Let \(G\) be a bounded domain \(\mathbb R^n\) with a \(C^2\)-boundary \(\Gamma\) and \(b(x)\) a symmetric, uniformly elliptic matrix-valued function on \(G\). A certain constraint on \(b\) is assumed as well as several other technical conditions. The author considers an equation \[ dw_t-(b^{ij}w_i)_j\,dt=F(w,w_t,\nabla w)\,dt+K(w)\,dB(t) \] on \(G\) with the homogeneous Dirichlet boundary condition on \(\Gamma\), where \(B\) is a standard Wiener process and \(F\), \(K\) are nonlinear functions satisfying certain growth and Lipschitz continuity conditions. It is assumed that this equation has a unique global solution in \(L^2(\Omega,C([0,T];H^1_0(G)))\cap L^2(\Omega,C^1([0,T];L^2(G)))\) for every initial condition \((w(0),w_t(0))=(w_0,w_1)\) in \(L^2(\Omega,\mathcal F_0,P;H^1_0(G)\times L^2(G))\).
A boundary and an internal apriori observability estimates are proved yielding, being applied to the equation in question, an estimation on \[ |(w(0),w_t(0))-(\hat w(0),\hat w_t(0))|_{L^2(\Omega,\mathcal F_0,P;H^1_0(G)\times L^2(G))} \] that is proved to be dominated by \[ C|\partial_\nu w-\partial_\nu\hat w|_{L^2(\Omega\times(0,T);L^2(\Gamma_0))} \] as a consequence of the boundary apriori observability estimate, as well as by \[ C|\nabla w-\nabla\hat w|_{L^2(\Omega\times(0,T);L^2(O_\delta(\Gamma_0)))} \] which is due to the internal apriori observability estimate. Here \(w\) and \(\hat w\) are arbitrary solutions of the above equation, \(\nu\) denotes the outer normal to the domain for the equation, \(\Gamma_0\) is a suitable part of the boundary \(\Gamma\) and \(O_\delta(\Gamma_0)\) is a closed \(\delta\)-neighbourhood of \(\Gamma_0\) intersected with \(G\) (i.e. a subset of \(G\)).