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Kiyosi Itô: Selected Papers

Kiyosi Itô
Publication Date: 
Number of Pages: 
Springer Collected Works in Mathematics
[Reviewed by
Michael Berg
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After the seminal work of a few decades ago by Edward Witten and Sir Michael Atiyah, introducing topological quantum field theory (or even QFT proper, the way the physicists think of it) into differential geometry in the broad sense, the Feynman path integral, now transplanted to pure mathematics, became much more than a means whereby to do physics calculations in quantum electrodynamics: it turned into a major player in a burgeoning area of mathematics per se. Thus the attendant questions of rigor and well-definition take on particular importance and the physicists’ pragmatic way of handling these things began to cause mathematicians a good deal of anguish.

Indeed, for those of us whose work actually intersects with these themes, the pain starts right off, with the mere realization that as a rule we have to live with the fact that the Feynman path integral, for all its fecundity, is in general not even well-defined. The point is made pithily by Atiyah himself on p. 52 of his seminal book, The Geometry and Physics of Knots, when he introduces the Feynman integral formalism (this being the title of his seventh chapter): “… we present Witten’s Feynman path integral approach. It is not mathematically rigorous, but it is conceptually simple, and provides a natural starting point for the theory.” Beware of physicists bearing gifts!

Is there anything that we can do? Well, there is: there is at least a partial fix. Within its area of applicability the Itô calculus is nothing less than a panacea, and this brings us to one of the showpieces in the book under review. In Itô’s elegant and remarkably compact 1960 article, “Wiener Integral and Feynman Integral,” covering pp. 275–286 of the present compendium, we read in the introduction that “[i]t is our purpose to define the generalized measure [posited heuristically by Feynman], that is [the according Feynman path integral], rigorously and to prove that [the latter] solves [Schrödinger’s wave equation].” He does this with considerable élan. In the context of probability theory he explicitly plays off of work by Marc Kač and develops stochastic integrals, which are of very far-reaching mathematical significance indeed. So, this remarkably short paper not only develops a central foundational point for Feynman’s original approach to quantum mechanics, but is of deep significance for mathematicians on two counts, namely the introduction of rigor into the mix and the indicated amplification of a new theme in probability theory.

Thus, it is indeed marvelous to have Itô’s paper at one’s disposal in all its clarity and concision. And it really is just one gem among many. Stochasticity is a famously important theme in modern mathematics, and the articles featured in the book under review address this theme in great abundance. Wiener integrals, and other objects studied and explored by Norbert Wiener , make their appearance in any number of places in this Selecta, as do such mainstays as Markov processes, Brownian motion, diffusion — the Gestalt is evident, and this is consonant with Itô’s prominence in this part of mathematics. Moreover, it is striking that these papers are uniformly written in a spare and to-the-point style, and are models of clear exposition: a master’s touch.

Speaking of the hand of a master, there is one final point in order: I don’t know if the manoeuvre I’m about to mention was premeditated on the part of the Springer editorial staff or not: in any case it is an interesting effect. On p. 213 of the book, i.e. on p. 221 (journal pagination) of Itô’s 1954 paper, “Stationary Random Distributions,” we find a photocopy of a handwritten marginal note, presumably in Itô’s own writing: “see the end of p. 223.” Going there we read, again in his presumed hand, an addendum that starts, “We can simplify the part from p. 221, line 4, to p. 222, line 4, as follows …” — and then we get two or three lines of hard analysis. A nice touch, even if it’s unintended. 

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.

[1] On Stochastic Processes (Infinitely divisible laws of probability)

[2] Differential Equations Determining a Markoff Process

[3] On the Ergodicity of a Certain Stationary Process

[4] A Kinematic Theory of Turbulence

[5] On the Normal Stationary Process with no Hysteresis

[7] Stochastic Integral

[9] On a Stochastic Integral Equation

[10] Stochastic Differential Equations in a Differentiable Manifold

[11] Brownian Motions in a Lie Group

[12] On Stochastic Differential Equations

[13] On a Formula Concerning Stochastic Differentials

[14] Multiple Wiener Integral

[15] Stochastic Differential Equations in a Differentiable Manifold

[16] Stationary Random Distributions

[17] Complex Multiple Wiener Integral

[18] Isotropic Random Current

[19] Spectral Type of the Shift Transformation of Differential Processes with Stationary Increments

[20] Potentials and the Random Walk

[21] Wiener Integral and Feynman Integral

[22] Construction of Diffusions

[23] The Brownian Motion and Tensor Fields on Riemannian Manifold

[24] Brownian Motions on a Half Line

[25] The Expected Number of Zeros of Continuous Stationary Gaussian Processes

[26] On Stationary Solutions of a Stochastic Differential Equation

[27] Transformation of Markov Processes by Multiplicative Functionals

[28] The Canonical Modification of Stochastic Processes

[29] On the Convergence of Sums of Independent Banach Space Valued Random Variables

[30] Generalized Uniform Complex Measures in the Hilbertian Metric Space with their Application to the Feynman Integral

[31] On the Oscillation Functions of Gaussian Processes

[32] Canonical Measurable Random Functions

[33] The Topological Support of a Gauss Measure on Hilbert Space

[34] Poisson Point Processes Attached to Markov Processes

[37] Stochastic Differentials

[38] Stochastic Parallel Displacement

[40] Extension of Stochastic Integrals

[44] Infinite Dimensional Ornstein-Uhlenbeck Processes

[45] Regularization of Linear Random Functionals (with M. Nawata)

[46] Distribution-Valued Processes Arising from Independent Brownian Motions