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.
Categories: math.PR
