Celebratio Mathematica

Joseph L. Doob

Joseph L. Doob, 1910–2004

by Donald Burkholder

Joseph Leo Doob, a pi­on­eer in the study of the math­em­at­ic­al found­a­tions of prob­ab­il­ity the­ory and its re­mark­able in­ter­play with oth­er areas of math­em­at­ics, died in June in Urb­ana, Illinois, where he had lived since 1935, ar­riv­ing as a new fac­ulty mem­ber of the Uni­versity of Illinois. He was 94.

Prob­ab­il­ity has been a sub­ject of math­em­at­ic­al ex­plor­a­tion for more than 300 years, but it was giv­en a firm found­a­tion in the 1930s by such math­em­aticians such as Kolmogorov in Rus­sia and Doob in Amer­ica. In his 1937 pa­per, Doob gave the math­em­at­ic­al frame­work for the study of con­tinu­ous para­met­er stochast­ic pro­cesses, that is, fam­il­ies of meas­ur­able func­tions in­dexed by a para­met­er such that the fam­ily is point­wise con­tinu­ous in the para­met­er.

After the pi­on­eer­ing work of Lèvy, Ville, and oth­ers, he began a wide-reach­ing de­vel­op­ment of mar­tin­gale the­ory about 1940, pub­lish­ing his pa­per on pro­cesses of type E. As Dav­id Black­well ob­served, Doob’s work did not catch on un­til he star­ted to use the more ex­cit­ing word mar­tin­gale which had already been used much earli­er by Bacheli­er in a math­em­at­ic­al con­text. Mar­tin­gale the­ory is the fo­cus of one of the chapters, nearly 100 pages long, in Doob’s 1953 book Stochast­ic Pro­cesses. This treat­ise of over 650 pages has been one of the most im­port­ant and in­flu­en­tial books on prob­ab­il­ity since Laplace’s 1812 book. One of Doob’s pa­pers, of­ten over­looked, gives his simple mar­tin­gale proof of the Strong Law of Large Num­bers, and his work on the al­most sure lim­its of in­verse prob­ab­il­it­ies; see his 1949 pa­per “Ap­plic­a­tion of the the­ory of mar­tin­gales”. Mar­tin­gale the­ory plays an es­sen­tial role not only in math­em­at­ic­al stat­ist­ics but also in the study of Markov pro­cesses, in­form­a­tion the­ory, fin­an­cial math­em­at­ics, com­bin­at­or­ics, and many oth­er parts of math­em­at­ics, sci­ence, and tech­no­logy.

In 1954, he showed how vari­ous clas­sic­al po­ten­tial the­ory con­cepts, such as the prop­er­ties of the Per­ron–Wien­er–Brelot solu­tion of the first bound­ary value prob­lem for har­mon­ic func­tions on an ar­bit­rary open set and ar­bit­rary as­signed bound­ary func­tion, cor­res­pond to prop­er­ties of su­per­har­mon­ic func­tions on Browni­an mo­tion paths. He ob­tained sim­il­ar res­ults for the heat equa­tion in 1955. By the age of 45, he had made sig­ni­fic­ant con­tri­bu­tions to com­plex func­tion the­ory, er­god­ic the­ory, Markov pro­cess the­ory, mar­tin­gale the­ory, bound­ary the­ory, and much else.

His work be­came enorm­ously in­flu­en­tial. Al­though he re­tired at 68, he con­tin­ued writ­ing, and pub­lished a num­ber of pa­pers and two books: one of these, Clas­sic­al Po­ten­tial The­ory and Its Prob­ab­il­ist­ic Coun­ter­part, is over 800 pages long. A glimpse of his per­son­al­ity and math­em­at­ic­al style can be found in his vivid auto­bi­o­graph­ic­al re­marks in re­sponse to Laurie Snell’s well-posed ques­tions in the Novem­ber 1997 is­sue of Stat­ist­ic­al Sci­ence.1

Joe was born in Cin­cin­nati, Ohio, Feb­ru­ary 27, 1910, the son of Leo Doob and Mol­lie Doer­fler Doob. The fam­ily moved to New York City be­fore he was three years old. His par­ents felt that he was un­der-achiev­ing in grade school and placed him in the Eth­ic­al Cul­ture School, from which he gradu­ated in 1926. He then went on to Har­vard where he re­ceived a BA in 1930, an MA in 1931, and a PhD in 1932. After postdoc­tor­al re­search at Columbia and Prin­ceton, he joined the De­part­ment of Math­em­at­ics of the Uni­versity of Illinois in 1935 and served un­til his re­tire­ment. Dur­ing World War II, he worked in Wash­ing­ton DC and Guam as a ci­vil­ian con­sult­ant to the Navy. He was a mem­ber of the Urb­ana cam­pus Cen­ter for Ad­vanced Study from its in­cep­tion in 1959.

He was mar­ried to Dr. Elsie Field for nearly sixty years; she died in 1991. She was the Med­ic­al Dir­ect­or of Planned Par­ent­hood of Cham­paign County and worked as a full-time vo­lun­teer after her re­tire­ment. He is sur­vived by his chil­dren Steph­en, Peter and De­borah, and four grand­chil­dren.

Doob served the math­em­at­ic­al pro­fes­sion as IMS Pres­id­ent in 1950, AMS Pres­id­ent 1963–64, and in many oth­er ca­pa­cit­ies. Among many hon­ors, he was elec­ted a mem­ber of the Na­tion­al Academy of Sci­ences in 1957, a mem­ber of the Amer­ic­an Academy of the Arts and Sci­ences in 1965, and a for­eign as­so­ci­ate of the French Academy of Sci­ences in 1975. He was awar­ded the Na­tion­al Medal of Sci­ence in 1979. In 1984 he was giv­en the Steele Prize for his out­stand­ing ca­reer and “con­tinu­ing pro­found in­flu­ence” by the Amer­ic­an Math­em­at­ic­al So­ci­ety.

He en­joyed, in all sea­sons and weath­ers, the ca­marader­ie of the Sat­urday Hikers, an in­form­al group with nearly 100 years of ex­ist­ence. He also liked to ca­noe on the nearby Salt Fork, the Ver­mil­lion, and the Sanga­mon rivers, rivers im­mor­tal­ized in the poem “Memory of a Schol­ar” by Rich­mond Lat­timore, the highly praised trans­lat­or of the Ili­ad and Odys­sey.

Ed­it­or’s note: Don­ald Burk­hold­er and Joseph Doob were long­time col­leagues in the De­part­ment of Math­em­at­ics at the Uni­versity of Illinois Urb­ana-Cham­paign. Rodrigo Bañuelos and Bur­gess Dav­is note Doob’s prob­able sig­ni­fic­ant in­flu­ence on Burk­hold­er’s own fo­cus of re­search in their In­tro­duc­tion (re­pro­duced here on Cel­eb­ra­tio) to the fest­s­chrift for the lat­ter in The An­nals of Prob­ab­il­ity 45:1 (2017).