[For the entire collection see Zbl 0552.00009.]
Let \(p=p(t,x,\omega,D)\), \(q_ i=q_ i(t,x,\omega,D)\), \(i=1,...,k\), be stochastic partial differential operators of constant order (respectively 2m and \(m_ i\leq m)\) and for an Itô process with values in \({\mathcal E}(O)\), i.e. the nuclear Frechet space of \(C^{\infty}\)-functions on the open domain O, we use the following (stochastic) differential notation: \(du_ t=D_+u_ t\) \(dt+\partial_{w^ i}u_ tdW^ 1_ d\). For any bounded stopping time T and E(O)-valued integrable Itô processes u and v, \(B_ T(u,v)\) denotes the following bilinear form: \[ B_ T(u,v)=E\int^{T}_{0}((D_++p_ s)u_ s,v_ s)_ 0ds+(1/2)E\int^{T}_{\quad 0}\sum_{i\leq k}(\partial_{w^ i}u_ s,\partial_{w^ i}v_ s)_ 0ds. \] Then we have the following proposition: Suppose that \((X_ t)\) is any D’(O)-valued (not necessarily integrable) solution of the following stochastic partial differential equation: \[ dX_ t=(-p+1/2(-1)^{m_ i+1}q^ 2_ i)X_ tdt+q_ iX_ tdW^ i_ t+dH\quad_ t \] where H is any Itô process with values in E(O). Suppose that \((p,q_ i)_ i\) satisfy the following condition: For any bounded stopping time T, there exists \(s>0\) such that for any KccO (i.e., compact), for any integrable Itô process u with values in \(D_ K(O)\) (i.e., the elements of D(O) whose supports are in K) we have \[ E\int^{T}_{0}\| u_ r\|^ 2_{m+s-1}dr\leq c[B_ T(u,u)+c'E\int^{T}_{0}\| u_ r\|^ 2_ 0dr \] where c and c’ are the constants depending only on T and K. Then any D’(O)-valued solution of the above equation has a modification which is an E(O)-valued semimartingale.