Ebook: Probability and Partial Differential Equations in Modern Applied Mathematics

"Probability and Partial Differential Equations in Modern Applied Mathematics" is devoted to the role of probabilistic methods in modern applied mathematics from the perspectives of both a tool for analysis and as a tool in modeling. There is a recognition in the applied mathematics research community that stochastic methods are playing an increasingly prominent role in the formulation and analysis of diverse problems of contemporary interest in the sciences and engineering. A probabilistic representation of solutions to partial differential equations that arise as deterministic models allows one to exploit the power of stochastic calculus and probabilistic limit theory in the analysis of deterministic problems, as well as to offer new perspectives on the phenomena for modeling purposes. There is also a growing appreciation of the role for the inclusion of stochastic effects in the modeling of complex systems. This has led to interesting new mathematical problems at the interface of probability, dynamical systems, numerical analysis, and partial differential equations.

This volume will be useful to researchers and graduate students interested in probabilistic methods, dynamical systems approaches and numerical analysis for mathematical modeling in the sciences and engineering.

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"Probability and Partial Differential Equations in Modern Applied Mathematics" is devoted to the role of probabilistic methods in modern applied mathematics from the perspectives of both a tool for analysis and as a tool in modeling. There is a recognition in the applied mathematics research community that stochastic methods are playing an increasingly prominent role in the formulation and analysis of diverse problems of contemporary interest in the sciences and engineering. A probabilistic representation of solutions to partial differential equations that arise as deterministic models allows one to exploit the power of stochastic calculus and probabilistic limit theory in the analysis of deterministic problems, as well as to offer new perspectives on the phenomena for modeling purposes. There is also a growing appreciation of the role for the inclusion of stochastic effects in the modeling of complex systems. This has led to interesting new mathematical problems at the interface of probability, dynamical systems, numerical analysis, and partial differential equations.

This volume will be useful to researchers and graduate students interested in probabilistic methods, dynamical systems approaches and numerical analysis for mathematical modeling in the sciences and engineering.

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"Probability and Partial Differential Equations in Modern Applied Mathematics" is devoted to the role of probabilistic methods in modern applied mathematics from the perspectives of both a tool for analysis and as a tool in modeling. There is a recognition in the applied mathematics research community that stochastic methods are playing an increasingly prominent role in the formulation and analysis of diverse problems of contemporary interest in the sciences and engineering. A probabilistic representation of solutions to partial differential equations that arise as deterministic models allows one to exploit the power of stochastic calculus and probabilistic limit theory in the analysis of deterministic problems, as well as to offer new perspectives on the phenomena for modeling purposes. There is also a growing appreciation of the role for the inclusion of stochastic effects in the modeling of complex systems. This has led to interesting new mathematical problems at the interface of probability, dynamical systems, numerical analysis, and partial differential equations.

This volume will be useful to researchers and graduate students interested in probabilistic methods, dynamical systems approaches and numerical analysis for mathematical modeling in the sciences and engineering.

Content:
Front Matter....Pages i-x
Nonnegative Markov Chains with Applications....Pages 1-9
Phase Changes with Time and Multi-Scale Homogenizations of a Class of Anomalous Diffusions....Pages 11-26
Semi-Markov Cascade Representations of Local Solutions to 3-D Incompressible Navier-Stokes....Pages 27-40
Amplitude Equations for Spdes: Approximate Centre Manifolds and Invariant Measures....Pages 41-59
Enstrophy and Ergodicity Of Gravity Currents....Pages 61-77
Stochastic Heat and Burgers Equations and Their Singularities....Pages 79-95
A Gentle Introduction to Cluster Expansions....Pages 97-115
Continuity of the It?-Map for Holder Rough Paths with Applications to the Support Theorem in Holder Norm....Pages 117-135
Data-Driven Stochastic Processes in Fully Developed Turbulence....Pages 137-150
Stochastic Flows on the Circle....Pages 151-162
Path Integration: Connecting Pure Jump and Wiener Processes....Pages 163-180
Random Dynamical Systems in Economics....Pages 181-195
A Geometric Cascade for the Spectral Approximation of the Navier-Stokes Equations....Pages 197-212
Inertial Manifolds for Random Differential Equations....Pages 213-236
Existence and Uniqueness of Classical, Nonnegative, Smooth Solutions of a Class of Semi-Linear Spdes....Pages 237-246
Nonlinear Pde’s Driven by L?vy Diffusions and Related Statistical Issues....Pages 247-258
Back Matter....Pages 259-262

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"Probability and Partial Differential Equations in Modern Applied Mathematics" is devoted to the role of probabilistic methods in modern applied mathematics from the perspectives of both a tool for analysis and as a tool in modeling. There is a recognition in the applied mathematics research community that stochastic methods are playing an increasingly prominent role in the formulation and analysis of diverse problems of contemporary interest in the sciences and engineering. A probabilistic representation of solutions to partial differential equations that arise as deterministic models allows one to exploit the power of stochastic calculus and probabilistic limit theory in the analysis of deterministic problems, as well as to offer new perspectives on the phenomena for modeling purposes. There is also a growing appreciation of the role for the inclusion of stochastic effects in the modeling of complex systems. This has led to interesting new mathematical problems at the interface of probability, dynamical systems, numerical analysis, and partial differential equations.

This volume will be useful to researchers and graduate students interested in probabilistic methods, dynamical systems approaches and numerical analysis for mathematical modeling in the sciences and engineering.

Content:
Front Matter....Pages i-x
Nonnegative Markov Chains with Applications....Pages 1-9
Phase Changes with Time and Multi-Scale Homogenizations of a Class of Anomalous Diffusions....Pages 11-26
Semi-Markov Cascade Representations of Local Solutions to 3-D Incompressible Navier-Stokes....Pages 27-40
Amplitude Equations for Spdes: Approximate Centre Manifolds and Invariant Measures....Pages 41-59
Enstrophy and Ergodicity Of Gravity Currents....Pages 61-77
Stochastic Heat and Burgers Equations and Their Singularities....Pages 79-95
A Gentle Introduction to Cluster Expansions....Pages 97-115
Continuity of the It?-Map for Holder Rough Paths with Applications to the Support Theorem in Holder Norm....Pages 117-135
Data-Driven Stochastic Processes in Fully Developed Turbulence....Pages 137-150
Stochastic Flows on the Circle....Pages 151-162
Path Integration: Connecting Pure Jump and Wiener Processes....Pages 163-180
Random Dynamical Systems in Economics....Pages 181-195
A Geometric Cascade for the Spectral Approximation of the Navier-Stokes Equations....Pages 197-212
Inertial Manifolds for Random Differential Equations....Pages 213-236
Existence and Uniqueness of Classical, Nonnegative, Smooth Solutions of a Class of Semi-Linear Spdes....Pages 237-246
Nonlinear Pde’s Driven by L?vy Diffusions and Related Statistical Issues....Pages 247-258
Back Matter....Pages 259-262
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