Translation, Centering and Reflection

holds true, thus (t, l) 7→ e−αP (t−)e−βtf (l) is in F1

p in the sense of [IW89, Defini-

tion (II.3.3)], and the result follows from [IW89, Equation between (II.3.7) and (II.3.8)] (see also [Çın11, Theorem 6.2]).

The reader may observe that for α = 0, that is without the term e−αP (t−), this theorem reduces to Campbell’s theorem, see, e.g., [CSKM13, Theorem 4.1]. However, in the case α > 0, the part P (t−) depends on all “marks” of the point process up to t, so the integrated function does not only depend on the current “mark” at t, and we had to apply the theory of stochastic integration with respect to Poisson point processes.

6.5. Translation, Centering and Reflection

Let X = Ω,G , (G_t)_t≥0, (Xt)t≥0, (Θt)t≥0, (Px)x∈E
be a Lévy Markov process on E = Rd.

The Markov property is a regularity condition on the process when translated in time, which is best described with the help of the (time) shift operators (Θ_t, t ≥ 0). Lévy Markov processes additionally feature regularity on space translations, so we introduce transformation operators which capture the “spatial shift” the Lévy process X:

(6.30) Definition. A family (γx, x ∈ E) is called translation operators for X, if it is a

collection of mappings γ_x: Ω → Ω, x ∈ E, satisfying

∀x ∈ E, t ≥ 0 : Xt◦ γx= Xt+ x,

∀x, y ∈ E : γx◦ γy = γx+y.

(6.31) Definition. A mapping Γ : Ω → Ω is called centering operator for X, if

∀t ≥ 0 : Xt◦ Γ = Xt− X0.

We assume the existence of translation operators (γx, x ∈ E) and of a centering

operator Γ for X for the rest of this subsection.

(6.32) Example. If we are in the context of the canonical coordinate process (Xt, t ≥ 0),

that is Xt(ω) = ω(t) for all ω ∈ Ω, t ≥ 0, on the path space

Ω = {ω : R+→ E | ω càdlàg},

then the existence of these operators is trivial: We can choose for all ω ∈ Ω γx(ω) := ω + x, x ∈ E,

Γ(ω) := ω − ω(0),

which exist, as translated càdlàg functions remain càdlàg. When also using the canonical shift operators (Θ_t, t ≥ 0), namely

Θt(ω) := ω(t + · ), t ≥ 0, ω ∈ Ω,

then the chosen translation and shift operators commute, as for all t ≥ 0, x ∈ E, ω ∈ Ω, γx◦ Θt(ω) = γx ω(t + · )

44 6. Lévy Processes

(6.33) Remark. Translation and shift operators always commute on process level, as

Xs◦ γx◦ Θt= Xs+t+ x = Xs◦ Θt◦ γx,

whereas translating a centered processes has no effect:

Xs◦ Γ ◦ γx= (Xs+ x) − (X0+ x) = Xs◦ Γ. 

We are going to show that, due to the spatial homogeneity, a translated Lévy Markov process behaves just like the original process with its starting point being translated, while centering lets it start at the origin. Of course, the following results only hold true if translation and centering operators exist for the Lévy Markov process X.

(6.34) Lemma. For all x, y ∈ E, t ≥ 0, f ∈ bE ,

Ex f (Xt) ◦ γy
= Ex+y f (Xt)
.

Proof. The set of all f ∈ bE for which the above identity holds true forms a MVS. By the MCT, it is therefore sufficient to prove that for all f =1A, A ∈E ,

Ex f (Xt) ◦ γy
= Ex f (Xt+ y)
= T_t f ( · + y)
(x) = T_t(x, A − y) = Tt(x + y, A) = Ttf (x + y) = Ex+y f (Xt)

holds, where we used1_A(x + y) = 1_A−y(x) for the third identity and the translation invariance of (Tt, t ≥ 0) for the forth identity.

As seen in subsection 2.1, the Markov property can be lifted from its standard definition with the help of shift operators to general functions, which are measurable with respect to the σ-algebra generated by the process. By employing the same standard techniques, we are able to show that this generalization also holds true for the spatial homogeneity, represented by translation and centering operators:

(6.35) Theorem. For all x, y ∈ E, F ∈ bF , the mapping F ◦ γy is in bF and satisfies

Ex F ◦ γy
= Ex+y F
.

Proof. As usual, it is sufficient to prove the above claim for functions of the form

F = f1(Xt1) · · · fn(Xtn),

with n ∈ N, 0 ≤ t1 < · · · < tn, f1, . . . , fn ∈ bE , which we are showing inductively over n ∈ N. The case n = 1 was already done in lemma (6.34). Assuming the assertion holds true for an n ∈ N, we are computing for n + 1 the expectation

Ex F ◦ γy
= Ex f1(Xt1 + y) · · · fn+1(Xtn+1+ y)
= Ex f1(Xt1 + y) · · · fn(Xtn + y) Ex fn+1(Xtn+1+ y)  F_tn

.

6.5. Translation, Centering and Reflection 45

The Markov property of X and lemma (6.34) applied on the last term yield Ex fn+1(Xtn+1+ y)  Ftn
= EXtn fn+1(Xtn+1−tn+ y)
= EXtn fn+1(Xtn+1−tn) ◦ γy
= EXtn+y fn+1(Xtn+1−tn)
,

so we get by renamingfe_i := f_i for i ∈ {1, . . . , n − 1} and fe_n:= f_nE· f_n+1(X_tn+1−tn)

: Ex F ◦ γy
= Ex fe₁(X_t1) · · ·fe_n(X_tn)
◦ γ_y
.

By using the inductive basis for n, which is applicable because fe₁, . . . ,fe_n∈ bE as well,

we conclude that Ex F ◦ γy
= Ex+y fe₁(X_t1) · · ·fe_n(X_t n)
= Ex+y f1(Xt1) · · · fn+1(Xtn) EXtn fn+1(Xtn+1−tn)

= Ex+y f1(Xt1) · · · fn(Xtn) fn+1(Xtn+1)
= Ex+y F
.

(6.36) Theorem. For all x ∈ E, F ∈ bF , the mapping F ◦ Γ is in bF and satisfies

Ex F ◦ Γ

= E0 F

.

Proof. Again, we only need to prove this for

F = f1(Xt1) · · · fn(Xtn),

with n ∈ N, 0 ≤ t1 < · · · < tn, f1, . . . , fn∈ bE . As X0 = x holds Px-a.s., we can directly

use theorem (6.35) to compute

Ex F ◦ Γ
= Ex f1(Xt1 − X0) · · · fn(Xtn − X0)
= Ex f1(Xt1 − x) · · · fn(Xtn− x)
= Ex f1(Xt1) · · · fn(Xtn)
◦ γ−x
= Ex+(−x) f1(Xt1) · · · fn(Xtn)
= E0 F
.

Some Lévy Markov processes are not only spatial homogeneous, but also invariant under the reflection at the origin. Just as with (time) shifts, (spatial) translation and centering, we can lift this reflection property from the semigroup up to the process level:

(6.37) Definition. X is called reflection invariant, if for all t ≥ 0, x ∈ E, A ∈E ,

46 6. Lévy Processes

(6.38) Example. The Brownian motion on R is reflection invariant, as for all t ≥ 0,

x ∈ R, A ∈ B(R), its semigroup satisfies Tt(x, A) = Z A 1 √ 2πte −(y−x)2 2t dy = Z −A 1 √ 2πte −(y+x)2 2t dy = T_t(−x, −A). _

(6.39) Definition. A mapping ι : Ω → Ω is called reflection operator for X, if it satisfies

∀t ≥ 0 : Xt◦ ι = −Xt.

(6.40) Example. If the process X is the canonical coordinate process on the path space

as given in example (6.32), then a reflection operator ι exists. It can be defined by

ι(ω) := −ω, ω ∈ Ω. 

The same course of discussion as for the centering operator also applies to the reflection operator. If the sample space admits a reflection operator ι for X, the following results hold true:

(6.41) Lemma. If X is reflection invariant, then for all x ∈ E, t ≥ 0, f ∈ bE ,

Ex f (Xt) ◦ ι
= E−x f (Xt)
.

Proof. Again using the MCT, it is sufficient to prove this for f =1A, A ∈E . We have

Ex f (Xt) ◦ ι
= Ex f (−Xt)
= Tt f (− · )
(x) = Tt(x, −A)

= Tt(−x, A) = Ttf (−x) = E−x f (Xt)
,

where we used 1A(−x) =1−A(x) for the third identity and the reflection invariance for the forth identity.

The proof of the following theorem proceeds exactly like the proof of theorem (6.35):

(6.42) Theorem. If X is reflection invariant, then for all x ∈ E, F ∈ bF , the mapping

F ◦ ι is in bF and satisfies

Ex F ◦ ι
= E−x F
.

Proof. As usual, it is sufficient to prove the above claim for F = f1(Xt1) · · · fn(Xtn),

with n ∈ N, 0 ≤ t1 < · · · < tn, f1, . . . , fn ∈ bE , which we are showing inductively over n ∈ N. The case n = 1 was already done in lemma (6.41). Assuming the assertion holds true for an n ∈ N, we are computing for n + 1 the expectation

Ex F ◦ ι
= Ex f1(−Xt1) · · · fn+1(−Xtn+1)
= Ex f1(−Xt1) · · · fn(−Xtn) Ex fn+1(−Xtn+1)  F_tn

.

6.5. Translation, Centering and Reflection 47

The Markov property of X and lemma (6.41) applied on the last term yield Ex fn+1(−Xtn+1)  F_tn
= EXtn fn+1(−Xtn+1−tn)
= EXtn fn+1(Xtn+1−tn) ◦ ι
= E−Xtn fn+1(Xtn+1−tn)
,

so we get by renamingfe_i := f_i for i ∈ {1, . . . , n − 1} and fe_n:= f_nE· f_n+1(X_t

n+1−tn)
: Ex F ◦ ι
= Ex fe₁(X_t1) · · ·fe_n(X_tn)
◦ ι
.

By using the inductive basis for n, which is applicable because fe₁, . . . ,fe_n∈ bE as well,

we conclude that Ex F ◦ ι
= E−x fe₁(X_t1) · · ·fe_n(X_tn)
= E−x f1(Xt1) · · · fn+1(Xtn) EXtn fn+1(Xtn+1−tn)

= E−x f1(Xt1) · · · fn(Xtn) fn+1(Xtn+1)
= E−x F
.

Chapter II.

Transformations

We prepare the different transformation methods for Markov processes which we will employ later for the characterization and construction of Brownian motions. We focus our attention on transformations which (under certain conditions) preserve the (strong) Markov property. Fortunately, most of the required methods are commonly known and we can resort to elaborate results in the literature, which we only adjust or extend slightly.

In section 7, path-space realizations, which are a standard “ad-hoc” technique in existence theorems, are treated. In section 8, we name special conditions which ensure that stopping a Markov process at a random time maintains its Markovian structure. The well-known transformations of time substitution via additive functionals and of killing via multiplicative functionals are merely reminded in sections 9 and 10. On the other hand, the concatenation of various Markov processes on different state spaces (forming a joint process which behaves like the first process until it dies, is revived as the second process, etc.) typically is treated not at all or only in a very specific fashion in the literature. In section 11, we assemble an extensive basis on this method, in order to establish, with the help of state space transformations given in section 12, a technique allowing us to concatenate alternating, independent copies of two underlying processes in section 13. This will be the main vehicle in the construction of chapter III, where we will join processes on different subgraphs to a Brownian motion on the complete graph.