The Martingale Problem for a Differential Operator with Piecewise Continuous Coefficients
Ping Gao · Birkhäuser Boston eBooks · 1993
Let L be a linear second order elliptic differential operator defined by 1 $$Lu = \sum\limits_{{i,j = 1}}^{n} {{{a}_{{i,j}}}} (x)\frac{{{{\partial }^{2}}u}}{{\partial {{x}_{i}}\partial {{x}_{j}}}} + \sum\limits_{{j = 1}}^{n} {{{b}_{j}}(x)\frac{{\partial u}}{{\partial {{x}_{j}}}}} $$ with b=(b j),a=(a ij) bounded and measurable, and a symmetric. Suppose a is uniformly positive definite, i.e. there exist positive numbers µand v such that for all y=(y 1,…,yn), and every $$ x \in \mathbb{R}^n $$ 2 $$ \mu \sum\limits_{i = 1}^n {y_i^2 } \leqslant \sum\limits_{i,j = 1}^n {a_{ij} (x)y_i y_j \leqslant u \sum\limits_{i = 1}^n {y_i^2 .} } $$