Transformation of measures in infinite-dimensional spaces by the~flow induced by a~stochastic differential equation

A. Yu. Pilipenko

Sbornik. Mathematics, Tome 194 (2003) no. 4, pp. 551-573

Résumé

Let $\mu$ be a Gaussian measure in the space $X$ and $H$ the Cameron–Martin space of the measure $\mu$. Consider the stochastic differential equation \begin{gather*} d\xi(u,t)=a_t(\xi(u,t))\,dt+\sum_n\sigma^n_t(\xi(u,t))\,d\omega_n(t), \quad t\in[0,T], \\ \xi(u,0)=u, \end{gather*} where $u\in X$, $a$ and $\sigma_n$ are functions taking values in $H$, $\omega_n(t)$, $n\geqslant 1$ are independent one-dimensional Wiener processes. Consider the measure-valued random process $\mu_t:=\mu\circ\xi(\,\cdot\,,t)^{-1}$. It is shown that under certain natural conditions on the coefficients of the initial equation the measures $\mu_t(\omega)$ are equivalent to $\mu$ for almost all $\omega$. Explicit expressions for their Radon–Nikodym densities are obtained.