Brownian motion on nested fractals.

*(English)*Zbl 0688.60065
Mem. Am. Math. Soc. 420, 128 p. (1990).

Before I give a brief description of the chapter contents, it is my impression that the author has done a very commendable job in presenting this topic in a clear, well motivated and concise manner. I enjoyed particularly the author’s informal writing style. If more books, memoirs, papers and the like were written in informal styles and allowed to appear in print, then topics such as this might be more easily assimilated by neophytes. This may be an important step in popularizing any mathematics that involves, even slightly, nonstandard analysis. This volume of the AMS Memoirs is a welcome addition to the literature on this subject. Even though it is not essential to the flow of the book, I did notice that two pages in my copy are out of order. Page 115 should be 116 and page 116 should be 115. Since these pages only involved some open questions all one needs to do is to renumber the questions. Now to the contents.

The author is concerned with a subclass of the class of all self-similar fractals, the nested fractals, as he axiomatically defines them. In Chapter 1, the author briefly mentions the Hausdorff dimension, but this chapter is mainly composed of the usual introductory remarks. Fractals in physics and mathematics are examined more closely beginning with Chapter 2. The author presents sufficient diagrams and an intuitive discussion of fractal behaviour exhibited by physically real attemps to measure physical quantities such as mass for homogeneous, but irregular, objects relative to a fixed geometric concept. This leads pictorially to considerations of such geometric configurations as the Sierpinski gasket, Koch-curve and nested cubes as geometric examples of the Hausdorff dimension log \(\mu/\log\nu\) where \(\mu\) is the volume scaling factor and \(\nu\) the length scaling factor.

The author introduces the snowflake fractal S in Chapter 3, as an example of a Brownian motion B on S as a limit of random walks. With this he gives a nice intuitive development for the transition probabilities. Along with this, transition times, time scaling factors and other concepts are introduced. The author then gives an interesting intuitive description for his modified random walk, from which he intends to obtain, as a type of limit, the actual Brownian motion. In Chapter 4, the nested fractals are axiomatically presented along with certain technical matters. It appears that the usual fractals encountered within the physical sciences are of this type.

Chapter 5 is concerned with certain technical ideas. Relative to the intuitive concepts displayed by the construction of the snowflake fractal, he defines the set \({\mathcal P}\) of basic transition probabilities each member being related to a homogeneous Markov chain \(B_ n\) for each n. If \((p_ 1,...,p_ r)\in {\mathcal P}\), then the author defines the composite transition probabilities \(\tilde p_ 1,...,\tilde p_ r\) for \(B_ 1\). The major aspect of this chapter is to show that the map defined by \(\tilde p(p_ 1,...,p_ r)=(\tilde p_ 1,...,\tilde p_ r)\) is a continuous map from \({\mathcal P}\) into \({\mathcal P}\) with a fixed point.

At the beginning of Chapter 6, the author points out that the processes \(B_ n\) thus far introduced are just Markov chains moving one step at each unit of time. In order to construct a limit process, one needs to known how to rescale time as n grows large. The author formulates this in terms of random transition times. He first uses the set \({\bar {\mathbb{R}}}_+=\{x\in {\mathbb{R}}|\) \(x\geq 0\}\cup \{\infty\}\) and considers a set of basic transition times as a sequence \((\tau_ 1,...,\tau_ r)\) of completed Borel probability measures on \({\bar {\mathbb{R}}}_+\)- one for each equivalence class \(c_ i\). Now these classes are composed of ordered pairs of essentially fixed points such that if (x,y), \((x',y')\in c_ i\), then \(| x-y| =| x'-y'|.\) Intuitively, if \((x,y)\in c_ i\), then one should think of \(\tau_ i\) as the distribution of the time a random particle uses to get from x to y. Using a set of additional conditions the author defines a process \(\hat B_ n(t,\omega)\) he calls a random walk induced by \((p_ 1,...,p_ r)\) and \((\tau_ 1,...,\tau_ r)\). After more machinery is introduced, the composition transition times (\({\tilde \tau}{}_ 1,...,{\tilde \tau}_ r)\) are defined. A major parameter \(\lambda\), the time scaling factor, is introduced and appears to be as significant as \(\mu\) and \(\nu\). One of the major results of this chapter is that there exists a sequence \((\tau_ 1,...,\tau_ r)\) of basic transition times such that the composition transition times (\({\tilde\tau}_ 1,...,{\tilde \tau}_ r)\) are just \((\tau_ 1,...,\tau_ r)\) scaled by the time scaling factor \(\lambda\). Further, the sequence \((\tau_ 1,...,\tau_ r)\) is unique up to a scaling factor.

Chapter 7 is where the author first introduces nonstandard methods. These are the customary methods of Robinson rather than Internal Set Theory. In brief, the actual standard process he selects is obtaind by first considering an infinitely large \(N\in^*{\mathbb{N}}\). In the set \(\{\hat B_ n|\) \(n\in {\mathbb{N}}\}\) there corresponds the process associated with N and denoted by the author by \(^*\hat B_ N(\lambda^ Nt,\omega)\) where time has been rescaled. The actual process is \(B=st(^*\hat B_ N(\lambda^ Nt,\omega))\), where st is the standard part operator. He calls this a Brownian motion and establishes that it is a strong Markov process with continuous paths and a law that is independent of N.

Chapter 8 has a very significant purpose. I let the author explain. “I argued in the introduction that one of the main reasons for studying Brownian motion on fractals is to get a better understanding of the physics of fractal media. Having studied the construction above, and having, in particular, observed the extreme care with which I have chosen my transition probabilities and transition times, you have every reason to doubt that my processes have anything to do with physics - why should one expect Nature to choose her transition properties according to abstract fixed point theorems? The purpose of this section is to relieve your doubts by showing that the choice of basic transition properties is not as crucial as the previous sections may have led you to believe; in fact, a large variety of basic transition probabilities and transition times give rise to the same Brownian motion in the limit.”

In Chapter 9, the author investigates a Laplace type operator \(\Delta\) associated with a stable point (\(\vec p,{\vec \tau})\), where \(\Delta\) is considered to be the infinitesimal generator of an appropriate semigroup. This operator is a self-adjoint non-positively definite operator on an appropriate \(L^ 2\)-space. The author completes this chapter with an in depth study of the asymptotic distribution of the eigenvalues of \(\Delta\).

The book concludes with a list of some 20 open questions and a detailed list of references. Most certainly this volume is a worthwile addition to the literature on this subject. I strongly recommend that all individuals interested in fractal behavior add this volume to their library as well as prepare themselves for what I believe will be revolution in physical modeling, the use of the nonstandard technique. It is not the fact that standard techniques may also lead to the same results, but, rather, it is the use of a differentiated language that lends itself to a much better intuitive understanding for many, if not all, natural processes.

The author is concerned with a subclass of the class of all self-similar fractals, the nested fractals, as he axiomatically defines them. In Chapter 1, the author briefly mentions the Hausdorff dimension, but this chapter is mainly composed of the usual introductory remarks. Fractals in physics and mathematics are examined more closely beginning with Chapter 2. The author presents sufficient diagrams and an intuitive discussion of fractal behaviour exhibited by physically real attemps to measure physical quantities such as mass for homogeneous, but irregular, objects relative to a fixed geometric concept. This leads pictorially to considerations of such geometric configurations as the Sierpinski gasket, Koch-curve and nested cubes as geometric examples of the Hausdorff dimension log \(\mu/\log\nu\) where \(\mu\) is the volume scaling factor and \(\nu\) the length scaling factor.

The author introduces the snowflake fractal S in Chapter 3, as an example of a Brownian motion B on S as a limit of random walks. With this he gives a nice intuitive development for the transition probabilities. Along with this, transition times, time scaling factors and other concepts are introduced. The author then gives an interesting intuitive description for his modified random walk, from which he intends to obtain, as a type of limit, the actual Brownian motion. In Chapter 4, the nested fractals are axiomatically presented along with certain technical matters. It appears that the usual fractals encountered within the physical sciences are of this type.

Chapter 5 is concerned with certain technical ideas. Relative to the intuitive concepts displayed by the construction of the snowflake fractal, he defines the set \({\mathcal P}\) of basic transition probabilities each member being related to a homogeneous Markov chain \(B_ n\) for each n. If \((p_ 1,...,p_ r)\in {\mathcal P}\), then the author defines the composite transition probabilities \(\tilde p_ 1,...,\tilde p_ r\) for \(B_ 1\). The major aspect of this chapter is to show that the map defined by \(\tilde p(p_ 1,...,p_ r)=(\tilde p_ 1,...,\tilde p_ r)\) is a continuous map from \({\mathcal P}\) into \({\mathcal P}\) with a fixed point.

At the beginning of Chapter 6, the author points out that the processes \(B_ n\) thus far introduced are just Markov chains moving one step at each unit of time. In order to construct a limit process, one needs to known how to rescale time as n grows large. The author formulates this in terms of random transition times. He first uses the set \({\bar {\mathbb{R}}}_+=\{x\in {\mathbb{R}}|\) \(x\geq 0\}\cup \{\infty\}\) and considers a set of basic transition times as a sequence \((\tau_ 1,...,\tau_ r)\) of completed Borel probability measures on \({\bar {\mathbb{R}}}_+\)- one for each equivalence class \(c_ i\). Now these classes are composed of ordered pairs of essentially fixed points such that if (x,y), \((x',y')\in c_ i\), then \(| x-y| =| x'-y'|.\) Intuitively, if \((x,y)\in c_ i\), then one should think of \(\tau_ i\) as the distribution of the time a random particle uses to get from x to y. Using a set of additional conditions the author defines a process \(\hat B_ n(t,\omega)\) he calls a random walk induced by \((p_ 1,...,p_ r)\) and \((\tau_ 1,...,\tau_ r)\). After more machinery is introduced, the composition transition times (\({\tilde \tau}{}_ 1,...,{\tilde \tau}_ r)\) are defined. A major parameter \(\lambda\), the time scaling factor, is introduced and appears to be as significant as \(\mu\) and \(\nu\). One of the major results of this chapter is that there exists a sequence \((\tau_ 1,...,\tau_ r)\) of basic transition times such that the composition transition times (\({\tilde\tau}_ 1,...,{\tilde \tau}_ r)\) are just \((\tau_ 1,...,\tau_ r)\) scaled by the time scaling factor \(\lambda\). Further, the sequence \((\tau_ 1,...,\tau_ r)\) is unique up to a scaling factor.

Chapter 7 is where the author first introduces nonstandard methods. These are the customary methods of Robinson rather than Internal Set Theory. In brief, the actual standard process he selects is obtaind by first considering an infinitely large \(N\in^*{\mathbb{N}}\). In the set \(\{\hat B_ n|\) \(n\in {\mathbb{N}}\}\) there corresponds the process associated with N and denoted by the author by \(^*\hat B_ N(\lambda^ Nt,\omega)\) where time has been rescaled. The actual process is \(B=st(^*\hat B_ N(\lambda^ Nt,\omega))\), where st is the standard part operator. He calls this a Brownian motion and establishes that it is a strong Markov process with continuous paths and a law that is independent of N.

Chapter 8 has a very significant purpose. I let the author explain. “I argued in the introduction that one of the main reasons for studying Brownian motion on fractals is to get a better understanding of the physics of fractal media. Having studied the construction above, and having, in particular, observed the extreme care with which I have chosen my transition probabilities and transition times, you have every reason to doubt that my processes have anything to do with physics - why should one expect Nature to choose her transition properties according to abstract fixed point theorems? The purpose of this section is to relieve your doubts by showing that the choice of basic transition properties is not as crucial as the previous sections may have led you to believe; in fact, a large variety of basic transition probabilities and transition times give rise to the same Brownian motion in the limit.”

In Chapter 9, the author investigates a Laplace type operator \(\Delta\) associated with a stable point (\(\vec p,{\vec \tau})\), where \(\Delta\) is considered to be the infinitesimal generator of an appropriate semigroup. This operator is a self-adjoint non-positively definite operator on an appropriate \(L^ 2\)-space. The author completes this chapter with an in depth study of the asymptotic distribution of the eigenvalues of \(\Delta\).

The book concludes with a list of some 20 open questions and a detailed list of references. Most certainly this volume is a worthwile addition to the literature on this subject. I strongly recommend that all individuals interested in fractal behavior add this volume to their library as well as prepare themselves for what I believe will be revolution in physical modeling, the use of the nonstandard technique. It is not the fact that standard techniques may also lead to the same results, but, rather, it is the use of a differentiated language that lends itself to a much better intuitive understanding for many, if not all, natural processes.

Reviewer: R.A.Herrmann

##### MSC:

60J65 | Brownian motion |

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

03H05 | Nonstandard models in mathematics |

35P20 | Asymptotic distributions of eigenvalues in context of PDEs |

60J60 | Diffusion processes |