Infinite dimensional stochastic differential equations of   Ornstein-Uhlenbeck type
  We consider the operator $$\sL f(x)=\tfrac12 \sum_{i,j=1}^\infty a_{ij}(x)\frac{\del^2 f}{\del x_i \del x_j}(x)-\sum_{i=1}^\infty \lam_i x_i b_i(x) \frac{\del f}{\del x_i}(x).$$ We prove existence and uniqueness of solutions to the martingale problem for this operator under appropriate conditions on the $a_{ij}, b_i$, and $\lam_i$. The process corresponding to $\sL$ solves an infinite dimensional stochastic differential equation similar to that for the infinite dimensional Ornstein-Uhlenbeck process. 
Siva R. Athreya, Richard F. Bass, Maria Gordina, Edwin A. Perkins