## The Annals of Probability

- Ann. Probab.
- Volume 3, Number 1 (1975), 41-48.

### Functionals of Markov Processes and Superprocesses

#### Abstract

It is well known that a contraction multiplicative functional $\alpha_t, t \geqq 0$ on some Markov process with transition $P_t, t \geqq 0$, yields another Markov process whose semigroup $Q_t(x, A) = E_x(\alpha_t, X_t \in A)$ is subordinate to $P_t, t \geqq 0$. The second process results from the original one by adding a killing operation at a rate of $-d\alpha_t/\alpha_t$. This paper deals with expansion multiplicative functionals (satisfying $\alpha_t \geqq 1$ and $E_x(\alpha_t) < \infty)$. It is proved that such functionals yield a Markov process with creation and annihilation of mass. Relations to the original process are established. Finally the results are generalized to, so-called, conditionally monotone functionals.

#### Article information

**Source**

Ann. Probab., Volume 3, Number 1 (1975), 41-48.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176996446

**Digital Object Identifier**

doi:10.1214/aop/1176996446

**Mathematical Reviews number (MathSciNet)**

MR400411

**Zentralblatt MATH identifier**

0302.60043

**JSTOR**

links.jstor.org

**Keywords**

6062 6067 Expansions multiplicative functionals dominating semigroup Markov processes with creation and annihilation

#### Citation

Leviatan, Talma. Functionals of Markov Processes and Superprocesses. Ann. Probab. 3 (1975), no. 1, 41--48. doi:10.1214/aop/1176996446. https://projecteuclid.org/euclid.aop/1176996446