## An Innovation Approach to Random Fields: Application of White Noise TheoryA random field is a mathematical model of evolutional fluctuatingcomplex systems parametrized by a multi-dimensional manifold like acurve or a surface. As the parameter varies, the random field carriesmuch 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 elementalstochastic process with a basic and simple way of dependence, and thenexpress the given field as a function of the innovation. Theytherefore establish an infinite-dimensional stochastic calculus, inparticular a stochastic variational calculus. The analysis offunctions of the innovation is essentially infinite-dimensional. Theauthors use not only the theory of functional analysis, but also theirnew tools for the study |

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### Contents

Introduction | 1 |

5 Of course there are many applications where a person can receive | 5 |

and we may say that the book is elemental but not elementary However | 11 |

Poisson Noise | 33 |

Random Fields | 55 |

Gaussian Random Fields | 65 |

Some NonGaussian Random Fields | 85 |

Variational Calculus for Random Fields | 97 |

Innovation Approach | 117 |

### Other editions - View all

An Innovation Approach to Random Fields: Application of White Noise Theory Takeyuki Hida,Si Si Limited preview - 2004 |

An Innovation Approach to Random Fields: Application of White Noise Theory Takeyuki Hida,Si Si Limited preview - 2004 |

### Common terms and phrases

assume assumption bilinear form Brownian bridge Brownian motion canonical kernel canonical representation Chapter characteristic functional compound Poisson noise compound Poisson process conditional expectation consider covariance function d-parameter defined definition denoted dimensional parameter dimensional rotation group discussed Euclidean Example expressed formula func functional derivative Gaussian process Gaussian random field Hence higher dimensional Hilbert space homogeneous chaos infinite dimensional rotation infinitesimal innovation introduced invariant kernel function Let X(C Levy Brownian motion manifold Markov field Markov property martingale multi-dimensional parameter multiple Markov property non-random notation Note nuclear space obtained operator orthogonal ovaloid parameter Poisson noise parameterized polynomials probability distribution Proof Proposition proved random complex systems random field X(C random functions random measure random variables reproducing kernel restriction S-transform sample function Section stochastic process stochastic variational equation subspace Theorem theory topology unique variational calculus vector white noise analysis white noise functionals white noise measure Wick product