The submartingale problem for a class of degenerate elliptic operators

Abstract

We consider the degenerate elliptic operator acting on $C^2$ functions on $[0,\infty {)}^d$: \[ L f(x)=\sum_{i=1}^d a_i(x) x_i^{α_i} \frac{\partial^2 f}{\partial x_i^2} (x) +\sum_{i=1}^d b_i(x) \frac{\partial f}{\partial x_i}(x), \] 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 ${\alpha }_i\in (0,1)$. We impose Neumann boundary conditions on the boundary of $[0,\infty {)}^d$. There will not be uniqueness for the submartingale problem corresponding to $L$. 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 $L$ holds within this class. Our result is equivalent to establishing weak uniqueness for the system of stochastic differential equations \[ dX_t^i=\sqrt{2a_i(X_t)} (X_t^i)^{α_i/2} dW^i_t+b_i(X_t) dt +dL_t^{X^i}, where X^i_t\geq 0, \] where $W_t^i$ are independent Brownian motions and $L_t^{X_i}$ is a local time at 0 for $X^i$.