Independence and L ́evy Processes in Quantum Probability 25
Proposition 1.4.5 If(Xst)is a L ́evy process with values in a topological
semigroup G, then its marginal distributionsμt=PX 0 tform a continuous
convolution semigroup.
Exercise 1.4.6 Prove this Proposition.
Conversely, given a continuous convolution semigroup(μt)t≥ 0
of probability measures on a topological semigroupG, one can
construct a Levy process with values in ́ Gwhose marginals are
equal to the convolution semigroup(μt)t≥ 0.
1.4.4 The De Finetti–Levy–Khintchine formula on ́ (R+,+)
Let us start with a description of infinitely divisible probability
measures on the semigroup(R+,+).
Theorem 1.4.7 A probability measureμonR+is infinitely divisible
if and only if there exist b≥ 0 andνa measure onR+with
∫∞
0 1 ∧
xdν(x)<∞such that the Laplace transform
ψμ(λ) =
∫∞
0
e−λxdμ(x)
ofμhas the form
ψμ(λ) =exp
(
Φ(λ)
)
for allλ≥ 0 , with
Φ(λ) =bλ+
∫∞
0
( 1 −e−λx)dν(x).
The pair(b,ν)is uniquely determined byμ.
Proof See [Ber98, p. 72].
Levy processes with values in ́ (R+,+)have increasing trajectories,
they calledsubordinators. The pair(b,ν)is called thecharacteristics
or thecharacteristic pair ofμ.
Corollary 1.4.8 Every infinitely divisible probability measure onR+is
embeddable into a continuous convolution semigroup.
1.4.5 Levy–Khintchine formulae on cones ́
We also have the following generalization for proper closed cones
in finite-dimensional vector spaces.