Spatial branching processes, random snakes and partial differential equations
著者
書誌事項
Spatial branching processes, random snakes and partial differential equations
(Lectures in mathematics ETH Zürich)
Birkhäuser Verlag, c1999
- : Basel
- : Boston
大学図書館所蔵 全22件
  青森
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  福島
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  埼玉
  千葉
  東京
  神奈川
  新潟
  富山
  石川
  福井
  山梨
  長野
  岐阜
  静岡
  愛知
  三重
  滋賀
  京都
  大阪
  兵庫
  奈良
  和歌山
  鳥取
  島根
  岡山
  広島
  山口
  徳島
  香川
  愛媛
  高知
  福岡
  佐賀
  長崎
  熊本
  大分
  宮崎
  鹿児島
  沖縄
  韓国
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注記
Includes bibliographical references and index
内容説明・目次
内容説明
This book introduces several remarkable new probabilistic objects that combine spatial motion with a continuous branching phenomenon and are closely related to certain semilinear partial differential equations (PDE). The Brownian snake approach is used to give a powerful representation of superprocesses and also to investigate connections between superprocesses and PDEs. These are notable because almost every important probabilistic question corresponds to a significant analytic problem.
目次
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 Ito 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 representation theorem.- VII.4 Further developments.- VIII Levy Processes and the Genealogy of General Continuous-state Branching Processes.- VIII.1 The discrete setting.- VIII.2 Levy processes.- VIII.3 The height process.- VIII.4 The exploration process.- VIII.5 Proof of Theorem 2.- Bibliographical Notes.
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