Feldman–Hájek theorem

In probability theory, the Feldman–Hájek theorem or Feldman–Hájek dichotomy is a fundamental result in the theory of Gaussian measures. It states that two Gaussian measures ${\displaystyle \mu }$ and ${\displaystyle \nu }$ on a locally convex space ${\displaystyle X}$ are either equivalent measures or else mutually singular:[1] there is no possibility of an intermediate situation in which, for example, ${\displaystyle \mu }$ has a density with respect to ${\displaystyle \nu }$ but not vice versa. In the special case that ${\displaystyle X}$ is a Hilbert space, it is possible to give an explicit description of the circumstances under which ${\displaystyle \mu }$ and ${\displaystyle \nu }$ are equivalent: writing ${\displaystyle m_{\mu }}$ and ${\displaystyle m_{\nu }}$ for the means of ${\displaystyle \mu }$ and ${\displaystyle \nu }$, and ${\displaystyle C_{\mu }}$ and ${\displaystyle C_{\nu }}$ for their covariance operators, equivalence of ${\displaystyle \mu }$ and ${\displaystyle \nu }$ holds if and only if[2]

• ${\displaystyle \mu }$ and ${\displaystyle \nu }$ have the same Cameron–Martin space ${\displaystyle H=C_{\mu }^{1/2}(X)=C_{\nu }^{1/2}(X)}$;
• the difference in their means lies in this common Cameron–Martin space, i.e. ${\displaystyle m_{\mu }-m_{\nu }\in H}$; and
• the operator ${\displaystyle (C_{\mu }^{-1/2}C_{\nu }^{1/2})(C_{\mu }^{-1/2}C_{\nu }^{1/2})^{\ast }-I}$ is a Hilbert–Schmidt operator on ${\displaystyle {\bar {H}}}$.

A simple consequence of the Feldman–Hájek theorem is that dilating a Gaussian measure on an infinite-dimensional Hilbert space ${\displaystyle X}$ (i.e. taking ${\displaystyle C_{\nu }=sC_{\mu }}$ for some scale factor ${\displaystyle s\geq 0}$) always yields two mutually singular Gaussian measures, except for the trivial dilation with ${\displaystyle s=1}$, since ${\displaystyle (s^{2}-1)I}$ is Hilbert–Schmidt only when ${\displaystyle s=1}$.

References

1. Bogachev, Vladimir I. (1998). Gaussian Measures. Mathematical Surveys and Monographs. 62. Providence, RI: American Mathematical Society. doi:10.1090/surv/062. ISBN 0-8218-1054-5. (See Theorem 2.7.2)
2. Da Prato, Giuseppe; Zabczyk, Jerzy (2014). Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and its Applications. 152 (Second ed.). Cambridge: Cambridge University Press. doi:10.1017/CBO9781107295513. ISBN 978-1-107-05584-1. (See Theorem 2.25)