Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

N G van Kampen 1981 Amsterdam: North-Holland xiv + 419 pp price Dfl 180 This is a book which, at a lower price, could be expected to become an essential part of the library of every physical scientist concerned with problems involving fluctuations and stochastic processes, as well as those who just enjoy a beautifully written book. It provides an extensive graduate-level introduction which is clear, cautious, interesting and readable.

https://iopscience.iop.org/article/10.1088/0031-9112/34/4/040/pdf

Stochastic Processes in Physics and Chemistry

The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

Sphere Packings, Lattices and Groups

Stochastic differential equations and diffusion processes

We present several constructions of a ``censored stable process'' in an open set D⊂R

 n
 , i.e., a symmetric stable process which is not allowed to jump outside D. 
We address the question of whether the process will approach the boundary of D in a finite time – we give sharp conditions for such approach in terms of the stability index α and the ``thickness'' of the boundary. As a corollary, new results are obtained concerning Besov spaces on non-smooth domains, including the critical exponent case. 
We also study the decay rate of the corresponding harmonic functions which vanish on a part of the boundary. We derive a boundary Harnack principle in C

 1,1
 open sets.

Censored stable processes

On the “Hot Spots” Conjecture of J. Rauch

Suppose that X1, X2 and Y are independent standard Brownian motions starting from 0 and let 

$$ X\left( t \right) = \left\{ {\begin{array}{*{20}{c}} {{X^1}\left( t \right) if t \geqslant 0,} \\ {{X^2}\left( { - t} \right) if t < 0.} \end{array}} \right.$$

We will consider the process $ \left\{ {Z\left( t \right)\underline{\underline {df}} X\left( {Y\left( t \right)} \right),t \geqslant 0} \right\}$ which we will call “iterated Brownian motion” or simply IBM. Funaki (1979) proved that a similar process is related to “squared Laplacian.” Krylov (1960) and Hochberg (1978) considered finitely additive signed measures on the path space corresponding to squared Laplacian (there exists a genuine probabilistic approach, see, e.g., Mądrecki and Rybaczuk (1992). A paper of Vervaat (1985) contains a section on the composition of self-similar processes.

/pdf/some-path-properties-of-iterated-brownian-motion-1f097eru54.pdf

Some path properties of iterated Brownian motion

We consider a model with a large number N of particles which move according to independent Brownian motions A particle which leaves a domain D is killed; at the same time, a different particle splits into two particles For large N, the particle distribution density converges to the normalized heat equation solution in D with Dirichlet boundary conditions The stationary distributions converge as N→∞ to the first eigenfunction of the Laplacian in D with the same boundary conditions

A Fleming-Viot Particle Representation of the Dirichlet Laplacian

We construct a counterexample to the "hot spots" conjecture; there exists a bounded connected planar domain (with two holes) such that the second eigenvalue of the Laplacian in that domain with Neumann boundary conditions is simple and such that the corresponding eigenfunction attains its strict maximum at an interior point of that domain.

/pdf/a-counterexample-to-the-hot-spots-conjecture-2jnntgn4by.pdf

Krzysztof Burdzy

Papers

Censored stable processes

On the “Hot Spots” Conjecture of J. Rauch

Some path properties of iterated Brownian motion

A Fleming-Viot Particle Representation of the Dirichlet Laplacian

A counterexample to the “hot spots” conjecture