Celebratio Mathematica

Joseph L. Doob

Book review: Classical potential theory and its probabilistic counterpart, by J. L. Doob

by P. A. Meyer

It had been known for more than ten years that Doob was writ­ing a book on this sub­ject. Now that it has ap­peared, it en­tirely ful­fills our ex­pect­a­tions: it is a great work. Great by its di­men­sions, writ­ten with ex­treme love and care, con­cen­trat­ing the know­ledge of a gen­er­a­tion which was su­preme in the his­tory of po­ten­tial the­ory, it also rep­res­ents the achieve­ment of Doob’s own epoch-mak­ing re­search on the re­la­tions between clas­sic­al po­ten­tial the­ory and the the­ory of Browni­an mo­tion.

All this may be ob­vi­ous for the spe­cial­ist, but I sup­pose that for the lay­man who takes in hand this book for the first time, the ques­tion will be, How is it pos­sible to write such a book (840 pages!) on a sub­ject which now looks like a tiny place in the wide field of ana­lys­is, lost some­where between el­lipt­ic par­tial dif­fer­en­tial equa­tions and com­plex vari­ables? There­fore, it may be reas­on­able, be­fore giv­ing de­tails about the book, to say something about po­ten­tial the­ory it­self.1

The most typ­ic­al prob­lem in clas­sic­al po­ten­tial the­ory is that of the elec­tro­stat­ic con­dens­er. Ima­gine a hol­low con­duct­or \( C \) as the bound­ary of a bounded do­main \( \Omega \) in \( \mathbf{R}^3 \), in­side which is placed an­oth­er con­duct­or \( F \) (call­ing \( F \) a con­duct­or will not im­ply, in the math­em­at­ic­al rep­res­ent­a­tion, that \( F \) is con­nec­ted in the to­po­lo­gic­al sense: it is simply a closed set con­tained in \( \Omega \)). Then \( C \) is groun­ded, and \( F \) is wired to the pos­it­ive pole of an elec­tro­stat­ic gen­er­at­or. Ex­per­i­ence shows that neg­at­ive charges will ap­pear on the in­ner sur­face of \( C \) and pos­it­ive charges on the out­er sur­face of \( F \), bal­an­cing each oth­er in such a way that an equi­lib­ri­um po­ten­tial \( V \) is gen­er­ated with­in \( \Omega \), (as­sum­ing on \( C \) the con­stant value 0 (by con­ven­tion), on \( F \) some con­stant pos­it­ive value (for math­em­aticians, the value 1)) and har­mon­ic in \( \Omega\, \backslash\, F \). The total pos­it­ive charge on F is the con­dens­er’s ca­pa­city.

Turn­ing this ex­per­i­ment­al evid­ence in­to rig­or­ous math­em­at­ics has been a chal­lenge for more than a cen­tury, start­ing with Gauss (1840) and at­tract­ing the in­terest of such people as Riemann, Wei­er­strass, Di­rich­let, Schwarz, Poin­caré, Hil­bert, and later, Le­besgue, La Vallée Poussin, F. and M. Riesz, Wien­er, and many more. This prob­lem has been a test for every new dis­cov­ery in ana­lys­is: Hil­bert space, in­teg­ral equa­tions, Le­besgue’s in­teg­ral with re­spect to ar­bit­rary meas­ures, dis­tri­bu­tion the­ory, …. It has provided the mo­tiv­a­tion for an in­cred­ible amount of re­search in ana­lys­is (in­clud­ing such gen­er­al the­or­ies as Cho­quet’s ca­pa­cit­ies and in­teg­ral rep­res­ent­a­tions in con­vex cones), and it isn’t dead yet: sig­ni­fic­ant work has been done very re­cently on double-lay­er po­ten­tials, a fa­vor­ite tool of 19th cen­tury ana­lysts.

The com­plete solu­tion of the con­dens­er prob­lem was, after much pre­lim­in­ary work in the years around 1920, one of the achieve­ments of the great peri­od of clas­sic­al po­ten­tial the­ory, marked by the con­tri­bu­tions of Frost­man (1935), H. Cartan (act­ive on this sub­ject 1941–1946), M. Brelot2 (from about 1938 to 1950, after which his work shifts to more gen­er­al situ­ations), R. S. Mar­tin (1941; this pa­per went al­most un­noticed un­til much later). You will find all this in Doob’s book, which also con­tains some of Cho­quet’s ca­pa­city the­ory (1951–1955) and, of course, Doob’s own prob­ab­il­ist­ic in­ter­pret­a­tions (be­gin­ning 1954). After this peri­od, in ac­cord­ance with the spir­it of the times, po­ten­tial the­ory moved in the dir­ec­tion of gen­er­al­iz­a­tion and ax­io­mat­iz­a­tion: Cho­quet’s and Deny’s search for ker­nels sat­is­fy­ing the ba­sic “prin­ciples” of po­ten­tial the­ory, and, in par­tic­u­lar, Deny’s “ele­ment­ary ker­nels”, which are the ana­lyt­ic ver­sion of the po­ten­tial the­ory for Markov chains; Beurl­ing’s and Deny’s “Di­rich­let spaces”, Brelot’s “har­mon­ic spaces”, i.e., the ax­io­mat­iz­a­tion of po­ten­tial the­ory by means of sheaves of har­mon­ic func­tions, de­veloped in dif­fer­ent dir­ec­tions by Bauer and Con­stantin­es­cu and Cornea. One should add to this the great prob­ab­il­ist­ic syn­thes­is ac­com­plished by Hunt (1957–1958) and the work of Dynkin’s school, which is more ori­ented to­wards the prob­ab­il­ist­ic the­ory of Markov pro­cesses. Though Doob con­trib­uted per­son­ally to these de­vel­op­ments, his title is ex­pli­citly clas­sic­al po­ten­tial the­ory, and from all that he has in­cluded only some of the prob­ab­il­ist­ic ad­vances.3

On the oth­er hand, the solu­tion of the con­dens­er prob­lem and the gen­er­al meth­ods which are ne­ces­sary for it oc­cupy a cent­ral po­s­i­tion in the book. Let us roughly de­scribe them. The two con­duct­ors are treated in a some­what asym­met­ric way. One first for­gets about \( F \). A unit pos­it­ive charge at \( x \in \Omega \) would gen­er­ate in open space a po­ten­tial \( n(\,\cdot\,,x) \) (new­to­ni­an or Cou­lomb po­ten­tial), but, due to the hol­low con­duct­or \( C \), on the sur­face of which neg­at­ive charges will ap­pear, it gen­er­ates a smal­ler po­ten­tial \( g(\,\cdot\,, x) \) (Green po­ten­tial) in \( \Omega \), which still is su­per­har­mon­ic pos­it­ive in \( \Omega \), but in some sense should “van­ish at the bound­ary”. The main ad­vance is the de­scrip­tion of this without any ref­er­ence to bound­ary be­ha­vi­or: \( g(\,\cdot\,, x) \) is just \( n(\,\cdot\,, x) \) minus its greatest pos­it­ive har­mon­ic minor­ant in \( \Omega \), which is shown to ex­ist without any smooth­ness as­sump­tion on the do­main. Of course, the dif­fi­culty has been shif­ted to an­oth­er place: (1) Does this gen­er­al­ized Green func­tion really van­ish at the bound­ary in any reas­on­able sense? (2) How can one de­scribe ex­pli­citly the greatest har­mon­ic minor­ant which has been sub­trac­ted? Does it really cor­res­pond to a dis­tri­bu­tion of charges on the bound­ary?

The first prob­lem is solved by the dis­tinc­tion between reg­u­lar bound­ary points, at which the Green func­tion van­ishes in the or­din­ary sense, and ir­reg­u­lar points. These are char­ac­ter­ized as the points where the (closed) com­ple­ment of \( \Omega \) is “thin”, and they are shown to form a very scarce po­lar set. The ex­act de­scrip­tion of these ex­cep­tion­al sets has been one of the ma­jor steps in the de­vel­op­ment of clas­sic­al po­ten­tial the­ory: after pre­lim­in­ary de­scrip­tions they could be char­ac­ter­ized as sets of in­ner ca­pa­city 0. Then Cartan proved they had out­er ca­pa­city 0, a much stronger res­ult, which was fi­nally un­der­stood when Cho­quet proved that in­ner and out­er ca­pa­city are the same for nice (ana­lyt­ic and, in par­tic­u­lar, bore­li­an) sets and for much more gen­er­al “ca­pa­cit­ies”.

The second prob­lem can be solved in two ways. One can show that the greatest har­mon­ic minor­ant of \( n(\,\cdot\,, x) \) is the new­to­ni­an po­ten­tial of a prob­ab­il­ity meas­ure \( h(x, dy) \) on \( C \), the swept meas­ure of \( \varepsilon_x \), or har­mon­ic meas­ure at \( x \). It turns out that it is car­ried by the reg­u­lar points of \( C \) and is the es­sen­tial tool for the solu­tion of the Di­rich­let prob­lem in \( \Omega \) by the PWB (Per­ron–Wien­er–Brelot) meth­od, then \( -h(x, dy) \) de­scribes the neg­at­ive charges which ap­peared “by in­flu­ence” on the hol­low con­duct­or. On the oth­er hand, one may for­get about the open space and look for an in­teg­ral rep­res­ent­a­tion of all pos­it­ive har­mon­ic func­tions in \( \Omega \) by charges dis­trib­uted over an ideal bound­ary, which may be quite dif­fer­ent from the eu­c­lidean one. This leads, in par­tic­u­lar, to the Mar­tin com­pac­ti­fic­a­tion and Mar­tin bound­ary of \( \Omega \).

Two re­marks are in or­der here. The first one con­cerns para­bol­ic (heat equa­tion) po­ten­tial the­ory, to which Doob de­votes a fair amount of his book. Most ana­lysts be­fore Doob had lived with the simple idea that el­lipt­ic equa­tions are nat­ur­ally as­so­ci­ated with prob­lems of Di­rich­let type; para­bol­ic and hy­per­bol­ic equa­tions, with Cauchy prob­lems. Since his pa­per of 1955 on the heat equa­tion, Doob has in­sisted on the fact that from the prob­ab­il­ist­ic point of view, there is little dif­fer­ence between Laplace’s equa­tion and the heat equa­tion, and there­fore the heat equa­tion may be treated in par­al­lel with clas­sic­al po­ten­tial the­ory. The main dif­fer­ence is that the usu­al ex­cep­tion­al sets are now the so-called semi­polar sets, which are not so scarce and can­not be en­tirely ig­nored.

The second re­mark is the strik­ing in­ter­pret­a­tion of har­mon­ic meas­ure us­ing browni­an mo­tion: if you place a browni­an particle at \( x \), then the har­mon­ic meas­ure \( h(x, dy) \) can be de­scribed as the dis­tri­bu­tion of the (ran­dom) place where the particle first hits the bound­ary C This, of course, re­quires a lot of work for rig­or­ous jus­ti­fic­a­tion, but it gives an ex­tremely in­tu­it­ive con­tent to the ab­stract ana­lys­is of har­mon­ic minor­ants, etc. Prob­ab­il­ist­ic po­ten­tial the­ory as Doob sees it, however, is much more than an in­ter­pret­a­tion of clas­sic­al po­ten­tial the­ory. (The title says coun­ter­part, which is quite dif­fer­ent; we re­turn to this be­low).

We have not yet placed the second con­duct­or \( F \) in­side \( C \): we would like to prove the ex­ist­ence of an equi­lib­ri­um po­ten­tial \( V \) which van­ishes at the bound­ary of \( \Omega \) (hence, it is reas­on­able to ex­pect for it a rep­res­ent­a­tion as a Green po­ten­tial \( fg(\,\cdot\,, y) \mu(dy)) \), is har­mon­ic out­side \( F \) (hence, \( \mu \) is ex­pec­ted to be car­ried by \( F \)), and takes the value 1 on \( F \). By max­im­um prin­ciple con­sid­er­a­tions, any pos­it­ive su­per­har­mon­ic func­tion which dom­in­ates 1 on \( F \) should dom­in­ate \( V \) every­where in \( \Omega \). This led Brelot to study (for­get­ting again about po­ten­tials, van­ish­ing at the bound­ary, etc.) the re­duc­tion of 1 on \( F \), i.e., the in­fim­um of the class of all pos­it­ive su­per­har­mon­ic func­tions in \( \Omega \) lar­ger than 1 on \( F \). It turns out that this func­tion is in­deed har­mon­ic in \( \Omega \backslash F \) and is equal to the equi­lib­ri­um po­ten­tial \( V \) we are try­ing to con­struct, ex­cept on the po­lar set con­sist­ing of those \( x \in F \) where \( F \) is “thin”. Again browni­an mo­tion provides a strik­ing in­ter­pret­a­tion: \( V(x) \) is just the prob­ab­il­ity that, for browni­an mo­tion start­ing from \( x \), \( F \) will be hit be­fore \( C \).

However, the key res­ult of prob­ab­il­ist­ic po­ten­tial the­ory is the fact that su­per­har­mon­ic func­tions, which for the ana­lyst are quite ir­reg­u­lar lower semi­con­tinu­ous func­tions, are seen by the prob­ab­il­ist as con­tinu­ous func­tions along browni­an paths. This was dis­covered by Doob in 1954, ex­ten­ded by him to the para­bol­ic case in 1955 (con­tinu­ity be­ing re­placed by right con­tinu­ity with left lim­its), and opened the main way of com­mu­nic­a­tion between con­tinu­ous time mar­tin­gale and su­per­martin­gale the­ory and po­ten­tial the­ory, through which they in­flu­enced each oth­er.

Slightly more than half of Doob’s book is de­voted to pure ana­lys­is: clas­sic­al po­ten­tial the­ory (a com­plete treat­ise in it­self) and para­bol­ic po­ten­tial the­ory (the only ex­ist­ing ac­count of this sub­ject). The oth­er half is the ”coun­ter­part”: a self-con­tained ex­pos­i­tion of the fun­da­ment­als of stochast­ic pro­cesses, mar­tin­gales and su­per­martin­gales (in­clud­ing the de­com­pos­i­tion the­ory), browni­an mo­tion, and re­lated pro­cesses. A lim­ited amount of stochast­ic in­teg­ra­tion and Markov pro­cesses is also in­cluded, suf­fi­cient for the pur­pose of the book. The stress in prob­ab­il­ist­ic po­ten­tial the­ory is laid on bound­ary be­ha­vi­or: ad­dit­ive func­tion­als are de­lib­er­ately left aside. It seems that some­how the over­lap with books de­voted to Hunt’s gen­er­al res­ults (like Chung’s Lec­tures from Markov pro­cesses to Browni­an mo­tion, to men­tion only a re­cent one) has been min­im­ized. On the oth­er hand, the de­vel­op­ment is re­mark­ably free of heavy tech­nic­al­it­ies, and the coun­ter­point with ana­lyt­ic po­ten­tial the­ory is fas­cin­at­ing. It os­cil­lates between full sym­metry and tiny ana­lo­gies which re­mind us (like the fin­ger rem­nants of a whale) that su­per­har­mon­ic func­tions and su­per­martin­gales have a com­mon ori­gin in su­per­aver­aging prop­er­ties.

The book is not only great as a whole, it also seems per­fect in every de­tail. The in­dex is un­usu­ally com­plete and pre­cise, the print­ing wide and beau­ti­ful. A malevol­ent search for mis­prints over many pages caught only two, so in­sig­ni­fic­ant that I would be ashamed to quote them.4 The style is, in the re­view­er’s opin­ion, very at­tract­ive, rather ex­plan­at­ory than dog­mat­ic.

Ed­it­or’s note: P. A. Mey­er (1934–2003) worked at the In­sti­tut de Recher­che Mathématique Avancée (IRMA), at the Uni­versité Louis Pas­teur in Stras­bourg at the time this art­icle was writ­ten.