泛函不等式,馬爾可夫半群與譜理論

內(nèi)容概要

本書(shū)的主要內(nèi)容涉及概率論、泛函分析、微分幾何和統(tǒng)計(jì)物理等多個(gè)學(xué)科，較系統(tǒng)的介紹了近十年有關(guān)泛函不等式及其近十年來(lái)的有關(guān)泛函不等式及其應(yīng)用的主要研究成果和研究方法。其中的一些成果和研究思想被國(guó)際同行專家大量引用，引發(fā)了一系列的后續(xù)工作。以泛函不等式為主要工具研究馬氏半群及其生成元的分析與概率性質(zhì)。特別的，使用弱不等式刻畫(huà)馬氏半群的各種收斂速度；使用超不等式刻畫(huà)半群一致可積性以及生成元的本征譜和高階特征值的估計(jì)；引入一般型可加性的不等式，刻畫(huà)馬氏半群在不同意義下的指數(shù)式收斂以及概論距離的上界估計(jì)。

作者簡(jiǎn)介

　　王風(fēng)雨，博士，1966年12月生于安徽省嘉山縣。北京師范大學(xué)教授，長(zhǎng)江學(xué)者特聘教授，博士生導(dǎo)師，國(guó)家杰出青年科學(xué)基金獲得者?！　≡鴳?yīng)邀訪問(wèn)英國(guó)Warwick大學(xué)，還應(yīng)邀訪問(wèn)過(guò)美國(guó)、法國(guó)、德國(guó)、俄羅斯、日本、新加坡、意大利和臺(tái)灣等國(guó)家和地區(qū)的20余所大學(xué)和研究所，并多次在國(guó)際學(xué)術(shù)會(huì)議上作邀請(qǐng)報(bào)告。目前，擔(dān)任中國(guó)概率統(tǒng)計(jì)學(xué)會(huì)常務(wù)理事，美國(guó)《數(shù)學(xué)評(píng)論》和德國(guó)《數(shù)學(xué)文摘》評(píng)論員，《應(yīng)用概率統(tǒng)計(jì)》等雜志的編委。曾經(jīng)作為洪堡學(xué)者在德國(guó)Bidefeld大學(xué)工作。曾經(jīng)獲得鐘嘉慶數(shù)學(xué)獎(jiǎng)，教育部科技進(jìn)步獎(jiǎng)一等獎(jiǎng)，國(guó)家自然科學(xué)三等獎(jiǎng)和教育部首屆高校青年教師獎(jiǎng)，霍英東青年教師獎(jiǎng)研究類一等獎(jiǎng)。獲得北京市五四青年獎(jiǎng)?wù)?，人選首批新世紀(jì)百千萬(wàn)工程國(guó)家級(jí)人才計(jì)劃，承擔(dān)國(guó)家重點(diǎn)基礎(chǔ)研究發(fā)展規(guī)劃“973”項(xiàng)目。研究方向涉及概率論、微分幾何、統(tǒng)計(jì)物理和泛函分析等多個(gè)學(xué)科領(lǐng)域，已發(fā)表論文近80篇，出版專著一部。

書(shū)籍目錄

Chapter 0 Preliminaries0.1 Dirichlet forms, sub-Markov semigroups and generators0.2 Dirichlet forms and Markov processes0.3 Spectral theory0.4 Riemannian geometryChapter 1 Poincaré Inequality and Spectral Gap1.1 A general result and examples1.2 Concentration of measures1.3 Poincaré inequalities for jump processes1.3.1 The bounded jump case1.3.2 The unbounded jump case1.3.3 A criterion for birth-death processes1.4 Poincaré inequality for diffusion processes1.4.1 The one-dimensional case1.4.2 Spectral gap for diffusion processes on R上標(biāo)d1.4.3 Existence of the spectral gap on manifolds and application to nonsymmetric elliptic operators1.5 NotesChapter 2 Diffusion Processes on Manifolds and Applications2.1 Kendall-Cranstons coupling2.2 Estimates of the first (closed and Neumann) eigenvalue2.3 Estimates of the first two Dirichlet eigenvalues2.3.1 Estimates of the first Dirichlet eigenvalue2.3.2 Estimates of the second Dirichlet eigenvalue and the spectralgap2.4 Gradient estimates of diffusion semigroups2.4.1 Gradient estimates of the closed and Neumann semigroups2.4.2 Gradient estimates of Dirichlet semigroups2.5 Harnack and isoperimetric inequalities using gradient estimates2.5.1 Gradient estimates and the dimension-free Harnack inequality2.5.2 The first eigenvalue and isoperimetric constants2.6 Liouville theorems and couplings on manifolds2.6.1 Liouville theorem using the Brownian radial process2.6.2 Liouville theorem using the derivative formula2.6.3 Liouville theorem using the conformal change of metric2.6.4 Applications to harmonic maps and coupling Harmonic maps2.7 NotesChapter 3 Functional Inequalities and Essential Spectrum3.1 Essential spectrum on Hilbert spaces3.1.1 Functional inequalities3.1.2 Application to nonsymmetric semigroups3.1.3 Asymptotic kernels for compact operators3.1.4 Compact Markov operators without kernels3.2 Applications to coercive closed forms3.3 Super Poincaré inequalities3.3.1 The F-Sobolev inequality3.3.2 Estimates of semigroups3.3.3 Estimates of high order eigenvalues3.3.4 Concentration of measures for super Poincaré inequalities3.4 Criteria for super Poincaré inequalities3.4.1 A localization method3.4.2 Super Poincaré inequalities for jump processes3.4.3 Estimates of β for diffusion processes3.4.4 Some examples for estimates of high order eigenvalues3.4.5 Some criteria for diffusion processes3.5 NotesChapter 4 Weak Poicaré Inequalities and Convergence of Semigroups4.1 General results4.2 Concentration of measures4.3 Criteria of weak Poincaré inequalities4.4 Isoperimetric inequalities4.4.1 Diffusion processes on manifolds4.4.2 Jump processes4.5 NotesChapter 5 Log-Sobolev Inequalities and Semigroup Properties5.2 Spectral gap for hyperbounded operators5.3 Concentration of measures for log-Sobolev inequalities5.4 Logarithmic Sobolev inequalities for jump processes5.4.1 Isoperimetric inequalities5.4.2 Criteria for birth-death processes5.5 Logarithmic Sobolev inequalities for one-dimensional diffusion processes5.6 Estimates of the log-Sobolev constant on manifolds5.6.1 Equivalent statements for the curvature condition5.6.2 Estimates of α(V) using Bakry-Emerys criterion5.6.3 Estimates of α(V) using Harnack inequality5.6.4 Estimates of α(V) using coupling5.7 Criteria of hypercontractivity, superboundedness and ultraboundedness5.7.1 Some criteria5.7.2 Ultraboundedness by perturbations5.7.3 Isoperimetric inequalities5.7.4 Some examples5.8 Strong ergodicity and log-Sobolev inequality5.9 NotesChapter 6 Interpolations of Poincaré and Log-Sobolev Inequalities6.1 Some properties of (6.0.3)6.2 Some criteria of (6.0.3)6.3 Transportation cost inequalities6.3.1 Otto-Villanis coupling6.3.2 Transportation cost inequalities6.3.3 Some results on (I下標(biāo)p)6.4 NotesChapter 7 Some Infinite Dimensional Models7.1 The (weighted) Poisson spaces7.1.1 Weak Poincaréinequalities for second quantization Dirichlet forms7.1.2 A class of jump processes on configuration spaces7.1.3 Functional inequalities for ε上標(biāo)Г下標(biāo)J7.2 Analysis on path spaces over Riemannian manifolds7.2.1 Weak Poincaré inequality on finite-time interval path spaces7.2.2 Weak Poincaré inequality on infinite-time interval path spaces7.2.3 Transportation cost inequality on path spaces with L上標(biāo)2-distance7.2.4 Transportation cost inequality on path spaces with the intrinsic distance7.3 Functional and Harnack inequalities for generalized Mehler semigroups7.3.1 Some general results7.3.2 Some examples7.3.3 A generalized Mehler semigroup associated with the Dirichlet heat semigroup7.4 NotesBibliographyIndex

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