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.