Skip to main content
MathOverflow

Return to Answer

Replaced double-backslash-comma with single-backslash-comma, to fix rendering.
Source Link
edit approved Feb 15, 2018 at 23:03
jeq
edit approved Feb 15, 2018 at 23:03
jeq
  • 1.2k
  • 5
  • 16
  • 21

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*}\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*}

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*}

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*}

Source Link
Jason Swanson
Jason Swanson
  • 363
  • 3
  • 10

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*}