Abstract:
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.
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