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Large Viscous Boundary Layers for Noncharacteristic Nonlinear Hyperbolic Problems

Guy Métivier University of Bordeaux, Talence, France
Kevin Zumbrun , Bloomington, IN
Available Formats:
Electronic ISBN: 978-1-4704-0427-7
Product Code: MEMO/175/826.E
List Price: $67.00 MAA Member Price:$60.30
AMS Member Price: $40.20 Click above image for expanded view Large Viscous Boundary Layers for Noncharacteristic Nonlinear Hyperbolic Problems Guy Métivier University of Bordeaux, Talence, France Kevin Zumbrun , Bloomington, IN Available Formats:  Electronic ISBN: 978-1-4704-0427-7 Product Code: MEMO/175/826.E  List Price:$67.00 MAA Member Price: $60.30 AMS Member Price:$40.20
• Book Details

Memoirs of the American Mathematical Society
Volume: 1752005; 107 pp
MSC: Primary 35;

This paper studies two types of integral transformation associated with fractional Brownian motion. They are applied to construct approximation schemes for fractional Brownian motion by polygonal approximation of standard Brownian motion. This approximation is the best in the sense that it minimizes the mean square error. The rate of convergence for this approximation is obtained. The integral transformations are combined with the idea of probability structure preserving mapping introduced in [48] and are applied to develop a stochastic calculus for fractional Brownian motions of all Hurst parameter $H\in (0, 1)$. In particular we obtain Radon-Nikodym derivative of nonlinear (random) translation of fractional Brownian motion over finite interval, extending the results of [48] to general case. We obtain an integration by parts formula for general stochastic integral and an Itô type formula for some stochastic integral. The conditioning, Clark derivative, continuity of stochastic integral are also studied. As an application we study a linear quadratic control problem, where the system is driven by fractional Brownian motion.

• Chapters
• 1. Introduction
• 2. Linear stability: the model case
• 3. Pieces of paradifferential calculus
• 4. $L^2$ and conormal estimates near the boundary
• 5. Linear stability
• 6. Nonlinear stability
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Volume: 1752005; 107 pp
MSC: Primary 35;

This paper studies two types of integral transformation associated with fractional Brownian motion. They are applied to construct approximation schemes for fractional Brownian motion by polygonal approximation of standard Brownian motion. This approximation is the best in the sense that it minimizes the mean square error. The rate of convergence for this approximation is obtained. The integral transformations are combined with the idea of probability structure preserving mapping introduced in [48] and are applied to develop a stochastic calculus for fractional Brownian motions of all Hurst parameter $H\in (0, 1)$. In particular we obtain Radon-Nikodym derivative of nonlinear (random) translation of fractional Brownian motion over finite interval, extending the results of [48] to general case. We obtain an integration by parts formula for general stochastic integral and an Itô type formula for some stochastic integral. The conditioning, Clark derivative, continuity of stochastic integral are also studied. As an application we study a linear quadratic control problem, where the system is driven by fractional Brownian motion.

• Chapters
• 1. Introduction
• 2. Linear stability: the model case
• 3. Pieces of paradifferential calculus
• 4. $L^2$ and conormal estimates near the boundary
• 5. Linear stability
• 6. Nonlinear stability
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