Harmonic Analysis of Local Times and Sample Functions of Gaussian Processes
Simeon M. Berman · Transactions of the American Mathematical Society · 1969
Septemberthen / is representable as (1.1) f(u) = j exp (iux(t)) dp(t)because x=x(0 is a measure-preserving transformation on (/, p.) to (F, v).If fis square integrable, then v is absolutely continuous and the local time is square integrable (cf.[2]).We denote the latter by B, where F is a Borel subset of /: it is the local time of the restriction of x to B. It is a consequence of the definition of the Radon-Nikodym derivative that (x) for almost all x; hence, B is square integrable if is.If satisfies the equation (1.2), then (pB satisfies a similar equation with F in place of / as the domain of integration.As a derivative with respect to Borel measure, the local time is Borel measurable; thus, from the definition of the relative local time it follows that if is square integrable, then, by (1.2) (1.3) f <pB(x(t)) dp.(t) = P <pB(x)