## Proceedings of the Centre for Mathematics and its Applications

- Proc. Centre Math. Appl.
- Miniconference on Operators in Analysis. Ian Doust, Brian Jefferies, Chun Li, and Alan McIntosh, eds. Proceedings of the Centre for Mathematical Analysis, v. 24. (Canberra AUS: Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University, 1990), 126 - 134

### 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 $x \in \mathbb(R)^d$.

#### Article information

**Dates**

First available in Project Euclid:
18 November 2014

**Permanent link to this document**

https://projecteuclid.org/
euclid.pcma/1416335067

**Mathematical Reviews number (MathSciNet)**

MR1060118

**Zentralblatt MATH identifier**

0704.60078

#### Citation

Jefferies, Brian. The uniqueness of diffusion semigroups. Miniconference on Operators in Analysis, 126--134, Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University, Canberra AUS, 1990. https://projecteuclid.org/euclid.pcma/1416335067