Apr 2, 2013

Obituary: Bob Blumenthal, 1931–2012

Robert McCallum Blumenthal passed away on November 8, 2012 at the age of 81, after a long illness. Bob, as he was known to one and all, was a mathematician whose main interest was probability theory, especially continuous parameter Markov processes on a topological state space. He received his PhD in 1956 at Cornell under the direction of G. A. Hunt. At the time Hunt was in the process of developing the relationships between Markov processes and potential theory to be published in a monumental work in three installments in 1957 and 1958. This subject was to become one of the main topics of research in probability for the next 20–25 years. The class of processes for which Hunt developed his theory came to be called Hunt processes in later years. The term process will mean a Hunt process in what follows, unless explicitly mentioned otherwise.

Already in his thesis Bob had established two of the basic principles of this topic: the strong Markov property (also established independently and more or less simultaneously by Dynkin and Yushkevich in the Soviet Union) and the quasi-left continuity of the sample paths of the process. In addition, his thesis contained what became known as the Blumenthal zero-one law. After obtaining his degree in 1956, he accepted a position as an Instructor in the Mathematics Department at the University of Washington (such positions were typical for new PhD’s in those days). He remained at UW until he retired in 1997, aside from two sabbatical years: in 1961–62 at the Institute for Advanced Study in Princeton, and 1966–67 in Germany.

I also accepted a position in the UW mathematics department beginning in September 1956. Shortly thereafter he and I began a long and fruitful collaboration. Perhaps the most important of our early papers was “Sample Functions of Stochastic Processes with Stationary Independent Increments” which appeared in 1961. At least it was the one that stimulated the most further research.

During the academic year 1961–62 we studied Hunt’s fundamental papers in great detail. To the best of my knowledge the only other person to have carefully read these papers at that time was P. A. Meyer in France. After completing the reading of Hunt’s papers, we began to work on the following problem: To what extent is a process characterized by its geometric paths? A few years earlier Feller had conjectured that a process should be determined by its “road map” and its “speed” as it travels along its paths. He had proved this for a class of Markov chains where “road map” was interpreted as the distribution of the position upon first exit from a point. Then Itô and McKean in 1956 established the appropriate result for one-dimensional diffusions. In this case the “road map” is given by the intrinsic scale and the “speed” by the speed measure of the diffusion. These concepts had been introduced by Feller in his description of the generator of a one-dimensional diffusion. Itô and McKean only published their results much later in their well-known 1965 book, Diffusion Processes and Their Sample Paths. Meanwhile the results had been obtained independently by the Russian School. Bob and I were very happy when we were able to prove the conjecture for a general Hunt process subject to a minor additional condition. Although it was fairly clear how to attack the problem, there were formidable technical difficulties to overcome. It was Bob’s deep probabilistic insight that was the key to our success. In early 1962 we sent our paper to J. L. Doob who was the editor of the Illinois Journal. At roughly the same time McKean had proved the result for general diffusions and had sent his paper to Doob also. Doob suggested that we combine our results, which yielded the result for general Hunt processes and our joint paper appeared in the Illinois Journal in 1962. In the ensuing years other proofs, simplifications, and extensions of our result have appeared.

Undoubtedly, our best known work was our 1968 book, Markov Processes and Potential Theory, which was reprinted by Dover in 2007. This book contained an expanded version of the theory developed by Hunt and later developments of its writing, especially the representation of excessive functions as the potentials of additive functionals. In general, we attempted to prove for standard processes what was previously known for Hunt processes. This extension is true most of the time, but sometimes quite difficult.

Bob continued his research on Markov processes during the 1970s and 80s, in particular on various construction problems and excursion theory. This led to his 1992 book, Excursions of Markov Processes, an excellent introduction to excursion theory as it existed at that time.

Bob was an excellent athlete. He captained his college tennis team and won the Ohio Conference singles title. After moving to Seattle he gave up tennis and became an accomplished mountaineer and skier. He obtained his professional ski instructor certificate and taught skiing weekends at Stevens Pass near Seattle for many years. He was a many faceted, friendly and gracious person with a wonderful sense of humor. He will be missed by who all who knew him. He is survived by his wife of many years, Sarah, and two sons, Joel and Jabe.

Ronald Getoor, Professor (Emeritus), University of California, San Diego

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