by Takeyuki Hida (Meijo University, Japan) & Si Si (Aichi Prefectural University, Japan)

A random field is a mathematical model of evolutional fluctuating complex systems parametrized by a multi-dimensional manifold like a curve or a surface. As the parameter varies, the random field carries much information and hence it has complex stochastic structure.

The authors of this book use an approach that is characteristic: namely, they first construct innovation, which is the most elemental stochastic process with a basic and simple way of dependence, and then express the given field as a function of the innovation. They therefore establish an infinite-dimensional stochastic calculus, in particular a stochastic variational calculus. The analysis of functions of the innovation is essentially infinite-dimensional. The authors use not only the theory of functional analysis, but also their new tools for the study.

• Introduction
• White Noise
• Poisson Noise
• Random Fields
• Gaussian Random Fields
• Some Non-Gaussian Random Fields
• Variational Calculus for Random Fields
• Innovation Approach
• Reversibility
• Applications