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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*}