Some properties of planar Brownian motion.

*(English)*Zbl 0779.60068
Probabilités, Ec. d’Eté XX, Saint-Flour/Fr. 1990, Lect. Notes Math. 1527, 111-235 (1992).

[For the entire collection see Zbl 0759.00009.]

This work is a welcome introduction to the recent developments in the theory of planar Brownian motion with emphasis on geometric properties of sample paths. An important conception is the notion of local time used in a very wide sense. Although the proofs based on local times are not always the shortest ones, such approach permits the author to arrange the text in a good logical scheme and state it in uniform style.

The work consists of 11 chapters. Chapter I is introduction. In Chapter II the conformal invariance of Brownian path and skew-product representation is used to prove several asymptotic theorems. Chapters III and IV deal with sample path properties: the existence and non-existence of one- and two-sided cone points, the smoothness of boundary of the convex hull and Hausdorff measure properties. Chapter V treats the Burdzy’s theorem on twist points. Chapter VI deals with asymptotic behavior of the areas of Wiener sausages. In Chapter VII Mandelbrot’s conjecture about the number of connected components of complement of a planar Brownian path is proved. In Chapters VIII and IX the existence of points of finite and infinite multiplicity is proved with the help of intersectional local times. Chapters X and XI are devoted to the further investigations of renormalizations of self-intersections and asymptotic expansions of the areas of Wiener sausages.

The work contains a lot of results appearing in book form for the first time and can be used as a good base for a modern course on Brownian motion.

This work is a welcome introduction to the recent developments in the theory of planar Brownian motion with emphasis on geometric properties of sample paths. An important conception is the notion of local time used in a very wide sense. Although the proofs based on local times are not always the shortest ones, such approach permits the author to arrange the text in a good logical scheme and state it in uniform style.

The work consists of 11 chapters. Chapter I is introduction. In Chapter II the conformal invariance of Brownian path and skew-product representation is used to prove several asymptotic theorems. Chapters III and IV deal with sample path properties: the existence and non-existence of one- and two-sided cone points, the smoothness of boundary of the convex hull and Hausdorff measure properties. Chapter V treats the Burdzy’s theorem on twist points. Chapter VI deals with asymptotic behavior of the areas of Wiener sausages. In Chapter VII Mandelbrot’s conjecture about the number of connected components of complement of a planar Brownian path is proved. In Chapters VIII and IX the existence of points of finite and infinite multiplicity is proved with the help of intersectional local times. Chapters X and XI are devoted to the further investigations of renormalizations of self-intersections and asymptotic expansions of the areas of Wiener sausages.

The work contains a lot of results appearing in book form for the first time and can be used as a good base for a modern course on Brownian motion.

Reviewer: N.M.Zinchenko (Kiev)

##### MSC:

60J65 | Brownian motion |

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

60G17 | Sample path properties |

60G55 | Point processes (e.g., Poisson, Cox, Hawkes processes) |