limit results. They are now a standard tool of probability theory, and in many cases are used to define processes.

A further important achievement of Stroock and Varadhan was their work on the support of diffusion processes. They described the support of the measure on path space governing a diffusion process in terms of the closure of a family of solutions of deterministic differential equations. In doing so they showed a type of continuity principle, for the correspondence between “noise” and “response” of the system. This approach prepared the ground for the later development of Malliavin calculus, where an appropriate notion of smoothness of the map “noise gives response” became a central theme.

Varadhan had previously gained international recognition for his early work (in the mid- to late-sixties) on large deviations. Varadhan had studied the small time behavior of the heat kernel of elliptic diffusions. He had showed that certain largely deviant paths of the diffusion drove the off-diagonal behavior of the heat kernel at small times. He had exhibited the crucial role of a certain quadratic action functional, involving the Riemannian distance attached to the inverse of the matrix of second order coefficients of the generator of the diffusion. These works were the precursors of a very thorough study by probabilistic methods of the small time behaviors of heat kernels by authors such as Bismut, Stroock, Ben Arous, Léandre and others. At the same time Varadhan had laid the foundations of the modern theory of large deviations. He had developed the abstract theory and applied it to striking examples.

About ten years later in a series of articles with Monroe Donsker, from the mid-seventies until the beginning of the eighties, Varadhan properly revolutionized large deviation theory by launching a systematic investigation of large deviations effects for the long time behavior of Markov processes. The idea was to measure the cost attached to the production of various behaviors of a Markov process, when it deviates from the predictions provided by the ergodic theorem. Such predictions were investigated on successively “higher and higher constructs”, culminating with a large deviation theory at the level of empirical processes; large deviation results on the “lower constructs” being revisited by means of so-called contraction principles. The striking character of this revolution was the fact that it was going back and forth between abstract theory and many applications to concrete examples. Donsker and Varadhan for instance applied their theory to large time asymptotics of the Wiener sausage, thereby solving a conjecture of M. Kac and J.M. Luttinger. A further striking application was the solution of a conjecture of S. Pekar concerning the Polaron, a model from quantum statistical mechanics. As mentioned above Donsker and Varadhan had developed an abstract general theory, which was able to crack a full palette of hard problems.

Varadhan has also been one of the key players in the investigation of the asymptotic behavior of diffusions or random walks in random media. The topic was a natural extension of periodic homogenization theory, where the asymptotic behavior of diffusions driven by periodic coefficients is considered. After proper scaling, one can obtain in this classical framework a description of the motion, singling out slow-varying variables and fast-varying variables. Averaging out the fast variables yields a limit autonomous evolution rule for the slowly varying macroscopic variables. This averaging procedure is typically not naïve, since some oscillating drifts are approximated by purely diffusive terms, and this step involves the construction of so-called correctors.

Random media came as a way to model microscopic disorder, and extend the rigid framework of periodic media. This created a serious mathematical challenge. Whereas the slow variables were quite clear in this new set-up, and remained attached to the position in macroscopic units of the particle, it took a conceptual leap to realize that the fast-varying variables corresponded to the whole random environment translated at the current position of the traveler. This idea emerged around the same time, at the end of the seventies, beginning of the eighties, in the work of Sergei Kozlov in the Soviet Union, and in the work of George Papanicolaou and Varadhan. Being able to tackle these fast-varying variables, which were now infinite dimensional, required the construction of an adequate invariant measure for the motion of the environment viewed from the particle, in order to average out the fast variables. This construction could be done in a number of physically relevant situations, where such a measure was easily accessible. One still had to develop the construction of correctors, or approximate correctors, to handle the oscillating drifts and replace them with diffusive terms. Progress came in the mid-eighties with an article by Claude Kipnis and Varadhan, which handled this task under self-adjointness assumptions. Their method had immediate implications for numerous models. This was the starting point of a long series of developments involving many authors. Varadhan’s name is thus intimately attached to the “method of the environment viewed from the particle”, one of the rare general tools that is used in the field to this day.

Varadhan’s work at the end of the eighties and beginning of the nineties had a deep impact on the question present in many problems of statistical mechanics of relating the microscopic point of view involving many interacting particles, with a macroscopic description by means of a partial differential equation (or a system of them). A fundamental problem corresponds to the question of relating Hamilton’s equations at the microscopic level, to the equations of hydrodynamics (for instance Euler’s equations) at the macroscopic level, when describing a fluid or a gas. Analogous questions arise in a broad palette of other models. The heart of the matter is to track down what happens to various locally-conserved quantities in the system (depending on the models this can, for instance, be the local number of particles, the local fluid velocities, the local energy density, etc…). These few quantities are precisely supposed to obey in the appropriate scaling limit the partial differential equations one is after. The difficulty stems from the fact that the equations for these quantities are not autonomous, and controlling the limit procedure requires being able to perform averages at a microscopic level of certain local quantities. Roughly one needs to see that in a suitable sense the particle systems get locally organized according to equilibria (such as Gibbs measures) with parameters corresponding to the current local value of the locally-conserved quantities. This is a hard question because it involves estimates over very large dimensional systems and all quantities under consideration wildly oscillate when going to small scales. In his work Varadhan developed a collection of methods that systematically exploit controls given by entropy and time-derivative of the entropy. These techniques are powerful enough to extract sufficient information on the microscopic picture to extract such limiting equations, and track down such local equilibria, with parameters that remain slowly varying. Varadhan’s work has triggered a large body of research in mathematical physics, involving among others students and collaborators of Varadhan, notably with important developments due to Horng-Tzer Yau. This has lead to a much better understanding, checked on numerous examples, of the relation between microscopic and macroscopic points of view. Particularly impressive was the treatment by Varadhan of the so-called non-gradient systems, where averaging out of the microscopic currents driving the locally conserved quantities was especially delicate.

Throughout his scientific activity Varadhan has kept an intimate link to questions of large deviations. During the last decade he has authored and coauthored many gems in a broad palette of subjects including random walks in random environments, homogenization of Hamilton-Jacobi-Bellman equations, and random graphs, to name a few.

Finally here are some elements concerning Raghu Varadhan’s biography. He was born in Chennai, India, in 1940. He received his doctoral degree in 1963 from the Indian Statistical Institute in Calcutta. He came to the Courant Institute at NYU, as a post-doctoral fellow in 1963. He became Assistant Professor in 1966, Associate Professor in 1968, Professor in 1972, and spent his entire professional carrier at the Courant Institute, except for visiting positions. He has received numerous awards including a Plenary Lecture at the International Congress of Mathematicians (1994), the Birkhoff Prize (1994), the AMS Steele Prize (1996), and the Abel Prize (2007). He is married to Vasu Varadhan, who is Associate Faculty at NYU’s Gallatin School. They have a son and two grandchildren; their elder son perished in the attack on the twin towers in September 2001.

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