Noncommutative Mathematics for Quantum Systems

102 Noncommutative Mathematics for Quantum Systems

(i) Increment property: We have

jrs?jst = jrt for all 0≤r≤s≤t≤∞,

jtt = ε (^1) A for all 0≤t≤∞.
(ii) Independence of increments: The family {jst} 0 ≤s≤t≤∞ is
tensor-independent (resp. boolean, monotonically,
anti-monotonically independent, or free) w.r.t.Φ, that is, the
n-tuple(js 1 t 2 ,... ,jsntn)is tensor-independent (resp. boolean,
monotonically, anti-monotonically independent, or free) for
alln∈Nand all 0≤s 1 ≤t 1 ≤s 2 ≤···≤tn≤∞.
(iii) Stationarity of increments: The distributionφst=Φ◦jstofjst
depends only on the differencet−s.
(iv) Weak continuity: The quantum random variablesjstconverge
tojssin distribution fort↘s.
Remark 1.9.4 The independence property depends on the
products and therefore for boolean, monotone, and anti-monotone
Levy processes on the choice of a decomposition ́ B=C 1 ⊕B^0. In
order to show that the convolutions defined by (φ 1 φ 2 )◦∆,
(φ 1 .φ 2 )◦∆, and(φ 1 /φ 2 )◦∆are associative and that the counitε
acts as unit element w.r.t. these convolutions, one has to use the
universal property [BGS99, Condition (P4)], which in our setting is
only satisfied for morphisms that respect the decomposition.
Therefore, we are forced to choose the decomposition given by
B^0 =kerε.
The marginal distributionsφt−s := φst = Φ◦jstform again a
convolution semigroup {φt}t∈R+, with respect to the tensor
(boolean, monotone, anti-monotone, or free, respectively)
convolution defined by(φ 1 ⊗ ̃φ 2 )◦∆((φ 1 φ 2 )◦∆,(φ 1 .φ 2 )◦∆,
(φ 1 /φ 2 )◦∆, or(φ 1 ∗φ 2 )◦∆, respectively). It has been shown that
the generating functionalψ:B →C,
ψ(b) =lim
t↘ 0
1
t
(
φt(b)−ε(b)
)
is well defined for all b ∈ B and uniquely characterizes the
semigroup{φt}t∈R+,cf.[Sch95b, BGS99, Fra01].
Schurmann and Voss [SV14] have given a new proof of the ̈
Schoenberg correspondence, using ideas from [SSV10], that