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 |
Common terms and phrases
assume bilinear form Brownian motion C₁ canonical kernel canonical representation Chapter characteristic functional compound Poisson noise conditional expectation defined definition denoted diffeomorphism dimensional parameter dimensional rotation group discussed Example expressed formula func functional of Poisson Gaussian process Gaussian random field given Hence Hilbert space homogeneous chaos infinite dimensional rotation innovation integral introduced invariant kernel function L²(Rd Laplacian Lévy Brownian motion Lévy process Lévy's linear lower dimensional manifold Markov property martingale multi-dimensional parameter multiple Markov property Note nuclear space obtained operator ovaloid parameter white noise parameterized polynomials probability distribution Proof Proposition proved random complex systems random field X(C random measure random variables Rd-parameter reproducing kernel restriction rotation group S-transform sample function Section stochastic process stochastic variational equation subspace Theorem theory topology variational calculus vector white noise analysis white noise functionals white noise measure Wick product