Spatial Branching Processes, Random Snakes and Partial Differential Equations

1999
ISBN: 978-3-7643-6126-6
Verlag: Springer

Buch, Englisch, 163 Seiten, Paperback, Format (B × H): 170 mm x 244 mm, Gewicht: 315 g

Reihe: Lectures in Mathematics. ETH Zürich

In these lectures, we give an account of certain recent developments of the theory of spatial branching processes. These developments lead to several fas­ cinating probabilistic objects, which combine spatial motion with a continuous branching phenomenon and are closely related to certain semilinear partial dif­ ferential equations. Our first objective is to give a short self-contained presentation of the measure­ valued branching processes called superprocesses, which have been studied extensively in the last twelve years. We then want to specialize to the important class of superprocesses with quadratic branching mechanism and to explain how a concrete and powerful representation of these processes can be given in terms of the path-valued process called the Brownian snake. To understand this representation as well as to apply it, one needs to derive some remarkable properties of branching trees embedded in linear Brownian motion, which are of independent interest. A nice application of these developments is a simple construction of the random measure called ISE, which was proposed by Aldous as a tree-based model for random distribution of mass and seems to play an important role in asymptotics of certain models of statistical mechanics. We use the Brownian snake approach to investigate connections between super­ processes and partial differential equations. These connections are remarkable in the sense that almost every important probabilistic question corresponds to a significant analytic problem.

Zielgruppe

Research

Autoren/Hrsg.

Le Gall, Jean-Francois

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Differentialrechnungen und -gleichungen
• Mathematik | Informatik Mathematik Stochastik Mathematische Statistik
• Mathematik | Informatik Mathematik Stochastik Wahrscheinlichkeitsrechnung
• Technische Wissenschaften Technik Allgemein Mathematik für Ingenieure
• Naturwissenschaften Physik Mechanik Klassische Mechanik, Newtonsche Mechanik
• Interdisziplinäres Wissenschaften Wissenschaften: Forschung und Information Datenanalyse, Datenverarbeitung

Inhaltsverzeichnis

I An Overview.- I.1 Galton-Watson processes and continuous-state branching processes.- I.2 Spatial branching processes and superprocesses.- I.3 Quadratic branching and the Brownian snake.- I.4 Some connections with partial differential equations.- I.5 More general branching mechanisms.- I.6 Connections with statistical mechanics and interacting particle systems.- II Continuous-state Branching Processes and Superprocesses.- II.1 Continuous-state branching processes.- II.2 Superprocesses.- II.3 Some properties of superprocesses.- II.4 Calculations of moments.- III The Genealogy of Brownian Excursions.- III.1 The Itô excursion measure.- III.2 Binary trees.- III.3 The tree associated with an excursion.- III.4 The law of the tree associated with an excursion.- III.5 The normalized excursion and Aldous’ continuum random tree.- IV The Brownian Snake and Quadratic Superprocesses.- IV.1 The Brownian snake.- IV.2 Finite-dimensional marginals of the Brownian snake.- IV.3 The connection with superprocesses.- IV.4 The case of continuous spatial motion.- IV.5 Some sample path properties.- IV.6 Integrated super-Brownian excursion.- V Exit Measures and the Nonlinear Dirichlet Problem.- V.1 The construction of the exit measure.- V.2 The Laplace functional of the exit measure.- V.3 The probabilistic solution of the nonlinear Dirichlet problem.- V.4 Moments of the exit measure.- VI Polar Sets and Solutions with Boundary Blow-up.- VI.1 Solutions with boundary blow-up.- VI.2 Polar sets.- VI.3 Wiener’s test for the Brownian snake.- VI.4 Uniqueness of the solution with boundary blow-up.- VII The Probabilistic Representation of Positive Solutions.- VII.1 Singular solutions and boundary polar sets.- VII.2 Some properties of the exit measure from the unit disk.- VII.3 The representationtheorem.- VII.4 Further developments.- VIII Lévy Processes and the Genealogy of General Continuous-state Branching Processes.- VIII.1 The discrete setting.- VIII.2 Lévy processes.- VIII.3 The height process.- VIII.4 The exploration process.- VIII.5 Proof of Theorem 2.- Bibliographical Notes.