Ciprian A Tudor 
STOCHAST PARTIAL DIFFERENT EQUATION ADDITIVE GAUSSIAN NOISE 
Analysis and Inference

The stochastic partial differential equations (SPDEs) arise in many applications of the probability theory. This monograph will focus on two particular (and probably the most known) equations: the stochastic heat equation and the stochastic wave equation.

The focus is on the relationship between the solutions to the SPDEs and the fractional Brownian motion (and related processes). An important point of the analysis is the study of the asymptotic behavior of the p-variations of the solutions to the heat or wave equations driven by space-time Gaussian noise or by a Gaussian noise with a non-trivial correlation in space.

The book is addressed to public with a reasonable background in probability theory. The idea is to keep it self-contained and avoid using of complex techniques. We also chose to insist on the basic properties of the random noise and to detail the construction of the Wiener integration with respect to them. The intention is to present the proofs complete and detailed.

Contents:

• Gaussian Processes and Sheets:
  
  
  
  • Gaussian Processes
  
  
  • Fractional and Bifractional Brownian Motion
  
  
  • Multiparameter Gaussian Processes
  
  
  • Isonormal Processes and Wiener Integral
  
  


• Stochastic Heat and Wave Equations with Additive Gaussian Noise:
  
  
  
  • The Stochastic Heat Equation with Space-Time White Noise
  
  
  • The Stochastic Heat Equation with Correlated Noise in Space
  
  
  • The Stochastic Wave Equation with Space-Time White Noise
  
  


• Power Variation and Statistical Inference for Solutions to SPDEs:
  
  
  
  • Variations of the Solution to the Stochastic Heat Equation
  
  
  • Parameter Estimation for the Stochastic Heat Equation via Power Variations
  
  
  • Power Variations and Inference for Stochastic the Wave Equation
  
  

Readership: Undergraduate, graduate students and researchers working in various areas of probability theory and mathematical statistics.

Key Features:

• The book is self-contained and it is accessible to graduate and undergraduate students and even to professionals


• It contains a quasi-complete description of some multiparameter Gaussian processes


• Several aspects in this book have not been treated before in any monograph: the full description of the relationships between solution to SPDEs and fractional processes, the limit behavior of the p-variation for the solutions to stochastic heat and wave equations, the statistical inference for SPDEs