#### 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.*