S·R·S·瓦拉德汉Sathamangalam Ranga Iyengar Srinivasa Varadhan

S·R·S·瓦拉德汉简称S R S 瓦拉德汉，朋友和同事称他为Raghu。他的父亲瓦拉德汉 瓦拉德汉是一位科学教师，后来成为Ponneri的董事会中学的校长，Ponneri是距离马德拉斯（现称金奈）约30公里的一个小镇。瓦拉德汉在[12]中感谢他的父母在他成长过程中的支持：-

在我们家，教育始终被放在优先位置，我不断从父母双方得到鼓励。

他在10中解释了自己在高中时如何第一次对数学产生兴趣：-

在高中，我有一位出色的数学老师，他让一些较好的学生在周末，星期六或星期日，到他家去，给他们额外的题目做。我们把这些题目当作我们玩的智力游戏，不像考试；更多是为了乐趣。这让我觉得数学是一种可以像下棋或解谜一样享受的事情。这种态度使数学成为一门友好得多的学科，而不是令人害怕的东西，这在我对数学产生兴趣的原因中起了作用。

1955年高中毕业后，瓦拉德汉进入马德拉斯大学院长学院。在那里，他选择攻读统计学学位而不是数学学位。原因是数学学位包括纯数学和应用数学，而统计学学位用统计学取代了应用数学，但除此之外提供相同的纯数学课程。在他职业生涯的这个阶段，瓦拉德汉认为自己毕业后会进入工业界，并觉得拥有统计学学位前景要好得多。他于1959年获得统计学荣誉学士学位，并继续第五年攻读硕士学位，于1960年获得。然后他去了加尔各答的印度统计研究所，尽管在这个阶段他仍然认为自己会在工业界找到工作[10]：-

有人告诉我应该研究统计质量控制，所以我花了将近6到8个月学习统计质量控制；最终，这让我完全不满意。然后我遇到了Varadarajan、Parthasarathy和瓦拉德汉 Rao，他们从完全数学的角度研究概率。他们说服我，我没有在有效利用时间，如果我想做任何事情，最好学一些数学。我产生了兴趣，我想在我在那里的第二年，我们对自己说：让我们研究一个问题。我们选了一个关于群上概率分布的问题。这让我们开始了；我们最终解决了这个问题，并在此过程中也学到了所需的工具。

在印度统计研究所，瓦拉德汉的导师是亚木普迪‧拉达克利西纳‧劳，但这只是一种形式上的安排，瓦拉德汉的论文源于与同事的互动。他于1963年因学位论文Convolution Properties of Distributions on Topological Groups获得博士学位。事实上，安德雷·柯尔莫哥洛夫于1962年在印度统计研究所待了一个月，并被任命为瓦拉德汉论文的审查人之一。他把论文带回莫斯科，并寄回了他的报告，报告中说[10]：-

这不是学生的作品，而是一位成熟大师的作品。

获得博士学位后，瓦拉德汉作为博士后访问学者前往纽约大学的理查·科朗特数学科学研究所，他在此职位上工作了三年（1963-66）。与他合作多年的Daniel Stroock在谈到瓦拉德汉的到来时写道[11]：-

瓦拉德汉，大家都叫他Raghu，于1963年秋天从祖国印度来到这片海岸。他乘飞机抵达Idlewild机场，然后乘巴士前往曼哈顿……他的目的地是那个名字谦逊的著名机构——理查·科朗特数学科学研究所，在那里，受Monroe Donsker之命，他获得了博士后奖学金。

1964年，他与Vasundra结婚，Vasundra于1947年出生在印度金奈，但生命的前十二年大部分时间都在纽约度过。Vasu和瓦拉德汉一样，成为纽约大学的教授。他们有两个孩子，Gopal（生于1969年）和他的弟弟Ashok。Gopal于2001年8月加入Cantor Fitzgerald，担任其美国利率衍生品业务的董事总经理，并于2001年9月11日在世界贸易中心北塔遇难。回到我们对瓦拉德汉职业生涯的概述，我们注意到他于1966年被任命为纽约大学助理教授。这是在路易·尼伦伯格于1965年致Monroe Donsker的信中作出推荐之后[12]：-

我对瓦拉德汉评价非常高，并预测他会有远大的未来。他非常年轻，我认为在许多方面，他可能是我们能做出的概率论助理教授的最佳任命。

瓦拉德汉于1968年晋升为副教授，并于1972年成为纽约大学正教授。

瓦拉德汉的研究广泛涉及概率论的不同领域，我们特别指出其研究的两个主要方向。他在这两个领域的工作在瓦拉德汉所获主要奖项的引文中得到认可，我们通过引用这些奖项的引文来简要介绍这些领域。1996年，瓦拉德汉与其合作者Daniel Stroock因对研究的基础性贡献而获得美国数学会的Leroy P 凯瑟琳·斯蒂尔奖。引文如下[1]：-

授予Daniel Stroock和瓦拉德汉，以表彰他们的四篇论文《具有连续系数的扩散过程 I 和 II》（1969年）、《关于扩散过程的支撑及其在强最大值原理中的应用》（1970年）、《多维扩散过程》（1979年），在这些论文中，他们引入了随机微分方程的鞅解这一新概念，使他们能够证明以前无法用纯分析方法处理的方程解的存在性、唯一性及其他重要性质；他们的表述已被广泛用于证明各种过程向扩散的收敛。

在获奖时，瓦拉德汉谈到了他在20世纪60年代在理查·科朗特研究所的同事和环境[1]：-

我非常高兴我的同事们选择挑出我在六十年代末与Dan Stroock的一些工作作为重要成果。理查·科朗特研究所，大部分工作是在那里完成的，为我们提供了理想的智力环境。我们得到了资深同事，特别是尼伦伯格和Monroe Donsker的积极鼓励和支持。由于Henry McKean和马克·卡茨在洛克菲勒，纽约对于有抱负的概率论学者来说确实是一个非常令人兴奋的地方。Dan和我在这一时期密切合作，对我来说非常令人兴奋且富有成果。我感谢他，不仅因为他是一个伟大的合作者，也因为多年的亲密友谊。我特别高兴能与他分享这个奖项。

我们挑出的瓦拉德汉工作的两个领域中的第二个是他对大偏差理论的杰出贡献。2007年，他被授予极具声望的尼尔斯·阿贝尔奖，该奖项的引文给出了一些他在这一领域工作的细节[12]：-

在他1966年的里程碑式论文《渐近概率与微分方程》和1969年对欧几里得量子场论中极化子问题的惊人解决中，瓦拉德汉开始塑造一个通用的大偏差理论，这远不止是收敛速率的定量改进。它解决了一个基本问题：如果一个随机系统偏离了由某种大数定律预测的遍历行为，或者如果它作为确定性系统的小扰动出现，它的定性行为是什么？答案的关键是一个强大的变分原理，它通过一个新的概率模型来描述这种意外行为，该模型最小化与初始概率测度的适当熵距离。在与Monroe D Donsker的一系列联合论文中，探索马尔可夫过程背景下的大偏差层级，瓦拉德汉展示了这种新方法的相关性和力量。一个引人注目的应用是他们解决了马克·卡茨关于布朗运动路径管状邻域大时间渐近性的猜想，即所谓的“维纳香肠”。瓦拉德汉的大偏差理论为阐明复杂随机系统中出现的丰富多样现象提供了一种统一而有效的方法，这些领域包括量子场论、统计物理、种群动力学、计量经济学和金融学以及交通工程等。它还极大地扩展了我们使用计算机模拟和分析稀有事件发生的能力。在过去四十年中，大偏差理论已成为现代概率论（无论是纯理论还是应用）的基石。

除上述奖项外，瓦拉德汉还荣获了以下荣誉：当选美国艺术与科学院（1988年）、第三世界科学院院士（1988年）以及国家科学院（1995年）。他于1991年当选为数理统计学会会士，并于1998年当选为伦敦皇家学会会士。1994年，他获得了由美国数学会和工业与应用数学学会联合颁发的乔治·戴维·伯克霍夫奖：-

……表彰其在扩散过程的鞅刻画、马尔可夫过程占位时泛函的大偏差理论以及随机介质研究方面的重要贡献……

并于1995年获得纽约大学艺术与科学学院的玛格丽特和赫尔曼·索科尔奖。

瓦拉德汉是纽约大学理查·科朗特数学科学研究所的Frank J Gould科学讲席教授和数学教授。他曾两度担任该研究所所长，即1980-84年和1992-94年。当瓦拉德汉于2007年被授予尼尔斯·阿贝尔奖时，纽约大学校长弗瑞兹·约翰 Sexton说[9]：-

我们为拉古感到无比高兴和自豪。他不仅是一位杰出的学者，还是一位友善而出色的同事、一位尽职尽责的教师，以及一位堪称楷模的“大学公民”，以奉献精神和专业素养担任理查·科朗特研究所所长，并在大学评议会等机构任职。这一殊荣对于一位谦逊谨慎几乎与其学术贡献齐名的教员来说，是当之无愧的荣誉。在拉古任职于纽约大学期间，我们的大学发生了巨大变化，但正是像他这样的学者的持续存在，使我们能够将纽约大学建设成今天的样子，并继续吸引顶尖学者和研究人员加入我们的行列。

2008年，瓦拉德汉获得了莲花士奖，该奖项由印度在人民庆祝第59个共和国日当天颁发。他是文学与教育领域九位获奖者之一。

现在让我们稍微谈谈瓦拉德汉写的一些书，其中许多是基于他在理查·科朗特研究所讲授的研究生课程讲义。他出版了Stochastic processes（1968年），该书由J R Kinney评论，他的描述是这样开始的：-

这些是1967-68年间在理查·科朗特研究所讲授的随机过程课程的讲义。作者打算给出一个综述，而不是深入特殊细节。……这套讲义是对其他地方只能以非常技术性形式获得材料的非常易读的综述。

《Mathematical statistics》（1974年）一书基于1973-1974学年的讲座，但他的专著（与Daniel Stroock合著）《Multidimensional diffusion processes》（1979年）是他获得Leroy P 斯蒂尔奖的作品之一。A D Wentzell的评论开头如下：-

这本书是关于马尔可夫过程理论的鞅方法。作者通过仔细处理一个问题而不是向读者展示大量截然不同的例子来展示这种方法。这一个问题就是沿着作者开创的路线为尽可能广泛的系数类构造扩散过程……

瓦拉德汉的书《Lectures on diffusion problems and partial differential equations》（1980年）从布朗运动开始，引导学生进入随机微分方程和扩散理论。François Bronner在评论瓦拉德汉的书《Large deviations and applications》（1984年）时解释说，它：-

……涵盖了在其他地方除期刊文章外未处理的大量材料。因此，阅读这位大偏差理论大师以愉快而清晰的风格写成的文本具有首要意义。在几页中，人们可以找到问题的设定、基本应用的综述，包括例如Wentzell和Freodlin理论，以及在马尔可夫情形下下界和上界的非常清晰的构造。

2001年出版的Probability theory（2001）一书基于1996年至1999年在理查·科朗特研究所开设的一年级研究生课程。这本极为成功的书中呈现的材料在Stochastic processes（2007）中得到了延续，后者同样基于在理查·科朗特研究所的课程。

现在让我们来看看瓦拉德汉在数学之外的兴趣[10]：-

我喜欢旅行。我喜欢参观新地方、看到新事物和获得新体验的乐趣和经历。在我们的职业中，你有机会旅行，我总是利用这一点。我喜欢音乐，包括印度古典音乐和一点西方古典音乐。如果有时间，我喜欢去听音乐会；我喜欢戏剧，纽约是一个绝佳的戏剧之地。我喜欢去看电影。我喜欢阅读泰米尔文学，我很享受。世界上没有多少人对泰米尔语这门语言熟悉。这是一门有2000年历史的语言，几乎和梵语一样古老。它也许是今天唯一一门与2000年前没有太大区别的语言。所以，我可以拿起一本2000年前写的诗集，我仍然能够阅读它。在我力所能及的范围内，我会这样做。

我们引用Daniel Stroock 1997年的文章[11]来结束这篇传记：-

我不会声称瓦拉德汉不喜欢他的成功；他确实喜欢。我也不会说他是某种圣人；如果他是的话，我们永远不可能成为朋友。尽管如此，瓦拉德汉与我遇到过的几乎所有其他有天赋的人的区别在于，他对自己天赋的非凡掌控。特别是，他学会了如何防止自己非凡的智力毒害他与智力较低者的关系。例如，瓦拉德汉能够容忍犯错，至少偶尔如此。此外，他不是那种认为自己对人类的所有义务都可以通过对数学研究的贡献来履行的众多数学王子之一。

Srinivasa Varadhan is known as S R S Varadhan for short and Raghu to his friends and colleagues. His father, Ranga Iyengar, was a science teacher who became the Principal of the Board High School in Ponneri, a small town about 30 km from Madras (now called Chennai). Varadhan, in [12], thanked his parents for their support as he grew up:-

Education always got high priority in our house and I received constant encouragement from both my parents.

He explained in [10] how he first became interested in mathematics while at high school:-

At high school I had an excellent mathematics teacher who asked some of his better students to come to his house during weekends, Saturday or Sunday, and gave them extra problems to work on. We thought of these problems just as intellectual games that we played, it was not like an exam; it was more for enjoyment. That gave me the idea that mathematics is something that you can enjoy doing like playing chess or solving puzzles. That attitude made mathematics a much more friendly subject, not something to be afraid of, and that played a role in why I got interested in mathematics.

After graduating from high school in 1955, Varadhan entered Presidency College of the University of Madras. There he chose to study for the degree in Statistics rather than in Mathematics. The reason for this was that the Mathematics degree consisted of Pure and Applied Mathematics while the Statistics degree replaced the applied mathematics with statistics but otherwise provided the same pure mathematics courses. At this stage in his career Varadhan thought he would enter industry after graduating and felt that his prospects were much better with the Statistics degree. He was awarded his B.A. with honours in Statistics in 1959 and continued for a fifth year to study for his Master's Degree which he received in 1960. He then went to the Indian Statistical Institute at Kolkata (Calcutta) although at this stage he still had the idea that he would find a job in industry [10]:-

I was told that I should work on statistical quality control, so I spent close to 6 or 8 months studying statistical quality control; in the end, that left me totally unsatisfied. Then I met Varadarajan, Parthasarathy, and Ranga Rao, who worked in probability from a totally mathematical point of view. They convinced me that I was not spending my time usefully, and that I better learn some mathematics if I wanted to do anything at all. I got interested, and I think in the second year I was there, we said to ourselves: Let us work on a problem. We picked a problem concerning probability distributions on groups. That got us started; we eventually solved the problem and in the process also learned the tools that were needed for it.

At the Indian Statistical Institute Varadhan's advisor was Calyampudi Radhakrishna Rao but this was only a formal arrangement and Varadhan's thesis grew out of interactions with his colleagues. He was awarded his doctorate for his dissertation Convolution Properties of Distributions on Topological Groups in 1963. In fact Andrei Nikolaevich Kolmogorov had spent a month at the Indian Statistical Institute in 1962 and was appointed as one the examiners of Varadhan's thesis. He took the thesis back with him to Moscow and sent in his report which said [10]:-

This is not the work of a student, but of a mature master.

After the award of his doctorate, Varadhan went as a post-doctoral visitor to the Courant Institute of Mathematical Sciences at New York University, a position he held for three years (1963-66). Daniel Stroock, who collaborated with him over many years, wrote about Varadhan's arrival [11]:-

Varadhan, whom everyone calls Raghu, came to these shores from his native India in the fall of 1963. He arrived by plane at Idlewild Airport and proceeded to Manhattan by bus .... His destination was that famous institution with the modest name, The Courant Institute of Mathematical Sciences, where, at the behest of Monroe Donsker, he had been given a postdoctoral fellowship.

In 1964 he married Vasundra, who was born in 1947 in Chennai in India but had spent most of the first twelve years of her life in New York. Vasu, like Srinivasa Varadhan, became a professor at New York University. They had two children, Gopal (born in 1969) and his younger brother Ashok. Gopal joined Cantor Fitzgerald as the Managing Director of its interest rate derivatives business in the United States in August 2001 and died in the north tower of the World Trade Center on September 11, 2001. Returning to our summary of Varadhan's career, we note he was appointed as an assistant professor at New York University in 1966. This followed the recommendation which Louis Nirenberg made in a letter to Monroe Donsker in 1965 [12]:-

I think very highly indeed of Varadhan and predict a great future for him. He is very young, and I think in many ways he might be the best appointment as assistant professor in probability we could make.

Varadhan was promoted to associate professor in 1968 and became a full professor at New York University in 1972.

Although Varadhan's research has ranged widely over different areas of probability theory we single out two main streams of this research. His work in these two areas has been recognised in the citations for the major prizes which Varadhan has been awarded and we give brief details of these areas by quoting from the citations for these awards. In 1996 Varadhan won the American Mathematical Society's Leroy P Steele Prize for fundamental contribution to research along with his collaborator Daniel Stroock. The citation reads [1]:-

To Daniel Stroock and Srinivasa Varadhan for their four papers 'Diffusion processes with continuous coefficients I and II' (1969), 'On the support of diffusion processes with applications to the strong maximum principle (1970), Multidimensional diffusion processes (1979), in which they introduced the new concept of a martingale solution to a stochastic differential equation, enabling them to prove existence, uniqueness, and other important properties of solutions to equations which could not be treated before by purely analytic methods; their formulation has been widely used to prove convergence of various processes to diffusions.

On receiving the award Varadhan spoke about his colleagues and the environment at the Courant Institute in the 1960s [1]:-

I am very pleased that my colleagues have chosen to single out some of my work with Dan Stroock in the late sixties as important. The Courant Institute, where most of the work was done, provided us with an ideal intellectual environment. We had the active encouragement and support of our senior colleagues, particularly Louis Nirenberg and Monroe Donsker. With the presence of Henry McKean and Mark Kac at Rockefeller, New York was indeed a very exciting place to be for an aspiring probabilist. Dan and I worked closely during this period, and to me it was very exciting and fruitful. I thank him, not just because he was a great person to work with, but for the years of close friendship as well. I am particularly pleased to be sharing this prize with him.

The second of the two areas of Varadhan's work which we single out is his remarkable contributions to the theory of large deviations. In 2007 he was awarded the highly prestigious Abel Prize and the citation for that award gives some details of his work in this area [12]:-

In his landmark paper 'Asymptotic probabilities and differential equations' in 1966 and his surprising solution of the polaron problem of Euclidean quantum field theory in 1969, Varadhan began to shape a general theory of large deviations that was much more than a quantitative improvement of convergence rates. It addresses a fundamental question: what is the qualitative behaviour of a stochastic system if it deviates from the ergodic behaviour predicted by some law of large numbers or if it arises as a small perturbation of a deterministic system? The key to the answer is a powerful variational principle that describes the unexpected behaviour in terms of a new probabilistic model minimizing a suitable entropy distance to the initial probability measure. In a series of joint papers with Monroe D Donsker exploring the hierarchy of large deviations in the context of Markov processes, Varadhan demonstrated the relevance and the power of this new approach. A striking application is their solution of a conjecture of Mark Kac concerning large time asymptotics of a tubular neighbourhood of the Brownian motion path, the so-called 'Wiener sausage'. Varadhan's theory of large deviations provides a unifying and efficient method for clarifying a rich variety of phenomena arising in complex stochastic systems, in fields as diverse as quantum field theory, statistical physics, population dynamics, econometrics and finance, and traffic engineering. It has also greatly expanded our ability to use computers to simulate and analyze the occurrence of rare events. Over the last four decades, the theory of large deviations has become a cornerstone of modern probability, both pure and applied.

Varadhan has been honoured, in addition to the awards mentioned above, with his election to the American Academy of Arts and Sciences (1988), the Third World Academy of Sciences (1988), and the National Academy of Sciences (1995). He was elected a Fellow of the Institute of Mathematical Statistics (1991), and of the Royal Society of London (1998). In 1994 he received the George David Birkhoff Prize, jointly awarded by the American Mathematical Society and the Society for Industrial and Applied Mathematics:-

... for important contributions to the martingale characterization of diffusion processes, to the theory of large deviations for functionals of occupation times of Markov processes, and to the study of random media ...

and he received the Margaret and Herman Sokol Award of the Faculty of Arts and Sciences, New York University in 1995.

Varadhan is Frank J Gould Professor of Science and professor of mathematics at the Courant Institute of Mathematical Sciences at New York University. He had served for two terms as director of the Institute, namely 1980-84 and 1992-94. When Varadhan was awarded the Abel Prize in 2007, the President of New York University John Sexton said [9]:-

We are so happy and proud of Raghu. Not only is he an outstanding scholar, he is also a kind and wonderful colleague, a devoted teacher, and an exemplary 'University citizen,' serving with dedication and professionalism as director of the Courant Institute and on such bodies as the University Senate. This distinction is a well-deserved honour for a faculty member whose modesty and discretion are almost as great as his scholarly contributions. In the time that Raghu has been at NYU, our University has changed a great deal, but it is the persistent presence of scholars such as he that has enabled us to build NYU into what it is today and to continue to attract top scholars and researchers to our midst.

In 2008 Varadhan received the Padma award, given by India on the day that the people celebrated their 59th Republic Day. He was one of nine winners in the field of Literature and Education.

Let us now say a little about some of the books written by Varadhan, many of which have been based on lecture notes of graduate courses he has given at the Courant Institute. He published Stochastic processes (1968) which was reviewed by J R Kinney whose begins his description as follows:-

These are lecture notes of a course on stochastic processes given at the Courant Institute during 1967-68. The author intends to give a survey rather than going into special details. ... This set of notes is a very readable survey of material available elsewhere only in very technical form.

The book Mathematical statistics (1974), was based on lectures given during the academic year 1973-1974, but his monograph (written jointly with Daniel Stroock) Multidimensional diffusion processes (1979) was one of the works for which he received the Leroy P Steele Prize. A D Wentzell begins a review as follows:-

The book is about the martingale approach to the theory of Markov processes. The authors demonstrate this approach by treating carefully just one problem rather than showing the reader a great number of widely different examples. This one problem is the construction of diffusion processes for the widest possible class of coefficients along the lines initiated by the authors ...

Varadhan's book Lectures on diffusion problems and partial differential equations (1980) starts from Brownian motion and leads the students to stochastic differential equations and diffusion theory. François Bronner, reviewing Varadhan's book Large deviations and applications (1984), explains that it:-

... covers a great deal of material not treated anywhere else outside of journal articles. So it is of prime interest to read this text, written in a pleasant and clear style, by a master of the theory of large deviations. In a few pages one can find the setting of the problem, a survey of the fundamental applications including, for example, the Wentzell and Freodlin theory, and a very clearly presented construction of the lower and upper bounds in the Markov case.

The 2001 book Probability theory (2001) was based on a first year graduate course given from 1996 to 1999 at the Courant Institute. The material presented in this highly successful book was continued in Stochastic processes (2007), also based on courses at the Courant Institute.

Let us now look at Varadhan's interests outside mathematics [10]:-

I like to travel. I like the pleasure and experience of visiting new places, seeing new things and having new experiences. In our profession, you get the opportunity to travel, and I always take advantage of it. I like music, both classical Indian and a little bit of classical Western music. I like to go to concerts if I have time; I like the theatre, and New York is a wonderful place for theatre. I like to go to movies. I like reading Tamil literature, which I enjoy. Not many people in the world are familiar with Tamil as a language. It is a language which is 2,000 years old, almost as old as Sanskrit. It is perhaps the only language which today is not very different from the way it was 2,000 years ago. So, I can take a book of poetry written 2,000 years ago, and I will still be able to read it. To the extent I can, I do that.

We end this biography by quoting from Daniel Stroock's 1997 article [11]:-

I am not going to claim that Varadhan does not enjoy his success; he does. Nor am I about to say that he is some sort of saint; we could never have become friends if he were. Nonetheless, what distinguishes Varadhan from nearly all the other gifted people whom I have met is the remarkable command he exercises over his own gift. In particular, he has learned how to prevent his unusual intellectual powers from poisoning his relations with lesser intellects. For example, Varadhan can tolerate being wrong, at least occasionally. In addition, he is not one of the many mathematical princes who espouse the notion that all their obligations to humanity can be met through their contributions to mathematical research.