Independence and L ́evy Processes in Quantum Probability 271.4.7 The Markov semigroup of a Levy process ́
Markov processes are thought of as memoryless. Roughly
speaking, a stochastic processes (Xt)t≥ 0 indexed by R+ is a
Markov process if for anytthe past and the future with respect to
tare independent conditionally onXt. More precisely, if we denote
by
Ft=σ(Xs; 0≤s≤t)theσ-algebra generated by the past ofXt, then, for 0≤s≤t, the
conditional expectationE(f(Xt)|Fs)of a function ofXtshould be a
function ofXsand not depend on the values of the process strictly
befores. Here we assume that the process(Xt)t≥ 0 takes values in
some measurable space(Ω,F)and thatfis a bounded measurable
function onΩ.
This allows to define transition operatorsTs,t:L∞(Ω)→L∞(Ω)
for 0≤s≤tby requiring
(Ts,tf)(Xs) =E(
f(Xt)∣
∣Fs)for f ∈ L∞(Ω), where Ts,tf is determined only PXs almost
everywhere. The Markov process (Xt)t≥ 0 is called
time-homogeneous, if the transition operatorsTs,tdepend only on
the differencet−s. In this case the operatorsTt=T0,t,t≥0, form
a semigroup, called the Markov semigroup of(Xt)t≥ 0.
Levy processes are time-homogeneous Markov processes, they ́
are even Feller processes, that is, their Markov semigroup maps
C 0 (G)to itself. The independence of the increments allows for a
simple description of their Markov semigroup(Tt)t≥ 0 ,
(Ttf)(g) =E(
f(gXt))
=∫Gf(gg′)dPXt(g′), g∈G,fort≥0,f∈C 0 (G).
For more information on Markov processes, Markov
semigroups, and also the special case of group-valued Levy ́
processes, see [RW00a, RW00b].
1.4.8 Hunt’s formula
LetGbe a Lie group with Lie algebrag.
For a Levy process in ́ Gwe have its Markov semigroup(Tt)t≥ 0
and its infinitesimal generatorL. Hunt’s theorem describes Levy ́
processes inGin terms of their generators.