Criteria for Recurrence and Existence of Invariant Measures for Multidimensional Diffusions

Let $L = \frac{1}{2} \sum^k_{i,j=1} a_{ij}(x)(\partial^2/\partial x_i \partial x_j) + \sum^k_{i=1} b_i(x)(\partial/\partial x_i)$ be an elliptic operator such that $a_{ij}(\bullet)$ are continuous and $b_i(\bullet)$ are measurable and bounded on compacts. Criteria for transience, null recurrence, and positive recurrence of diffusions on $R^k$ governed by $L$ are derived in terms of the coefficients of $L$.