Noncommutative Mathematics for Quantum Systems

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.