On the Carleman inequality for the stochastic parabolic equations with multiplicative noise

Abstract

The stochastic parabolic equation

\centerline{$d_tX-\D X\,dt+aX\,dt+(b\cdot\na
X)dt =\dd\sum^N_{k=1}X\mu_k\,d\b_k,$}

\noindent in $(0,T)\times\calo$ with Dirichlet homogeneous
boundary conditions is\break\mbox{exactly} observable on a domain $(0,T)\times\calo_0$ via Carleman type inequality.   Here
$\calo_0\subset\calo\subset\rr^d$ are open, bounded sub\-sets,
$\{\b_k\}^N_{k=1}$ are linearly independent Brownian motions in a
   pro\-ba\-bi\-lity space $\{\ooo,\P,\calf\}$, $a=a(t,\xi),$ $b=b(t,\xi)$,
   $\mu=\mu(t,\xi)$  are given con\-ti\-nuous functions on $[0,T]\times\ov\calo$. Here, we give a simple proof to this well known result via rescaling process which reduces  the equation to a random parabolic equation.