We consider the operator

Lf(x) = \frac{1}{2} ∑_{i,j=1}^{∞} a_{ij}(x)\frac{∂^{2}f}{∂x_{i}∂x_{j}} (x) − ∑_{i=1}^{∞} λ_{i}x_{i}b_{i}(x)\frac{∂f}{∂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 λ_{i}. The process corresponding to L solves an infinite-dimensional stochasticdifferential equation similar to that for the infinite-dimensional Ornstein-Uhlenbeck process.