Uniqueness

Abstract

We are going to show in this chapter that if a: [0, ∞) × R^d → S _d and b: [0, ∞) × R^d→ R^d are bounded measurable functions and if a is uniformly positive definite on compacts and satisfies:

$$ \mathop {\lim {\text{ }}}\limits_{y \to x{\text{ }}} \mathop {\sup }\limits_{0 \leqslant s \leqslant T} ||a(s,y) - a(s,x)|| = 0 $$

((0.1))

for all T > 0 and x ∈ R^d, then the martingale problem for a and b is well-posed and the associated family of solutions {P_s,x: (s, x) ∈ [0, ∞) × R^d} is (strongly) Feller continuous. Most of the work involved is devoted to proving this result for the situation in which b ≡ 0 and a is very nearly independent of x. Once this case has been thoroughly understood, the general case follows quite easily with the aid of the reduction procedures developed in Chapter 6. Since the details may obscure the idea, we now outline the reasoning behind our analysis.