Radon-Nikodym Derivatives of Gaussian Measures

Abstract

We give simple necessary and sufficient conditions on the mean and covariance for a Gaussian measure to be equivalent to Wiener measure. This was formerly an unsolved problem [26]. An unsolved problem is to obtain the Radom-Nikodym derivative $d\mu/du$ where $\mu$ and $u$ are equivalent Gaussian measure [28]. We solve this problem for many cases of $\mu$ and $u$, by writing $d\mu/du$ in terms of Fredholm determinants and resolvents. The problem is thereby reduced to the calculation of these classical quantities, and explicit formulas can often be given. Our method uses Wiener measure $\mu_w$ as a catalyst; that is, we compute derivatives with respect to $\mu_w$ and then use the chain rule: $d\mu/du = (d\mu/d\mu_w)/(du/d\mu_w)$. Wiener measure is singled out because it has a simple distinctive property--the Wiener process has a random Fourier-type expansion in the integrals of any complete orangeade system. We show that any process equivalent to the Wiener process $W$ can be realized by a linear transformation of $W$. This transformation necessarily involves stochastic integration and generalizes earlier nodulation transformations studied by Legal [21] and others [4], [27]. New variants of the Wiener process are introduced, both conditioned Wiener processes and free $n$-fold integrated Wiener processes. We given necessary and sufficient conditions for a Gaussian process to be equivalent to any one of the variants and also give the corresponding Radon-Niels (R-N) derivative. Last, some novel uses of R-N derivatives are given. We calculate explicitly: (i) the probability that $W$ cross a slanted line in a finite time, (ii) the first passage probability for the process $W(T + 1) - W(t)$, and (iii) a class of function space integrals. Using (iii) we prove a zero-one law for convergence of certain integrals on Wiener paths.