The Martingale Problem for a Differential Operator with Piecewise Continuous Coefficients

Abstract

Let L be a linear second order elliptic differential operator defined by

$$Lu = \sum\limits_{{i,j = 1}}^{n} {{{a}_{{i,j}}}} (x)\frac{{{{\partial }^{2}}u}}{{\partial {{x}_{i}}\partial {{x}_{j}}}} + \sum\limits_{{j = 1}}^{n} {{{b}_{j}}(x)\frac{{\partial u}}{{\partial {{x}_{j}}}}} $$

(1)

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,…,y_n), and every $x \in \mathbb{R}^n$

$$ \mu \sum\limits_{i = 1}^n {y_i^2 } \leqslant \sum\limits_{i,j = 1}^n {a_{ij} (x)y_i y_j \leqslant \nu \sum\limits_{i = 1}^n {y_i^2 .} } $$

(2)