Bernsteins theorem for completely excessive measures

Abstract

Bernstein’s theorem states that the following properties are equivalent for a function ψ:]0, ∞ [→ R (which then is called completely monotone):moreover, the measure σ in iii) is uniquely determined.

On temporally completely monotone functions for Markov processes

“…In particular, Itô-Suzuki [13] and Ben Saad-Janßen [1] studied a dual problem, namely the integral representation of completely superharmonic ( [13]) or completely excessive ( [1]) measures. In [13], the method consists, modulo some additional hypotheses, in applying the classical Choquet representation theorem, whereas in [1], the authors use a result of Getoor [10] on the representation of pseudo-kernels as kernels (and the classical Bernstein theorem) to give a simple proof of a general representation theorem of completely excessive measures.…”

“…Our "strategy" to obtain an integral representation of TCM (or completely excessive) functions is: given a completely excessive function, to associate with it a completely excessive measure, and to use the theorem in [1]. For this, we shall need an absolute continuity hypothesis (similarly as in [4]), and, following [24] closely, to prove under this hypothesis the existence of a dual semi-group.…”

“…In particular, we determine the extremal rays of the cone of completely excessive functions. • In Section 3, under an absolute continuity hypothesis, we state a representation theorem of completely excessive functions, which is deduced from the corresponding theorem in [1] for the completely excessive measures. • In Section 4, we express, in a Markovian set-up, the TCM property in terms of complete superharmonicity with respect to the extended generator L of Y .…”

“…a converse of Proposition 2.3. We shall use a theorem due to Ben Saad and Janßen [1] which bears on measures. For this purpose, we need to introduce a reference measure, and a dual semi-group, which will allow to use the functions -measures duality.…”

“…Remark In order to apply the results in [1], we need a further topological hypothesis, e.g: E) is a Polish space, endowed with its Borel σ-field.…”