Additive Lévy Processes
Introduction
Let X 1 � � � � X N denote N independent Lévy processes on R� . We can conN , as follows:
struct an N-parameter stochastic process �, indexed by R+
�� := X�11 + · · · + X�NN
We might also write
N
�
for every � := (�1 � � � � � �N ) ∈ R+
� := X 1 ⊕ · · · ⊕ X N �
And in this way, it follows that if � 1 � � � � � � N are independent additive Lévy
processes, then � 1 ⊕ · · · ⊕ � N is an additive Lévy process as well, notation
being more or less obvious.
It is not hard to convince yourself that if Ψ1 � � � � � ΨN denote the respective Lévy exponents of X 1 � � � � � X N , then
where
Ψ(ξ)
Ee�ξ·�� = e−�·Ψ
N
for all � ∈ R+
and ξ ∈ R� �
�
�
Ψ (ξ) := Ψ1 (ξ) � � � � � ΨN (ξ) �
And that Ψ determines uniquely the finite-dimensional distributions of �.
Definition 1. The N-parameter stochastic process � is called the additive
Lévy process corresponding to X 1 � � � � � X N . The function Ψ is called the
Lévy exponent of �.
�
I mention a simple example of additive Lévy process. Exercise 2 shows
us how to create other types of additive Lévy processes from independent
Lévy processes.
91
92
14. Additive Lévy Processes
Example 2. Let X 1 � � � � � X N denote N independent �-dimensional Brownian motions. Then the N-parameter Gaussian process � is called additive
Brownian motion. More generally, if X 1 � � � � � X N are independent isotropic
stable processes with the same index α ∈ (0 � 2], then � is an additive stable process with index α. Note that Ψ(ξ) ∝ �ξ�α (1 � � � � � 1)�
�
Let us define
(�� �)(�) := E�(� + �� )�
(�1 �)(�) :=
�
N
R+
e−
�N
�=1 ��
(�� �)(�) d��
N , but there is
[We could just as easily define �λ for λ > 0, or even λ ∈ R+
no pressing need for doing this here.]
You should check the following; it states that there are natural, and
easy-to-understand, analogues of semigroups and resolvents in the present
N-parameter setting.
Lemma 3. If P 1 � � � � � P N denote the respective semigroups of X 1 � � � � � X N ,
π(1)
π(N)
then �� = Pπ(1) · · · Pπ(N) for every permutation (π(1) � � � � � π(N)) of (1 � � � � � �).
And if R11 � � � � � R1N respectively denote the 1-resolvents of X 1 � � � � � X N , then
�1 = Rπ(1) · · · Rπ(N) .
And, not surprisingly, we have also potential measures:
Definition 4. The 1-potential measure �1 of � is defined as
�
�N
�1 (A) := E
e− �=1 �� 1lA (�� ) d��
N
R+
for all A ∈ �(R� ).
�
Lemma 5. If U11 � � � � � U1N denote the respective 1-potential measures of
X 1 � � � � � X N , then �1 = U11 ∗ · · · ∗ U1N .
An addition theorem
Theorem 6 (Khoshnevisan and Xiao, 2009; Khoshnevisan et al., 2003; Yang,
N . Then, E|�(R N ) � G| > 0 if
2007). Choose and fix a Borel set G ⊆ R+
+
and only if there exists a Borel probability measure ρ on G such that
�
�
� �
N
1
Re
|ρ̂(ξ)|2 dξ < ∞�
(1)
1 + Ψ� (ξ)
R�
�=1
I will prove the sufficiency of (1); that is the easier half of Theorem 6;.
You can find the details of the [much] more difficult half in Khoshnevisan
and Xiao (2009).
An addition theorem
Define for all probability densities � : R� → R and � ∈ R� ,
�
�N
(J�)(�) :=
e− �=1 �� �(� + �� ) d��
93
N
R+
Then, a careful computation, using Theorem 5 (page 86), reveals the following multiparameter analogue of Lemma 6 (page 88):
Lemma 7. For all measurable probability densities � : R� → R+ ,
�
�
�
E (J�)(�) d� = 1�
�
R�
R�
�
2
E |(J�)(�)|
�
1
d� =
(2π)�
�
N
�
R� �=1
Re
�
1
1 + Ψ� (ξ)
�
ˆ 2 dξ�
|�(ξ)|
Proof of half of Theorem 6. If (J�)(�) > 0 for some probability density �,
then certainly � + X• has hit the support of � at some time. That is,
�
�
P � ∈ supp(�) � X(R+ ) ≥ P {(J�)(�) > 0} �
In particular,
�
� �
�
�
E �supp(�) � X(R+ )� ≥
R�
P {(J�)(�) > 0} d��
Lemma 7 and the Paley–Zygmund inequality (page 89) together imply that

−1
�
�
� �
N
�
�
1
1
�
�
ˆ 2 dξ  �
E �supp(�) � X(R+ )� ≥ 
Re
|�(ξ)|
�
1 + Ψ� (ξ)
(2π) R�
�=1
where 1/∞ := 0. Now we approximate G by the support of a probability
density of the form � := ρ ∗ �� , where ρ ∈ M1 (G) and �� is a bounded probˆ
ability density with support in B(0 � �). Since |�(ξ)|
≤ |ρ̂(ξ)|, the preceding
shows that if there exists a probability measure ρ on G that satisfies (1),
then E|X(R+ ) � G| = E|G � X(R+ )| > 0.
�
�
Definition 8. We say that � is absolutely continuous if �1 (A) = A υ(�) d�
for some measurable υ. The function υ is called the 1-potential density of
�.
�
It is not hard to see that υ can always be chosen to be a probability
density. Moreover, if U 1 � � � � � U N have 1-potential densities �1 � � � � � �N respectively, then υ = �1 ∗ · · · ∗ �N . The following is proved similarly to
Theorem 10 (page 82).
94
14. Additive Lévy Processes
Theorem 9. Suppose � is absolutely continuous with a 1-potential denN
sity υ such that υ(0) > 0. Then, for all G ∈ �(R� ), P{�(R+
) ∩ G �= ∅} > 0
if and only if there exists a probability measure ρ on G that satisfies (1).
Example 10. Let � be an additive stable process on R� with N parameters
and index α ∈ (0 � 2]. Then,
�
�
�
|ρ̂(ξ)|2
N
P �(R+ ) ∩ G �= ∅ > 0 iff
dξ < ∞ for some ρ ∈ M1 (G)�
Nα
R� 1 + �ξ�
When α = 2, this says something about additive Brownian motion.
�
A connection to Hausdorff dimension
Definition 11. The Hausdorff dimension dim� G of a Borel set G ⊆ R�
is defined as
�
�
�
|ρ̂(ξ)|2
∃
dim� G := inf � ∈ (0 � �) : ρ ∈ M1 (G) such that
dξ < ∞ �
�−�
R� 1 + �ξ�
[The preceding is well defined, provided that we set inf ∅ := �.]
�
[This is not the usual definition, rather the consequence of a famous
theorem called “Frostman’s theorem” of classical potential theory.]
In particular, Example 10 tells us the following.
Proposition 12. If � is an M-parameter additive stable process with
index α ∈ (0 � 2], then for all G ∈ �(R� ):
M
(1) If dim� G > � − Mα, then P{�(R+
) ∩ G �= ∅} > 0; whereas
M
(2) If dim� G < � − Mα, then P{�(R+
) ∩ G �= ∅} = 0.
Now let � be an N-parameter additive Lévy process on R� with Lévy
exponent Ψ := (Ψ1 � � � � � ΨN ), independent from �. Then, � := � ⊕ � is an
(N + M)-parameter additive Lévy process on R� . It follows from Theorem
9 that
�
�
� �
N
�
�
1
dξ
N+M
P 0 ∈ �(R+ ) > 0 iff
Re
< ∞�
1 + Ψ� (ξ) 1 + �ξ�Nα
R�
�=1
But 0 is in the closure of the range of � if and only if the closures of
the ranges of � and � intersect! Therefore, Proposition 12 implies the
following:
Theorem 13 (Khoshnevisan et al., 2003; Yang, 2007). With probability one,


�
�
� �
N


�
�
1
dξ
N
dim� �(R+
) = sup � ∈ (0 � �) :
Re
<
∞
�


1 + Ψ� (ξ) �ξ��−�
R�
�=1
An application to subordinators
where sup ∅ := 0.
95
[Why can we replace (1 + �ξ��−� )−1 by �ξ�−�+� ?]
It is not hard to see that if C is at most countable, then dim� (G ∪ C) =
dim� G for all G ∈ �(R� ). Therefore, one can use the fact that the X � ’s
are cadlag [hence have denumerably-many jumps] to prove that the closure
sign of the preceding theorem can be removed.
An application to subordinators
Let us now apply Theorem 13 to the case where X� := T� is a subordinator
with a Lévy exponent Φ [and Lévy exponent Ψ, still].
Theorem 14 (Horowitz, 1968). With probability one,
�
�
�
� ∞�
�
�
1
dλ
dim� T(R+ ) = sup � ∈ (0 � 1) :
<∞ �
1 + Φ(λ) λ 1−�
0
where sup ∅ := 0.
The following is a convenient method by which we can “transform” a
Lévy exponent to a Laplace exponent.
Proposition 15. For every λ > 0,
�
�
� ∞
1
1
d�
1
=
Re
�
1 + Φ(λ)
πλ −∞
1 + Ψ(�) 1 + (�/λ)2
Proof. Define {Cλ }λ>0 to be an independent linear Cauchy process, normalized so that E exp(��Cλ ) = exp(−λ|�|) for � ∈ R and λ > 0. By independence,
e−�Φ(λ) = Ee−λT� = Ee�Cλ T� = Ee−�Ψ(Cλ ) �
But the probability density of Cλ is �(�) = (πλ)−1 (1 + (�/λ)2 )−1 . Therefore,
� ∞
1
e−�Ψ(�)
e−�Φ(λ) =
d��
πλ −∞ 1 + (�/λ)2
Multiply both sides by exp(−�) and integrate [d�] to finish.
�
Proof of Theorem 14. Here is how we can apply Proposition 15. First,
note that if 0 < θ < 2 and � ∈ R, then
� ∞
dλ
�
� ∝ |�|−θ �
1+θ 1 + (�/λ)2
λ
0
96
14. Additive Lévy Processes
Therefore, if we multiply the equation in Proposition 15 by λ −θ and integrate [dλ], then we obtain
�
�
�
� ∞�
�
1
dλ
1
d�
∝
Re
�
θ
1 + Φ(λ) λ
1 + Ψ(�) |�|θ
0
R�
This and Theorem 13 together prove the theorem.
�
Let us conclude this section by applying Horowitz’s theorem to the set
of “increase times” of linear Brownian motion.
Proposition 16 (Lévy XXX). If B denotes standard Brownian motion, then
�
�
1
dim� � ≥ 0 : B� = sup B� =
a.s.
2
�∈[0��]
Proof. Define
T� := inf {� > 0 : B� = �}
for all � > 0�
Then, T is a stable subordinator with index√α = 1/2; see Exercise 2 [page
51]. And Horowitz’s theorem [ with Φ(λ) ∝ λ] tells us that the Hausdorff
dimension of the range of T is a.s. 1/2. On the other hand, it is a realvariable fact that
�
�
T(R+ ) =
� ≥ 0 : B� = sup B�
�∈[0��]
(Check!) Therefore, the proposition follows.
�
�
There is a theorem of Lévy which implies that {� ≥ 0 : B� = sup[0��] B}
has the same “law” as the zero set {� ≥ 0 : B� = 0} of Brownian motion.
Therefore, the preceding implies that the Hausdorff dimension of the zero
set of B is almost surely 1/2. Rather than study this particular problem
in greater depth, we study the zero set of a more general Lévy process in
the next lecture.
Problems for Chapter 14
1. Let {P� }�∈R denote the two-sided semigroup of a two-sided Lévy process.
(1) Is it true that P�+� = P� P� for all �� � ∈ R? [In other words, is
{P� }�∈R a semigroup of linear operators?]
(2) Define linear operators P̄� := P� P−� for all � ∈ R. Prove that
{P̄� }�∈R is a semigroup of linear operators.
2. Let X 1 and X 2 denote two independent Lévy processes on R� with
respective exponents Ψ1 and Ψ2 .
Problems for Chapter 14
97
(1) Verify that ���� := X� − Y� defines a 2-parameter additive Lévy
process on R� ; compute its Lévy exponent.
(2) Verify that ���� := (X� � Y� ) defines a 2-parameter additive Lévy
process on (R� )2 ; compute its Lévy exponent.
3. Derive Lemma 7.
4. Let X denote an isotropic stable process on R� with index α ∈ (0 � 2].
Compute dim� (X(R+ )). Indicate the changes to your formula if X(R+ ) is
replaced by �(R+ ), where � is an additive stable process on R� with index
α ∈ (0 � 2] and N parameters. Or more generally still if � := X 1 ⊕ · · · ⊕ X N ,
where X � ’s are independent symmetric stable processes on the line with
respective indices α1 � � � � � α� ∈ (0 � 2].
5. Compute the Hausdorff dimension of the range of Y� := (� � B� ), where
B denotes �-dimensional Brownian motion. The range of Y is called the
“graph” of Brownian motion. Indicate the changes made to your formula
if we replace “Brownian motion” by “isotropic stable process on R� with
index α ∈ (0 � 2].”