A duality analysis on stochastic partial differential equations (Q1188095)

The aim of the paper is to solve the duality equation of the following stochastic partial differential equation: \[ dq(t)=[A(t)q(t)+f(t)]dt+\sum^{d'}_{k=1} [M^ k(t)q(t)+g^ k(t)]dW_ k(t) \] with the initial condition \(q(0)=q_ 0\) where \(W\) is a Brownian motion and \(A(t)=A(t,\omega)\), \(M^ k(t)=M^ k(t,\omega)\) are random differential operators of order 2 and 1, respectively, and \(\omega\in\Omega\). It is shown that the duality equation of the above equation admits a unique solution in a Sobolev space \(L^ 2([0,1]\times\Omega; W^{m+1}_ 2({\mathbf R}^ d))\times L^ 2([0,1]\times\Omega; W^ m_ 2({\mathbf R}^ d))^{d'}\), and a corresponding estimate for the Sobolev norm is derived. The author considers first the case that the \(A(t)\) are nondegenerate parabolic operators. The degenerate case is treated under the additional assumption that the first order parts of the \(M^ k(t)\) are equal to zero.