Infinite dimensional stochastic differential equations of Ornstein–Uhlenbeck type

We consider the operator L f ( x ) = 1 2 ∑ i , j = 1 ∞ a ij ( x ) ∂ 2 f ∂ x i ∂ x j ( x ) - ∑ i = 1 ∞ λ i x i b i ( x ) ∂ 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 stochastic differential equation similar to that for the infinite dimensional Ornstein–Uhlenbeck process.