# Wolfgang Doeblin

### Born: 17 March 1915 in Berlin, Germany

Died: 21 June 1940 in Housseras, France

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**Wolfgang Doeblin**is arguably one of the four major contributors to probability theory in the first half of the 20

^{th}century up to World War II (the other three are Khinchin, Kolmogorov and Lévy). His work contains several profound results, and his importance is also due to his innovativeness and introduction of new methods. He laid many of the cornerstones of the modern theory for Markov chains and processes, to be developed after the war by others. The article in

*Mathematical Reviews*on the historical paper by Paul Lévy [8] from 1955 about Doeblin's work states that:-

Considering Doeblin's short career, it is remarkable that he published 13 papers and 13 contributions toIt is a pity that this tribute to Doeblin's genius was written without reference to later developments of his work. After all there can be no greater testimony to a man's work than its influence on others. Fortunately, for Doeblin, this influence has been visible and is still continuing. On limit theorems his work has been complemented and completed by Gnedenko and other Russian authors. On Markov processes it has been carried out mostly in the United States by Doob, T E Harris and the reviewer. Here his mine of ideas and techniques is still being explored.

*Comptes Rendus*; for bibliographies, cf. Paul Lévy [8] or Lindvall [9]. In his work on the theory of Markov chains and processes, his main field, we notice major contributions to: Markov chains with general state spaces, Jump Markov processes, the coupling method (innovation), and diffusions. The importance of these is to a large extent due to Doeblin's emphasis on path methods rather than analytical ones; much of what is standard approaches today stem from him. The book by Doob [3] from 1953 has been crucial for the development of probability theory; for a large part of its contents on Markov chains and processes, Doeblin's work is the base.

Concerning Markov chains with general state space, Orey [5] from 1971 states that:-

In Doeblin's mine of ideas, the coupling method was paid attention to by very few until the early 1970s; then the time was ripe to explore it, and the method is now a major tool in probability theory, with applications ranging from elementary theory to front research.It is essentially Doblin's theory as completed during the quarter of a century following the publication of his papers that is presented here.

The term "diffusion" was not coined until the 1950s. Nevertheless, the first steps were taken in the 1930s. This type of process were the main interest in the last phase of Doeblin's mathematical career, interrupted by World War II; the spare time he had as a soldier was spent on this. The results presented (without proofs) in

*Comptes Rendus*are remarkable, and so are the contents of the file that was sealed in February 1940 and not opened until May 2000. It contains pieces of what we now call stochastic calculus, including a version of Ito's formula.

On Doeblin's work concerning sums of independent random variables, Feller [4] writes:-

Doeblin also contributed to the theory of random chains with complete connection, some of which was used in a paper by him on ergodic properties of continued fractions.The interest in the theory was stimulated by W. Doblin's masterful analysis of the domain of attraction(1939). His criteria were the first to to involve regularly varying functions. The modern theory still carries the imprint...[This concerns weak convergence in arrays]. The last result was obtained by Doblin in a masterful study in1940, following previous work by Khinchin(1937).[This concerns the so called Doeblin's universal law.]

Doeblin's life and fate are remarkable and gripping. His father was Alfred Döblin, a medical doctor but best known as author; the novel

*Berlin Alexanderplatz*(1929) stands out among his many books. Wolfgang was born in Berlin, but he spent his first three years behind the German front in World War I, in Saargemünd where his father volunteered as an army doctor. At the peace in 1918, the family moved back to Berlin, where the reputation of the father, as author and left-wing participant in political and cultural debates, started to rise.

Warned by friends, Alfred Döblin left Berlin for Zürich the day after the Reichstag-fire in February 1933. He was on the black list of the Nazis: a Jew, controversial for his political views. Wolfgang stayed until May to finish school. The sojourn in Zürich lasted for the summer 1933 only; the family settled in Paris after that.

Doeblin immediately made a strong impression in Paris; Fréchet was his adviser, but Doeblin also got in touch with Paul Lévy, with whom he wrote his first note. He received his PhD from the Sorbonne in 1938, but at that time he was far into topics beyond or not related to those of his thesis. He was enrolled in the French army in November 1938 for military training, and called up for front service in September 1939. This means that his mathematical career comprised only five years of concentrated work.

In February 1940, the Nazi invasion was expected to come in the spring to follow, and Doeblin decided to file his work on diffusions at the Académie des Sciences in Paris. The part of Lorraine where he was commanded fell in June. With the Nazi troops just minutes away from the little village Housseras, he decided to take his own life there rather than giving himself up as a prisoner of war.

Housseras is located some 100 kilometres from Sarreguemines (French after World War I), the place where Doeblin had spent the first three years of his life.

The reader is referred to [1], [6], [7], [9], [10] for biographical texts on Doeblin.

**Article by:** *Torgny Lindvall*

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