Noncommutative Mathematics for Quantum Systems

Independence and L ́evy Processes in Quantum Probability 23

1.4 Infinite Divisibility in Classical Probability

Before we study quantum stochastic processes with independent
and stationary increments, that is, quantum Levy processes, let us ́
first recall some definitions and facts about infinite divisibility and
Levy processes in classical probability. See also [Sko91, Ber98, Sat99, ́
App04, App05, Kyp07].

1.4.1 Stochastic independence

Recall that two random variablesX 1 : (Ω,F,P) → (E 1 ,E 1 )and
X 2 :(Ω,F,P) → (E 2 ,E 2 )are calledindependentif their joint law
P(X 1 ,X 2 )is equal to the product of their marginal laws, that is,

P(X 1 ,X 2 )=PX 1 ⊗PX 2.

This means that

P(X 1 ∈A 1 ,X 2 ∈A 2 ) =P(X 1 ∈A 1 )P(X 2 ∈A 2 )

for allA 1 ∈E 1 ,A 2 ∈E 2.

1.4.2 Convolution

LetG be a topological semigroup with neutral element e and
multiplicationm:G×G→G. ThenGis a measurable space with
Borel-σ-algebraB(G). We can define theconvolution productμ 1 ?μ 2
of two probability measuresμ 1 ,μ 2 defined on (G,B(G))as the
image measurem∗(μ 1 ⊗μ 2 )of their productμ 1 ⊗μ 2 , that is,

(μ 1 ?μ 2 )(A) = (μ 1 ⊗μ 2 )({(g 1 ,g 2 )∈G×G;g 1 g 2 ∈A})

forA∈B(G).
IfX 1 ,X 2 :(Ω,F,P)→(G,B(G))are two independent random
variables with distributionsPX 1 =μ 1 ,PX 2 =μ 2 , then their product
has distribution

PX 1 X 2 =m∗(PX 1 ⊗PX 2 ) =PX 1 ?PX 2 =μ 1 ?μ 2.

1.4.3 Infinite divisibility, continuous convolution semigroups,
and Levy processes ́

Definition 1.4.1 A probability measure μ on a topological
semigroupGis calledinfinitely divisible, if for every integern≥ 1
there exists a probability measureμnsuch that