##
**From Brownian motion to Schrödinger’s equation.
2nd corrected printing.**
*(English)*
Zbl 1195.60002

Grundlehren der Mathematischen Wissenschaften 312. Berlin: Springer (ISBN 978-3-540-57030-1/hbk). xii, 287 p. (2001).

It is very hard to assess the impact of the œuvre, let alone a
single monograph, of a scientist who is such a prolific writer as Kai
Lai Chung. Kai Lai Chung (1917–2009) was an extraordinary researcher,
a great expositor and a truly outspoken mind. All of these qualities
show up in almost every single publication of Chung. Many of his most
important books are both research monographs and textbooks, some changed
their character as time went on and once novel material became a
‘standard’ topic. This is true for the Grundlehren volumes
Markov chains with stationary transition probabilities [Zbl 0092.34304, Zbl 0146.38401] and Lectures From Markov Processes to
Brownian Motion [Zbl 0503.60073, Zbl 1082.60001], while the
Lectures on Boundary Theory for Markov Chains [Zbl 0204.51003] and the volume under review, From Brownian Motion to
Schrödinger’s Equation (with Z. Zhao) [Zbl 0819.60068], are
probably too specialized and (still) too advanced for course
adoption. Brilliant examples of Chung’s abilities as an academic
teacher are his four textbooks A Course in Probability Theory [Zbl 0159.45701,
Zbl 0345.60003, Zbl 0980.60001], Elementary Probability Theory with Stochastic Processes
[Zbl 0293.60001, Zbl 0328.60001, Zbl 0404.60002, Zbl 1019.60001],
Introduction to Stochastic Integration [Zbl 0527.60058, Zbl 0725.60050],
A New Introduction to Stochastic Processes (in Chinese) [Zbl 0917.60001]
which are still in use after more than 40 years. So is his classic translation of Gnedenko’s and Kolmogorov’s Limit Distributions for
Sums of Independent Random Variables [Zbl 0056.36001] which he
extends and corrects.

Chung’s exposition is very clear, easy to follow and always mathematically reliable. Chung has a very personal style of writing; not all theorems, techniques or colleagues are equally important in his opinion—and he is rather blunt about this! His views on mathematics, in general, and probability theory, in particular, are most poignantly presented in Green, Brown and Probability [Zbl 0871.60001, Zbl 1014.60001] and Introduction to Random Times and Quantum Randomness [Zbl 0989.81500, Zbl 1021.81500, Zbl 1041.81074].

Also in mathematics Chung held firm beliefs. That a recent probabilistic proof of almost anything may be still further probabilisable (p. 190, Notes on Chapter 6) has been his programme throughout his life. Kai Lai Chung will be remembered as one of the really influential probabilists in the second half of the twentieth century. His first publication appeared in 1936 when he was an undergraduate of Tsinghua University and he worked literally until he died in June 2009. Chung wrote 133 research and expository papers. A selection of his œuvre is contained in the two volumes Chance and Choice: Memorabilia (edited by K.L. Chung, World Scientific 2004 [Zbl 1059.01017]) and Selected Works of Kai Lai Chung (with commentaries; edited by F. Ait Sahlia, E. Hsu, R. Williams, World Scientific 2008 [Zbl 1165.60302]).

The volume under review is his last research monograph. Originally published in 1995 as volume 312 of Springer’s Grundlehren series it is a compilation and refinement of the authors’ results on the Schrödinger equation in \(\mathbb{R}^d\)

\[ \frac 12\,\Delta u(x) + q(x)u(x) = 0 \eqno(1) \]

and the time-dependent Schrödinger equation

\[ \frac{\partial}{\partial t}\,w(t,x) = \frac 12\,\Delta w(t,x) + q(x)w(t,x) \eqno(2) \]

(where \(q:\mathbb{R}^d\to\mathbb{R}\) is a Borel-measurable function) up to 1995. Since the 2001 reprint contains only minor corrections, we refer to the original Zentralblatt review [Zbl 0819.60068] for a detailed description of the contents. Chung initiated this line of research in the late 1970s. His 1980 Séminaire de Probabilités paper On stopped Feynman-Kac functional [Zbl 0444.60061] started the interest in probabilistic solutions of (1) where \(q\) is not necessarily positive; Chung calls these solutions \(q\)-harmonic functions. These developments made it possible, in Chung’s own words (Preface, p. vii), to tackle the great problems of quantum potential theory: the representation of a \(q\)-harmonic function by its boundary values, the unique solvability of the Dirichlet boundary value problem, Poisson’s equation, Harnack’s inequality, …, and all the lush Green landscape now tinted with the ubiquitous \(q\) with the tools of probability theory. The central object is the gauge, i.e., the expression

\[ \mathbb E^x\left(\mathbf{1}_{\{\tau<\infty\}} \int_0^{\tau} q(X_s)\,ds\right) \]

where \(\tau\) is the first exit time of the stochastic process \(X_t\) associated with (1) from the domain \(D\). The gauge theorem asserts that for bounded potentials \(q\) and for domains \(D\) with finite Lebesgue measure this expression is either everywhere infinite or bounded. The same conclusion remains true if \(q\) is a Kato class function. If one uses conditioned Brownian motion (instead of Brownian motion) one can derive conditional gauge theorems, but in this case the boundary of the domain \(D\) needs to be sufficiently smooth, e.g. Lipschitz continuous. Considering different domains \(D\) with a fixed potential \(q\), it is possible to study various objects such as the principal eigenvalue and eigenfunction of (1) or the boundary Harnack principle.

Chung states in the preface to From Brownian Motion to Schrödinger’s Equation that this book (…) is a new departure with a new bent. (…) Here the focus is on a few central themes and their (…) developments. And what a departure this has been: since its publication, the notion of the Kato class is firmly established in the research literature and the use of stochastic methods in the study of the Harnack inequality, the boundary Harnack principle, eigenvalue estimates and subsequent developments originated from the ideas developed in this monograph.

Chung’s exposition is very clear, easy to follow and always mathematically reliable. Chung has a very personal style of writing; not all theorems, techniques or colleagues are equally important in his opinion—and he is rather blunt about this! His views on mathematics, in general, and probability theory, in particular, are most poignantly presented in Green, Brown and Probability [Zbl 0871.60001, Zbl 1014.60001] and Introduction to Random Times and Quantum Randomness [Zbl 0989.81500, Zbl 1021.81500, Zbl 1041.81074].

Also in mathematics Chung held firm beliefs. That a recent probabilistic proof of almost anything may be still further probabilisable (p. 190, Notes on Chapter 6) has been his programme throughout his life. Kai Lai Chung will be remembered as one of the really influential probabilists in the second half of the twentieth century. His first publication appeared in 1936 when he was an undergraduate of Tsinghua University and he worked literally until he died in June 2009. Chung wrote 133 research and expository papers. A selection of his œuvre is contained in the two volumes Chance and Choice: Memorabilia (edited by K.L. Chung, World Scientific 2004 [Zbl 1059.01017]) and Selected Works of Kai Lai Chung (with commentaries; edited by F. Ait Sahlia, E. Hsu, R. Williams, World Scientific 2008 [Zbl 1165.60302]).

The volume under review is his last research monograph. Originally published in 1995 as volume 312 of Springer’s Grundlehren series it is a compilation and refinement of the authors’ results on the Schrödinger equation in \(\mathbb{R}^d\)

\[ \frac 12\,\Delta u(x) + q(x)u(x) = 0 \eqno(1) \]

and the time-dependent Schrödinger equation

\[ \frac{\partial}{\partial t}\,w(t,x) = \frac 12\,\Delta w(t,x) + q(x)w(t,x) \eqno(2) \]

(where \(q:\mathbb{R}^d\to\mathbb{R}\) is a Borel-measurable function) up to 1995. Since the 2001 reprint contains only minor corrections, we refer to the original Zentralblatt review [Zbl 0819.60068] for a detailed description of the contents. Chung initiated this line of research in the late 1970s. His 1980 Séminaire de Probabilités paper On stopped Feynman-Kac functional [Zbl 0444.60061] started the interest in probabilistic solutions of (1) where \(q\) is not necessarily positive; Chung calls these solutions \(q\)-harmonic functions. These developments made it possible, in Chung’s own words (Preface, p. vii), to tackle the great problems of quantum potential theory: the representation of a \(q\)-harmonic function by its boundary values, the unique solvability of the Dirichlet boundary value problem, Poisson’s equation, Harnack’s inequality, …, and all the lush Green landscape now tinted with the ubiquitous \(q\) with the tools of probability theory. The central object is the gauge, i.e., the expression

\[ \mathbb E^x\left(\mathbf{1}_{\{\tau<\infty\}} \int_0^{\tau} q(X_s)\,ds\right) \]

where \(\tau\) is the first exit time of the stochastic process \(X_t\) associated with (1) from the domain \(D\). The gauge theorem asserts that for bounded potentials \(q\) and for domains \(D\) with finite Lebesgue measure this expression is either everywhere infinite or bounded. The same conclusion remains true if \(q\) is a Kato class function. If one uses conditioned Brownian motion (instead of Brownian motion) one can derive conditional gauge theorems, but in this case the boundary of the domain \(D\) needs to be sufficiently smooth, e.g. Lipschitz continuous. Considering different domains \(D\) with a fixed potential \(q\), it is possible to study various objects such as the principal eigenvalue and eigenfunction of (1) or the boundary Harnack principle.

Chung states in the preface to From Brownian Motion to Schrödinger’s Equation that this book (…) is a new departure with a new bent. (…) Here the focus is on a few central themes and their (…) developments. And what a departure this has been: since its publication, the notion of the Kato class is firmly established in the research literature and the use of stochastic methods in the study of the Harnack inequality, the boundary Harnack principle, eigenvalue estimates and subsequent developments originated from the ideas developed in this monograph.

Reviewer: René L. Schilling (Dresden)

### MSC:

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |

60J65 | Brownian motion |

60J45 | Probabilistic potential theory |

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

35J25 | Boundary value problems for second-order elliptic equations |

60G40 | Stopping times; optimal stopping problems; gambling theory |

60J25 | Continuous-time Markov processes on general state spaces |

60J35 | Transition functions, generators and resolvents |

60J55 | Local time and additive functionals |

35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |

35J08 | Green’s functions for elliptic equations |

35K05 | Heat equation |

35K15 | Initial value problems for second-order parabolic equations |

81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |

81Q15 | Perturbation theories for operators and differential equations in quantum theory |

81Q30 | Feynman integrals and graphs; applications of algebraic topology and algebraic geometry |