We sample from the powerful collection of mathematical tools for understanding evolving populations developed by Tom Kurtz over a period of more than forty years. We briefly discuss the Fleming–Viot process, the Donnelly–Kurtz lookdown process, and the generalised lookdown processes with which he captured an incredibly wide variety of population dynamics.
Stroock and Varadhan introduced the martingale problem, in the context of multidimensional diffusion processes, as a means of characterizing and studying Markov processes in terms of their generators. Kurtz saw the need for a treatment suited to more general Markov processes and provided one. Other work by Kurtz and his coauthors explored new applications of the martingale problem and alternative versions including the filtered martingale problem, the controlled martingale problem, and the constrained martingale problem. This article reviews these contributions.
Back in the day I was wary of Tom Kurtz. Every year there would be a good conference of Midwestern probabilists in Evanston, Illinois, organized by Mark Pinsky. People would come from Minnesota, Wisconsin, Illinois, Indiana, and even Iowa. Tom came from Madison, WI; he was the only person who always wore a tie and a sports jacket. He looked like he worked for the military, or the FBI, in his spare time. When my wife, Diane Rubenstein, finished her PhD and was hired at Madison (in Political Science), I asked Peter Ney if I could visit Wisconsin. Purdue was always happy to see me leave, it seemed. Peter arranged things, but when I got to Madison I discovered it was Tom who shared my intellectual interests. In particular, due to a question Darrell Duffie asked me (J. Math. Finance 2 (1992) 1–15), I was working on the weak convergence of stochastic integrals. Jean Jacod and Gilles Pagès in France had done some important work on the subject, but they never answered the questions I was interested in. I quickly learned Tom had the exact same interests. He told me he needed the results for problems stemming from the thesis of his student, Cristina Costantini (Probab. Theory Related Fields 91 (1992) 43–70). I also learned, rather quickly, that Tom was an amazing person. Despite his sartorial choices, he was one of the best persons I could ever hope to meet.
In this paper, we first present a short overview for the development of stochastic partial differential equations (SPDEs) and the three fundamental approaches. Then, we introduce the particle representation, which was initiated by Kurtz, as an alternative approach for the SPDEs. Finally, some applications are presented.
