This is the second edition of Stroock’s beefy introductory graduate text on probability theory. It’s a serious enterprise, coming in at over 500 pages; although it’s a second edition, Stroock notes in his Preface to the Second Edition that he has chosen not to do any sea-change updating from the 1994 first edition due to the fact that “for at least the last decade, the most exciting developments … have [had] a strong combinatorial component” and “combinatorics [is] a topic in which my abilities are woefully deficient.” Thus we’re getting 500 pages of probability à la Kolmogorov: analysis with total measure one, so to speak. Well, it’s of course a lot more than that, and Stroock’s book’s Table of Contents makes this abundantly clear ab initio: the first 100 pages or so are devoted to “Sums of independent random variables” and “The Central Limit Theorem.” But, to be sure, this is analysis with a vengeance (and rightly so): on p. 82 we read that “[i]n most modern treatments of The Central Limit Theorem, Fourier analysis plays a central role. Indeed, the Fourier transform makes the argument so simple that it can mask what is really happening” [Aha! The probabilist’s insight.] Stroock goes on to say that “now that we know Lindeberg’s argument, it is time to introduce Fourier techniques and begin to see how they facilitate reasoning involving independent random variables …” And then it’s on to FT’s of finite complex-valued Borel measures on real n-space.
But before I get too deeply involved with the nuts and bolts of Probability Theory: An Analytic View proper, I must share a little more of Stroock’s personal ruminations (in his Prefaces) because they reveal a great deal about the author — much more than is usual — and this information is in fact very relevant to the exposition that follows. Thus we find that Stroock’s
graduate education was anything but deprived [via McKean, Donsker, F. John, Nirenberg, and Varadhan at NYU] … On the other hand I [ = S ] had never had a proper introduction to my field, probability theory … I am not a dyed-in-the-wool probabilist … [e.g.] I have never been able to develop sufficient sensitivity to the distinction between a proof and a probabilistic proof …
Well, this is quite mysterious, no? Not only is Stroock refreshingly and usefully candid, he sets the stage for a discussion of probability theory “inextricably interwoven with other branches of mathematics … not … an entity unto itself.” Very tantalizing.
This dovetails well with the changes (not of the sea-change variety as alluded to above, but quite substantive nonetheless) Stroock introduced into the present second edition of Probability Theory: An Analytic View. These changes include an exciting new chapter devoted to “Gaussian measures in infinite dimensions from the perspective of the Segal-Gross school,” in connection with which he mentions conformal field theory, and a jettisoning of the erstwhile theme of singular integrals (viz. Calderón-Zygmund stuff, &c.) Note in the former connection that the late I. E. Segal was Stroock’s long-time colleague at MIT.
Beyond Chapter 2, i.e. after the first 100 pages, we get to “the study of infinitely divisible laws,” Lévy processes, conditional expectations (Ch.5: this is where “Doob’s basic theory of real-valued, discrete parameter martingales” appears), more martingale theory (including Burkholder’s inequality for martingales with values in a Hilbert space), martingales with a continuous parameter. Here’s a very cool result (p. 288): “By combining Theorem 7.2.3 with Theorem 7.2.1, one can show that, up to time reparameterization, all continuous martingales are Brownian motions.”
Subsequently we come to an extremely exciting theme (alluded to above): “Brownian motion in terms of its Gaussian, as opposed to its independent increment properties.” This is an ecumenical theme inasmuch as Stroock
attempt[s] to convince the reader that Wiener measure (i.e. the distribution of Brownian motion) would like to be the standard Gauss measure on the Hilbert space … of absolutely continuous paths with a square integrable derivative, but, for [deep!] technical reasons cannot live there and has to settle for a Banach space in which [the former space] is densely embedded
and this augurs for connections with one of the stickiest wickets in the fecund interphase between mathematics and quantum field theory (or even just quantum mechanics), namely the status of Feynman’s path integral formalism as being mathematically ill-defined, except in cases where Wiener-Kac measures come to the rescue. And indeed, already on p. 303 of the present (8th) chapter, dealing with “the classical Cameron-Martin space,” Stroock addresses this very issue: “In order to guess on which Hilbert space [the classical Wiener measure] would like to live, I will give R. Feynman’s highly questionable but remarkably powerful way of thinking about such matters,” and then Stroock goes on to discuss Feynman’s trickery in considerable detail.
At this stage we’re in the thick of it: the sequence of topics after Chapter 8 is thus: “convergence of measures on a Polish space,” more on Wiener measures — now vis à vis PDE (and it is here that we find the Dirichlet problem and a discussion of “other heat kernels,” including that of Feynman-Kac), and finally a chapter on “some classical potential theory.”
Probability Theory: An Analytic View is a well-written book, eminently suited to (very well-prepared) graduate students in probability with interests in this subject’s interphase with other parts of mathematics, both classical and contemporary. Stroock has included exercises, examples, and remarks in his narrative, and his idiosyncratic style (as already illustrated above) makes for very pleasant and even entertaining reading. But make no mistake, this is very serious mathematics.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.