A Minicourse on Stochastic Partial Differential Equations

By: Robert Dalang, Davar Khoshnevisan

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Paperback | 21 October 2008

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236 Pages

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In May 2006, The University of Utah hosted an NSF-funded minicourse on stochastic partial differential equations. The goal of this minicourse was to introduce graduate students and recent Ph.D.s to various modern topics in stochastic PDEs, and to bring together several experts whose research is centered on the interface between Gaussian analysis, stochastic analysis, and stochastic partial differential equations. This monograph contains an up-to-date compilation of many of those lectures. Particular emphasis is paid to showcasing central ideas and displaying some of the many deep connections between the mentioned disciplines, all the time keeping a realistic pace for the student of the subject.

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Non-Fiction Mathematics Calculus & Mathematical Analysis Differential Calculus & Equations Probability & Statistics Integral Calculus & Equations

A Primer on Stochastic Partial Differential Equations	p. 1
What is an SPDE?	p. 1
Gaussian Random Vectors	p. 2
Gaussian Processes	p. 2
Regularity of Random Processes	p. 8
Martingale Measures	p. 14
A Nonlinear Heat Equation	p. 23
From Chaos to Order	p. 32
References	p. 36
The Stochastic Wave Equation	p. 39
Introduction	p. 39
The Stochastic Wave Equation	p. 41
Spatially Homogeneous Gaussian Noise	p. 47
The Wave Equation in Spatial Dimension 2	p. 49
A Function-Valued Stochastic Integral	p. 56
The Wave Equation in Spatial Dimension d >= 1	p. 58
Spatial Regularity of the Stochastic Integral (d = 3)	p. 61
Holder-Continuity in the 3-d Wave Equation	p. 70
References	p. 71
Application of Malliavin Calculus to Stochastic Partial Differential Equations	p. 73
Introduction	p. 73
Malliavin Calculus	p. 73
Application of Malliavin Calculus to Regularity of Probability Laws	p. 83
Stochastic Heat Equation	p. 92
Spatially Homogeneous SPDEs	p. 99
References	p. 108
Some Tools and Results for Parabolic Stochastic Partial Differential Equations	p. 111
Introduction	p. 111
Basic Framework	p. 113
Duality	p. 115
Large Deviations for SPDEs	p. 125
A Comparison Theorem	p. 129
Applications	p. 131
References	p. 142
Sample Path Properties of Anisotropic Gaussian Random Fields	p. 145
Introduction	p. 145
Examples and General Assumptions	p. 148
Properties of Strong Local Nondeterminism	p. 160
Modulus of Continuity	p. 164
Small Ball Probabilities	p. 168
Hausdorff and Packing Dimensions of the Range and Graph	p. 170
Hausdorff Dimension of the Level Sets and Hitting Probabilities	p. 183
Local Times and Their Joint Continuity	p. 194
References	p. 207
List of Participants	p. 213
Index	p. 215
Table of Contents provided by Ingram. All Rights Reserved.

A Minicourse on Stochastic Partial Differential Equations

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