Linear stochastic partial differential equations with constant coefficients (Q1124210)

Consider stochastic partial differential equations with constant coefficients \[ (1)\quad du=A(D)u dt+B(D)u dw \] on \({\mathbb{R}}^ d\) with a scalar Wiener process. If the function \[ H_{\epsilon}(\xi):=2 {\mathfrak R} A(i\xi)-(1-\epsilon)({\mathfrak R} B(i\xi))^ 2+({\mathfrak I} B(i\xi))^ 2\quad (\xi \in {\mathbb{R}}^ d) \] is bounded above on \({\mathbb{R}}^ d\) for some \(\epsilon >0\) and \(u_ 0\) is in the intersection \(H^{\infty}\) of all Sobolev spaces, (1) admits a unique continuous \(H^{\infty}\)-valued solution. This solution is \(L^ 2\)-stable (on finite time-intervals) if and only if \(H_ 2(\xi)\) is bounded above. Finally, approximation of solutions by piecewise linear approximations of w and stability w.r.t. perturbations of the coefficients are considered.