Representation and construction of multiplicative noise
The authors study noise, i.e. random fields with independent values at distinct points, taking values in a topological semigroup. More precisely, let \(F\subset {\mathcal P}(M)\) be a ring, M any set, and let \(\Omega\) be a probability space. If S is a topological monoid (locally compact and separable) we let \(\xi: F\to RV(\Omega,S)\), the set of all random variables taking values in S, be an (F,S) noise if (i) \(A,B\in F,\) \(A\cap B=\emptyset \Rightarrow \xi (A),\xi (B)\) independent, (ii) \(A\cap B=\emptyset \Rightarrow \xi (A\cup B)=\xi (A)\xi (B)\) in distribution, (iii) \(\xi (\emptyset)=e,\) (iv) \(\xi\) continuous at \(\emptyset.\) First it is proved that (F,S) noise is equivalent to an (F,S) convolution semigroup, the latter being a family \((P_ A\), \(A\in F)\) satisfying \(P_{A\cup B}=P_ A*P_ B\) if \(A\cap B=\emptyset\), \(P_{\emptyset}=\delta_ e\) and \(P_ A\) continuous. The remaining part of the paper consists of a discussion of noise in various special situations. If S is assumed in addition to be an abelian group, one can give a complete characterization in terms of a Lévy-Khinchin type formula. If one furthermore assumes that \(\xi\) is invariant in the sense that \(\xi\) (A) and \(\xi\) (TA) are equal in distribution for all \(A\in F\) and all T in a family \(\Lambda\) of measurable maps T:M\(\to M\), one can prove that the semigroup \(P_ A\) is given by \(P_ A=P_{v(A)}\) where v(A) is the Haar measure and \(\Lambda\) is the family of all right translations. In the final section the authors study the case when the noise takes values in a Lie algebra. Using the exponential map one constructs noise with values in the Lie group.
scientific article; zbMATH DE number 4140906
scientific article