Thomas (Tom) Kurtz

Thomas G. Kurtz, renowned for his work on Markov processes and stochastic analysis, passed away on April 19, 2025, at the age of 83. He was born on July 14, 1941, in Kansas City, and earned a BA in mathematics from the University of Missouri in 1963 and a PhD in mathematics from Stanford University in 1967. That year, he joined the Department of Mathematics at the University of Wisconsin–Madison, where he remained for his entire career, retiring from teaching in 2008. He received a joint appointment with the Department of Statistics in 1985.

Tom served as the Department of Mathematics Chair (1985–88) and as the Director of the Center for Mathematical Sciences (1990–96). In 1996, he was awarded the WARF–University Houses Professorship, which he chose to identify as the Paul Lévy Professorship.

Tom had a year-long sabbatical at the Université de Strasbourg in France in 1977–78 (to learn “Strasbourgeois,” as he said, which is the stochastic analysis theory developed by the French school), and many shorter visiting positions around the world. He loved to travel.

Tom was a Fellow of the American Academy of Arts and Sciences, the Institute of Mathematical Statistics, and the American Mathematical Society. He served as the IMS President (2005–06) and as the Editor of The Annals of Probability (2000–02). He gave the Wald Memorial Lectures in 2014 at the IMS Annual Meeting in Sydney.

Tom supervised 29 PhD students and organized a Summer Internship Program in Probability in Madison for nearly a decade, significantly impacting a large number of young probabilists. He always took special care to encourage and support young mathematicians.

Tom’s PhD thesis, written under the supervision of James McGregor, was titled “Convergence of Operator Semigroups with Applications to Markov Processes.” It extended Trotter’s operator semigroup approximation theorem by providing necessary and sufficient conditions, which later led to perturbation and averaging theorems for operator semigroups, as well as to conditions for weak convergence to Markov processes. But the martingale problem, formulated and developed by D.W. Stroock and S.R.S. Varadhan in the late 1960s for finite-dimensional diffusion processes, came to be Tom’s preferred approach. The recognition of the need for a more general treatment led to Tom’s 1986 book, Markov Processes: Characterization and Convergence (with his former student Stewart Ethier), which has had nearly 10,000 citations. Subsequent work by Tom and his coauthors included martingale problems for conditional distributions of Markov processes, the filtered martingale problem, martingale problems for controlled and for constrained Markov processes, and a martingale problem formulation of the Markov mapping theorem.

Characteristic of Tom was his determination to always achieve maximum generality, which allowed him to apply his techniques to a great variety of fields: population genetics, chemical reaction networks, stochastic filtering and control, SPDEs, numerical methods, and others. A drawback of this generality is that potential users often found his papers difficult to understand. Tom recognized the problem, because he began a 2014 paper by writing:

This paper is essentially a rewrite of Kurtz (2007) following a realization that the general, abstract theorem in that paper was neither as abstract as it could be nor as general as it should be. The reader familiar with the earlier paper may not be pleased by the greater abstraction, but an example indicating the value of the greater generality will be given in Section 2.

That paper included a generalization of the Yamada–Watanabe and Englebert theorems concerning existence and uniqueness of strong and weak solutions for stochastic equations. Related important work by Tom, in this case with Philip Protter, includes necessary and sufficient conditions for the weak convergence of stochastic integrals, and semimartingales in general, that were particularly easy to verify in practice, and hence became very popular.

The impact of Tom’s work in population genetics has been broad. The most obvious example is the lookdown construction, which was developed in a series of papers with Peter Donnelly in the mid- to late 1990s. In a 1996 paper, they constructed an interacting particle system that carries both the Fleming–Viot superprocess, a probability-measure-valued diffusion process that generalizes the Wright–Fisher diffusion model of population genetics, and Kingman’s coalescent, a continuous-time Markov chain that tracks the genealogy of the population looking backwards in time. Until then, the Fleming–Viot process and the coalescent represented two distinct approaches. The approach was rapidly generalized to incorporate selection, recombination, fat-tailed offspring distributions, and a huge variety of measure-valued population processes. These papers are a treasure trove of results, such as the introduction of generalized Fleming– Viot processes and the multiple-merger coalescent duals that have come to be known as Lambda coalescents. Nowadays, the merger of forwards- and backwards-in-time models provided by lookdown constructions is rightly recognized as a powerful tool, especially important in the study of scaling limits.

Analogously, Tom’s work has had a deep and lasting impact in the field of (bio)chemical reaction networks. His early 1970s results established a rigorous connection between deterministic and stochastic models and his random timechange representation of Markov processes led—among other applications, for instance in statistical mechanics—to the strong diffusion approximation of density-dependent pure jump Markov processes. More recently, his averaging results for martingale problems allowed him and his collaborators to obtain multiscale approximations of reaction networks with fast and slow components. This body of work was disseminated widely through his 2015 monograph Stochastic Analysis of Biochemical Systems (with David Anderson).

Tom’s 2006 monograph Large Deviations for Stochastic Processes (with his former student Jin Feng) presents a unified approach to large deviations for Markov processes based on convergence of nonlinear semigroups. A derivative-free notion of viscosity solution and a functional analytic version of the maximum principle are developed. A convergence theory for the Hamilton–Jacobi equation in the space of probability measures was given for the first time in this work.

Tom was married to Carolyn, and had two children, Marci and Kevin, and six grandchildren. His family was deeply important to him.

Tom was known for his dedication to his profession and his students. He was always generous of his time and expertise with everybody, and he was an incredible resource for probabilists. He will be greatly missed.

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Written by Cristina Costantini, Alison Etheridge, Stewart Ethier, Jin Feng, Hye-Won Kang, and Richard Stockbridge