Noncommutative Mathematics for Quantum Systems

24 Noncommutative Mathematics for Quantum Systems

μ = μn?···?μn

︸ ︷︷ ︸

.

ntimes

Definition 1.4.2A family (μt)t≥ 0 of probability measures on a
topological semigroup is called acontinuous convolution semigroup
(ccs)if

(i) limt↘ 0 μt = δeweakly, that is, limt↘ 0

∫

Gfμt = f(e)for all

f∈Cb(G).

(ii)μs?μt=μs+tfor alls,t≥0.

Definition 1.4.3A probability measureμis calledembeddable into a
continuous convolution semigroup if there exists a continuous
convolution semigroup(μt)t≥ 0 such thatμ=μ 1.

Clearly, a probability measure that is embeddable into a
continuous convolution semigroup is also infinitely divisible. On
many groups, e.g.,(Rd,+)the converse is also true, but there exist
also groups where the converse does not hold.
Recall that astochastic process(Xı)ı∈Iis simply a family of random
variables indexed by some setI.

Definition 1.4.4A stochastic process(Xst) 0 ≤s≤twith values in a
topological semigroup is called a (right)L ́evy process, if

(i) (increment property)Xss = eandXstXtu = Xsua.s. for all

0 ≤s≤t≤u;

(ii) (independence) the increments Xs 1 t 1 ,... ,Xsntn are

independent for alln≥1 and alls 1 ≤t 1 ≤s 2 ≤···≤tn;

(iii) (stationarity)PXst =PXs+h,t+hfor allh>0 and all 0≤s≤t,

that is, the law ofXstdepends only ont−s;

(iv) (weak continuity)(Xst) 0 ≤s≤tis stochastically continuous, that

is, we haveXst

t↘s

−→Xssin probability.

We defineXt=X 0 t. IfGis a group, then the increments can be
recovered from(Xt)t≥ 0 byXst=X− 0 s^1 X 0 t.
A stochastic process(Xt)t≥ 0 indexed byR+and with values in
a group is called a Levy process, if its increment process ́ (Xst) 0 ≤s≤t
withXst=X−s^1 Xtis a L ́evy process in the sense of Definition 1.4.4.