The submartingale problem for a class of degenerate elliptic operators

We consider the degenerate elliptic operator acting on ${C^2_b}$ functions on [0,∞) ^d : $$mathcal{L}f(x)=sum_{i=1}^d a_i(x) x_i^{alpha_i} frac{partial^2 f}{partial x_i^2} (x) +sum_{i=1}^d b_i(x) frac{partial f}{partial x_i}(x), $$ where the a _i are continuous functions that are bounded above and below by positive constants, the b _i are bounded and measurable, and the ${alpha_iin (0,1)}$ . We impose Neumann boundary conditions on the boundary of [0,∞) ^d . There will not be uniqueness for the submartingale problem corresponding to ${mathcal{L}}$ . If we consider, however, only those solutions to the submartingale problem for which the process spends 0 time on the boundary, then existence and uniqueness for the submartingale problem for ${mathcal{L}}$ holds within this class. Our result is equivalent to establishing weak uniqueness for the system of stochastic differential equations $$ {rm d}X_t^i=sqrt{2a_i(X_t)} (X_t^i)^{alpha_i/2}{rm d}W^i_t + b_i(X_t) {rm d}t + {rm d}L_t^{X^i},quad X^i_t geq 0, $$ where ${W_t^i}$ are independent Brownian motions and ${L^{X_i}_t}$ is a local time at 0 for X ⁱ .