Noncommutative Mathematics for Quantum Systems

34 Noncommutative Mathematics for Quantum Systems

The continuity is an immediate consequence of the last condition
in Definition 1.5.3.

Lemma 1.5.6 The convolution semigroup of states characterizes a L ́evy
process on an involutive bialgebra up to equivalence.

Proof This follows from the fact that the increment property and
the independence of increments allow to express all joint moments
in terms of the marginals. E.g., for 0≤s≤t≤u≤vanda,b,c∈B,
the momentΦ

(

jsu(a)jst(b)jsv(c)

)

becomes

Φ

(

jsu(a)jst(b)jsv(c)

)

=Φ

(

(jst?jtu)(a)jst(b)(jst?jtu?juv)(c)

)

=Φ

(

jst(a( 1 ))jtu(a( 2 ))jst(b)jst(c( 1 ))jtu(c( 2 ))juv(c( 3 ))

)

=Φ

(

jst(a( 1 )bc( 1 ))jtu(a( 2 )c( 2 ))juv(c( 3 ))

)

=φt−s(a( 1 )bc( 1 ))φu−t(a( 2 )c( 2 ))φv−u(c( 3 )).

It is possible to reconstruct a process(jst) 0 ≤s≤tdirectly from its
convolution semigroup via an inductive limit, see [Sch93, Section
1.9] or [FS99, Section 4.5]. Therefore, we even have a one-to-one
correspondence between equivalence classes of Levy processes on ́
Band convolution semigroups of states onB.
There are other constructions of Levy processes based on the ́
Schurmann triple, see, e.g., [Sch93, Chapter 2], [LS05], and ̈
[SSV10].

1.5.2 The generating functional of a Levy process ́

In this subsection we will meet two more objects that classify Levy ́
processes, namely their generating functionals and their triples
(called Schurmann triples by P.-A. Meyer, see [Mey95, Section ̈
VII.1.6]).
We begin with a technical lemma.

Lemma 1.5.7

(a) Letψ:C →Cbe a linear functional on some coalgebraC. Then the
series

exp?ψ(b)

def

= ∑

n= 0

ψ?n

n!

(b) =ε(b) +ψ(b) +

1

2

ψ?ψ(b) +···

converges for all b∈C.