The uniqueness of diffusion semigroups

Abstract

It is shown that if the highest order co-efficients of a uniformly elliptic second order differential operator $L$ on $\mathbb{R}^d$ are bounded and Hölder continuous, and the other coefficients are bounded and measurable, then there is at most one semigroup $S$ acting on bounded Borel measurable functions, such that $S$ is given by a transition function, and for all smooth functons $f$ with compact support in $\mathbb{R}^d, S(t)f(x) + \int_0^t S(s)Lf(x) ds$ for all $t > 0$ and $x \in \mathbb(R)^d$.