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From Brownian motion to Schrödinger’s equation. (English) Zbl 0819.60068
Grundlehren der Mathematischen Wissenschaften. 312. Berlin: Springer- Verlag. xii, 292 p. (1995).
The Schrödinger equation studied in this book is \[ \Delta u + 2qu = 0, \tag{*} \] where \(u\) and \(q\) are real-valued functions on a domain \(D \subset \mathbb R^ d\) \((d \geq 1)\), and \(\Delta = \sum^ d_{i=1} \partial^ 2/ \partial x^ 2_ i\) is the Laplacian. A continuous weak-sense solution of \((*)\) is called a \(q\)-harmonic function. As an extension of the classical harmonic theory, a theory of \(q\)-harmonicity is set up. By using probabilistic means there are solved problems as: the representation of a \(q\)-harmonic function by its boundary values, the uniqueness of the solution to the Dirichlet boundary value problem, Poisson’s equation, Harnack’s inequality, Green functions. The starting point is the Brownian motion process \(\{X_ t\}\) in \(\mathbb R^ d\) \((d \geq 1)\) enriched with a multiplicative functional \(\exp\,[\int^ t_ 0 q(X_ s)\,ds]\). A domain \(D\) is fixed and the paths are stopped at the exit of \(\{X_ t\}\) from \(D\) yielding the stopping time \(\tau_ D\).
The book is divided into nine chapters. To describe its contents we freely adapt from the preface (written by Chung). A substantial review of the classical theory \((q \equiv 0)\) is given in Chapters 1 and 2. Chapter 3 introduces the class \(J\) of functions \(q\) that is particularly amenable to the probabilistic treatment in both its old and new forms. Chapters 4 and 5 present a study of the so-called gauge function \(x \to E^ x (\tau_ D < \infty;\, \exp\,[\int^{\tau_ D}_ 0 q(X_ s)\, ds])\), \(x \in D\). The main result, the gauge theorem, is a dichotomy: if \(D\) has finite Lebesgue measure, then either the gauge is infinite everywhere in \(D\) or it is bounded in \(\overline D\). For any domain \(D\) and any \(q \in J\), the pair \((D,q)\) is said to be gaugeable when the gauge is bounded in \(D\). This is the main assumption under which all the problems mentioned above are solved by explicit expressions closely related to the gauge itself. In Chapters 6, 7, and 8 there are studied \(q\)-Green functions, the conditional gauge theorem (an extension of the gauge theorem for a bounded Lipschitz domain \(D)\) and their relationship, as well as the variation of the gaugeability of \((D,q)\) with \(D\) or with \(q\). Finally, Chapter 9 is devoted to the case \(d = 1\), where the special geometry leads to new questions and concepts. At the end of each chapter there is an extensive section of notes, written by Chung, which include historical remarks, comments on the results presented, and credit.
This is a masterly written self-contained monograph which includes much original research by the authors as well as detailed and improved versions of important results by other authors, not easily accessible in the existing literature. Even if it mainly addresses readers who are interested in probability theory as applied to analysis and mathematical physics, its potential readership should be much larger. In the reviewer’s opinion this is an important addition to the probability research literature.

MSC:
60J65 Brownian motion
60-02 Research exposition (monographs, survey articles) pertaining to probability theory
81Q15 Perturbation theories for operators and differential equations in quantum theory
81-02 Research exposition (monographs, survey articles) pertaining to quantum theory
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