In the case of white noise, if $\xi$ is deterministic, then
  \begin{align*}
  E\bigg|\iint\_{\phi(U)}\xi(y,s)W(dy,ds)\bigg|^2
    &= E\bigg|\iint\xi(y,s)1_{\phi(U)}(y,s)W(dy,ds)\bigg|^2\\\\
  &= \iint|\xi(y,s)1_{\phi(U)}(y,s)|^2\\,dy\\,ds\\\\
  &= \iint_{\phi(U)}|\xi(y,s)|^2\\,dy\\,ds\\\\
  &= \iint_U |\xi(\phi(x,t))|^2\\,|\det\phi|\\,dx\\,dt\\\\
  &= E\bigg|\iint_U \xi(\phi(x,t))\\,|\det\phi|^{1/2}\\,W(dx,dt)\bigg|^2.
  \end{align*}