Infinite dimensional stochastic differential equations of Ornstein–Uhlenbeck type

We consider the operatorLf(x)=12∑i,j=1∞aij(x)∂2f∂xi∂xj(x)-∑i=1∞λixibi(x)∂f∂xi(x).We prove existence and uniqueness of solutions to the martingale problem for this operator under appropriate conditions on the aij,biaij,bi, and λiλi. The process corresponding to LL solves an infinite dimensional stochastic differential equation similar to that for the infinite dimensional Ornstein–Uhlenbeck process.