The measure preserving and nonsingular transformations of the jump Levy processes
Let ξ(t), t belongs [0, 1], be a jump Levy process. By Pξ, we denote the law of ξ in the Skorokhod space D[0, 1]. Under some conditions on the Levy measure of the process, we construct the group of Pξ preserving transformations of D[0, 1]. For the Levy process that has only positive (or only negativ...
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| Cite this: | The measure preserving and nonsingular transformations of the jump Levy processes / N.V. Smorodina // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 1. — С. 144–154. — Бібліогр.: 8 назв.— англ.. |
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irk-123456789-45442009-11-26T12:00:45Z The measure preserving and nonsingular transformations of the jump Levy processes Smorodina, N.V. Let ξ(t), t belongs [0, 1], be a jump Levy process. By Pξ, we denote the law of ξ in the Skorokhod space D[0, 1]. Under some conditions on the Levy measure of the process, we construct the group of Pξ preserving transformations of D[0, 1]. For the Levy process that has only positive (or only negative) jumps, we construct the semigroup of nonsingular transformations. 2008 Article The measure preserving and nonsingular transformations of the jump Levy processes / N.V. Smorodina // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 1. — С. 144–154. — Бібліогр.: 8 назв.— англ.. 0321-3900 http://dspace.nbuv.gov.ua/handle/123456789/4544 519.21 en Інститут математики НАН України |
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Let ξ(t), t belongs [0, 1], be a jump Levy process. By Pξ, we denote the law of ξ in the Skorokhod space D[0, 1]. Under some conditions on the Levy measure of the process, we construct the group of Pξ preserving transformations of D[0, 1]. For the Levy process that has only positive (or only negative) jumps, we construct the semigroup of nonsingular transformations. |
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Smorodina, N.V. The measure preserving and nonsingular transformations of the jump Levy processes |
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Smorodina, N.V. |
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Smorodina, N.V. |
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The measure preserving and nonsingular transformations of the jump Levy processes |
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The measure preserving and nonsingular transformations of the jump Levy processes |
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The measure preserving and nonsingular transformations of the jump Levy processes |
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The measure preserving and nonsingular transformations of the jump Levy processes |
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The measure preserving and nonsingular transformations of the jump Levy processes |
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measure preserving and nonsingular transformations of the jump levy processes |
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Інститут математики НАН України |
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The measure preserving and nonsingular transformations of the jump Levy processes / N.V. Smorodina // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 1. — С. 144–154. — Бібліогр.: 8 назв.— англ.. |
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Theory of Stochastic Processes
Vol. 14 (30), no. 1, 2008, pp. 144–154
UDC 519.21
NATALYA V. SMORODINA
THE MEASURE PRESERVING AND NONSINGULAR
TRANSFORMATIONS OF THE JUMP LÉVY PROCESSES
Let ξ(t), t ∈ [0, 1], be a jump Lévy process. By Pξ, we denote the law of ξ in the
Skorokhod space � [0, 1]. Under some conditions on the Lévy measure of the process,
we construct the group of Pξ− preserving transformations of � [0, 1]. For the Lévy
process that has only positive (or only negative) jumps, we construct the semigroup
of nonsingular transformations.
1. Introduction.
Let ξ(t), t ∈ [0, 1], be a jump Lévy process. It is well known (see [5]) that trajectories
of this process with probability 1 belong to the space D[0, 1] of right-continuous functions
from [0, 1] into R with left limits. We equip D[0, 1] with the Skorokhod topology and
denote, by Pξ, the law of ξ in D[0, 1].
We use the Lévy-Khinchin representation of these processes ([5]), namely,
ξ(t) = at+
∫ t
0
∫
0<|x|�1
xν̃(ds, dx) +
∫ t
0
∫
|x|>1
xν(ds, dx). (1)
In this representation, a is a nonrandom constant, ν(ds, dx) is a Poisson random
measure on the space [0, 1]×R with the intensity measure Π of the form dΠ = dtΛ(dx),
where Λ(dx) is the Lévy measure of the process ξ. By ν̃(dt, dx) = ν(dt, dx)−Eν(dt, dx),
we denote the corresponding compensated measure.
It is well known that the Lévy measure Λ satisfies the following conditions:
1)Λ({0}) = 0,
2)
∫
R
min(x2, 1)Λ(dx) <∞.
We also suppose that Λ(R) = ∞, that is, Λ is the Lévy measure of a ”non-trivial”
(i.e., non-compound Poisson) process, and that Λ({a}) = 0 for any a ∈ R.
As a probability space for the Poisson random measure, we choose the space of con-
figurations on the set [0, 1]× R (in a special case of one-sided processes, we use the set
[0, 1]× (0,∞) for this purpose). On this space of configurations, we consider the Poisson
measure (see [3]) P with intensity measure Π of the form Π(dt, dx) = dtΛ(dx).
We consider the distribution of the process ξ in the space D[0, 1] as an image of the
Poisson measure P under the action of a mapping defined by (1). First, we consider
more general problems.
1) We construct a semigroup of nonsingular (with respect to the Poisson measure P )
transformations of the configuration space X (G), where G is of the form G = S× (0,∞)
2000 AMS Mathematics Subject Classification. Primary 28C20, 60H05, 60G57.
Key words and phrases. Configuration space, Poisson measure, group of transformations, measure
preserving transformations, nonsingular transformations, Lévy process..
This article was partially supported by DFG project 436 RUS 113/823.
144
THE MEASURE PRESERVING AND NONSINGULAR . . . 145
and S is a complete separable metric space. We suppose that the intensity measure Π of
the Poisson measure P is of the form
Π(dθ, dx) = π(dθ)γθ(dx), (2)
where π is a finite measure on S and, for every θ ∈ S, γθ is a σ−finite measure
on (0,∞) (depending on θ). We suppose that, for every ε > 0, Π(S × (ε,∞)) =∫
S γθ((ε,∞))π(dθ) <∞.
2) We construct a group of P−preserving transformations of the configuration space
X (G), where G is of the form G = S×R, (here, R = R∪{∞}), and the intensity measure
Π of the Poisson measure P is of the form (2) (here, γθ is a σ−finite measure on R). We
suppose that, for every ε > 0, Π(S × (R \ [−ε, ε])) <∞.
We consider the transformations F : X (G)→ X (G) generated by a mapping ϕ : G→
G such that F maps each configuration X = {x}x∈X to a configuration Y = F (X) of
the form
F (X) = {ϕ(x)}x∈X . (3)
It is easy to show that an image of a Poisson measure with intensity measure Π
under the action of (3) is a Poisson measure with intensity measure Πϕ−1. The absolute
continuity conditions for Poisson measures with different intensity measures were first
obtained by Skorokhod [4] (see also [1,3,7]).
Later on, Vershik and Tsilevich [8] considered the absolute continuous transformations
of the so-called gamma measure which is a Poisson measure with the intensity measure
Π of the form
Π(dθ, dx) = π(dθ)
e−x
x
dx, θ ∈ S, x ∈ (0,∞).
(Here as above, S is a complete separable metric space, and π is a finite measure on
S.) It was proved in [8] that the gamma measure is quasi-invariant under a group of
transformations Fa, a ∈M, whereM is a set of all measurable functions a : S → (0,∞)
such that ∫
S
| log a(θ)|π(dθ) <∞,
and Fa is a transformation of the form (3) with ϕ(θ, x) = (θ, a(θ)x). Note that the set
M is a commutative group with respect to the pointwise multiplication of functions and,
for every a1, a2 ∈M, we have Fa1a2 = Fa1 ◦ Fa2 .
In [6], the transformations of the so-called stable measures Pα, α > 0, which are the
Poisson measures with intensity measures of the form
π(dθ)
dx
x1+α
, θ ∈ S, x ∈ (0,∞)
were considered. It was proved there that, for every α > 0, a stable measure Pα is
quasi-invariant under a semigroup of transformations Φf , f ≥ 0, f ∈ L1(S, π), where
Φf is a transformation of the form (3) with
ϕ(θ, x) =
(
θ,
x
(1 + αf(θ)xα)1/α
)
. (4)
Further, in the same paper, it was proved that, for every α > 0, the Poisson measure
P̃α on the configuration space on S × R (here, R = R ∪ {∞}) with intensity measure
π(dθ) dx
|x|1+α is invariant under a group of transformations Φ̃f , where f is an arbitrary
measurable function on S, and Φ̃f is a transformation of the form (3) with the function
ϕ defined by (4) [in (4), the functions xα and x1/α are extended to the whole real line in
an odd way].
146 NATALYA V. SMORODINA
In the present paper, the similar results are proved for an arbitrary Poisson measure
with an intensity measure of the form (2).
The paper is organized as follows. Secton 2 contains the necessary definitions concern-
ing the configuration space and Poisson measures. In Section 3, we construct a semigroup
of nonsingular (with respect to the Poisson measure P ) transformations of the configu-
ration space X (G), where G is of the form S × (0,∞), and the intensity measure of the
Poisson measure P is of the form (2). In Section 4, we construct a group of P−preserving
transformations in the case where G = S×R. In Section 5, we construct the groups and
the semigroups of transformations of trajectories of the Lévy process. Finally, in Section
6, we consider one multidimensional generalization of the results of Section 4.
2. The space of configurations.
Let G be a metric space, B be its Borel σ−algebra, B0 be the ring of bounded Borel
subsets of G. Let Π be a σ−finite measure on G. Suppose that Π(V ) < ∞ for every
V ⊂ B0.
We denote, by X = X (G), the space of configurations on G. By definition,
X (G) = {X ⊂ G : |X ∩ V | <∞ for all V ⊂ B0},
where |A| denotes the cardinality of the set A. We equip X with the vague topology
O(X ), i.e., the weakest topology such that all functions X → R
X �→
∑
x∈X
f(x)
are continuous for all continuous functions f : G → R with bounded supports. The
Borel σ−algebra corresponding to O(X ) will be denoted by B(X ). This is the smallest
σ−algebra for which the mapping
X �→ |X ∩ V |
is measurable for any V ⊂ B0.
We say that a probability measure P on (X ,B(X )) is the Poisson measure with inten-
sity measure Π, if, for every V ∈ B0,
P (X : |X ∩ V | = k) = e−Π(V ) Π(V )k
k!
.
For more details, see, e.g., [2,3].
3. Nonsingular transformations of the Poisson measure.
In this section, we suppose that G = S × (0,∞), where S is a complete separable
metric space, and the measure Π is of the form
Π(dθ, dx) = π(dθ)γθ(dx), θ ∈ S, x ∈ (0,∞),
where π is a finite measure on S. We also suppose that, for π−a.e. θ ∈ S,
1) γθ((0,∞)) =∞,
2) for every x > 0 γθ((x,∞)) <∞, and the function
Uθ(x) = γθ((x,∞)) (5)
is continuous and strictly decreasing. Set Uθ(0) =∞, Uθ(∞) = 0.
Further, for θ ∈ S, t � 0, by T θ
t , we denote a mapping from (0,∞) into (0,∞) defined
by the formula
T θ
t (x) = U−1
θ (Uθ(x) + t). (6)
THE MEASURE PRESERVING AND NONSINGULAR . . . 147
Note that T θ
0 is the identity mapping. Let us show that, for any s, t ∈ [0,∞),
T θ
t+s = T θ
t ◦ T θ
s . (7)
Using (6), we get
T θ
t ◦ T θ
s (x) = T θ
t (U−1
θ (Uθ(x) + s)) = U−1
θ (Uθ(U−1
θ (Uθ(x) + s)) + t) =
U−1
θ (Uθ(x) + s+ t) = T θ
t+s.
Denote, by L+(S), the space of all Borel nonnegative functions on S, and, by L+
1 (S, π),
L+
1 (S, π) ⊂ L+(S), the space of all nonnegative integrable functions with respect to the
measure π.
For f ∈ L+(S), we define a mapping τf : G→ G by
τf (θ, x) = (θ, T θ
f(θ)(x)). (8)
Note that, for f ≡ 0, the mapping τf is the identity mapping. It follows easily from
(7) that, for every f, g ∈ L+(S),
τf+g = τf ◦ τg. (9)
Now, using the semigroup of transformations τf , f ∈ L+(S), we construct a semigroup
Φf , f ∈ L+(S) of transformations of X (S × (0,∞)). In accordance with (3), for f ∈
L+(S), we define a mapping Φf : X (S×(0,∞))→ X (S×(0,∞)) as a mapping generated
by τf ,
Φf (X) = Y = {(θ, T θ
f(θ)(x))}. (10)
It follows from (7) and (8) that, for every f, g ∈ L+(S),
Φf+g = Φf ◦Φg. (11)
Note that, in the case where the measure γθ(dx) has the form dx
x1+α , α > 0, we get,
for every θ ∈ S,
τf (θ, x) =
(
θ,
x
(1 + αf(θ)xα)1/α
)
,
that corresponds to the semigroup of transformations constructed in [6].
The main result of this section is the following.
Theorem 1. 1. The measure PΦ−1
f is absolutely continuous with respect to P for every
f ∈ L+
1 (S, π) and has the form
PΦ−1
f = e
�
S
fdπ · P ∣∣
Af
,
where Af is a measurable subset of X (S× (0,∞)), Φf maps the measure P to its restric-
tion P
∣∣
Af
on Af , e
�
S
fdπ = 1
P (Af ) is a natural normalizing coefficient.
2. For every f ∈ L+(S) such that
∫
S
fdπ = ∞, the measures PΦ−1
f and P are
orthogonal.
Proof. First, for fixed θ ∈ S, t > 0, we calculate the image of the measure γθ under the
action of T θ
t : (0,∞)→ (0,∞).
It is easy to check that T θ
t (x) is a strictly increasing function of the variable x, and it
follows from the equality T θ
t (+∞) = U−1
θ (t) that the support of the measure γθ(T θ
t )−1
is an interval (0, U−1
θ (t)). We now show that, on this interval, the measure γθ(T θ
t )−1
coincides with the measure γ, i.e.,
γθ(T θ
t )−1 = γθ
∣∣
(0,U−1
θ
(t))
. (12)
148 NATALYA V. SMORODINA
For an arbitrary interval [α, β] ⊂ (0, U−1
θ (t)), we have
γθ(T θ
t )−1([α, β]) = γθ([(T θ
t )−1(α), (T θ
t )−1(β)]) =
Uθ(U−1
θ (Uθ(α) − t))− Uθ(U−1
θ (Uθ(β)− t)) =
Uθ(α)− Uθ(β) = γθ([α, β]).
This completes the proof of (12). We deduce from (12) that the image of the measure
Π(dθ, dx) = π(dθ)γθ(dx) under the action of τf , is a measure of the form
Πτ−1
f (dθ, dx) = 1(0,U−1
θ (f(θ))(x)π(dθ)γθ(dx), (13)
where 1B denotes the indicator function of the set B. Consider a subset Gf of the set
G = S × (0,∞) such that the density
dΠτ−1
f
dΠ is equal to 1 on Gf , i.e.,
Gf = {(θ, x) ∈ G : 0 < x < U−1
θ (f(θ))}. (14)
It follows from (13) that the measure PΦ−1 is concentrated on a set Af of the form
Af = {X ∈ X (G) : X ∩ (G \Gf ) = ∅}.
Note that Af coincides with the space X (Gf ) of configurations on the space Gf . By
(13), we know that the intensity measure of the Poisson measure PΦ−1
f restricted on Gf
is equal to π(dθ)γθ(dx).
Hence, the measure PT−1
f is absolutely continuous with respect to P if P (Af ) > 0 and
PT−1
f is orthogonal to P if P (Af ) = 0. To complete the proof, it remains to calculate
P (Af ).
We have
P (Af ) = P (X :| X ∩ (G \Gf ) |= 0) = exp(−Π(G \Gf )) =
exp(−
∫
S
π(dθ)
∫ ∞
U−1
θ (f(θ))
γθ(dx)) = exp(−
∫
S
fdπ).
The last expression is positive iff
∫
S fdπ <∞. This completes the proof of the theorem.
4. The measure preserving transformations.
In this section, we suppose that the set G = S×R, where R = R∪{∞} is an extended
real line (by definition, the point ∞ has no sign, so that −∞ = +∞). We also suppose
that the intensity measure Π of the Poisson measure P is of the form
Π(dθ, dx) = π(dθ)γθ(dx),
where π is a finite measure on S, and, for every θ ∈ S , γθ is a σ−finite measure on R
(depending on θ).
For every fixed θ ∈ S, let Uθ : R→ R denote a function defined by
Uθ(x) =
{
γθ((x,∞)), if x > 0;
−γθ((−∞, x)), if x < 0.
(15)
Set Uθ(∞) = 0, Uθ(0) =∞. Note that, for x > 0, formula (15) coincides with (5).
We suppose that, for π−a.e. θ ∈ S,
1) γθ({0}) = γθ({∞}) = 0,
2) γθ((0,∞)) = γθ((−∞, 0)) = +∞,
3) for every ε > 0 γθ(R \ [−ε, ε]) <∞,
4) Uθ is continuous on R \ {0} and strictly decreasing on (−∞, 0) and (0,∞) .
First, for every fixed θ ∈ S, we construct a one-parameter group of γθ−preserving
transformations.
THE MEASURE PRESERVING AND NONSINGULAR . . . 149
For t ∈ (−∞,∞), let T θ
t be a map from R to R such that
T θ
t (x) = U−1
θ
(
Uθ(x) + t
)
. (16)
We preserve here the same notation T θ
t as in (6). We only note that, unlike (6), the
domain of mapping (16) is R, and the parameter t may be either positive or negative.
It is easily proved that T θ
t , t ∈ (−∞,∞), is a one-parameter group of transformations,
i.e., for any s, t ∈ R.
T θ
t+s = T θ
t ◦ T θ
s . (17)
Show that the group T θ
t , t ∈ (−∞,∞), is a group of γθ−preserving transformations,
i.e., for every t ∈ (−∞,∞),
γθ(T θ
t )−1 = γθ. (18)
Notice that T θ
t is a superposition of three transformations, namely, Uθ, a shift trans-
formation, and U−1
θ . The first transformation maps the measure γθ into a Lebesgue
measure, the shift transformation preserves the Lebesgue measure, and U−1
θ maps the
Lebesgue measure into γθ.
As before, we denote the set of all Borel functions on S by L(S). Given f ∈ L(S), we
define a mapping
τf : S × R→ S × R,
by
τf (θ, x) = (θ, T θ
f(θ)). (19)
It follows from (17) that, for every f, g ∈ L(S),
τf+g = τf ◦ τg, (20)
and it follows from (18) that, for every f ∈ L(S),
Πτ−1
f = Π.
Using the group of transformations τf , f ∈ L(S), we construct a group Φ̃f , f ∈ L(S)
of transformations of X (S × R) by analogy with (10). In accordance with (3), for f ∈
L(S), we define a mapping Φ̃f : X (S × R) → X (S × R) as a mapping generated by τf
and such that
Φ̃f (X) = Y = {(θ, T θ
f(θ)(x))}.
The main result of this section is the following
Theorem 2.
1. For every f, g ∈ L(S), we have
Φ̃f+g = Φ̃f ◦ Φ̃g.
2. For every f ∈ L(S),
P Φ̃−1
f = P,
i.e., the group Φ̃f , f ∈ L(S) is a group of P−preserving transformations.
Proof. Statement 1 of the theorem follows from (20). To prove statement 2, we note that
the measure P Φ̃−1
f is the Poisson measure with intensity measure Πτ−1
f = Π.
150 NATALYA V. SMORODINA
5. The transformations of the trajectories of Lévy processes.
First we consider the Lévy process that has only positive jumps. The Lévy measure
Λ of such a process is concentrated on (0,∞). Suppose that
1) Λ((0,∞)) =∞,
2) the function
U(x) = Λ((x,∞)) (21)
is continuous and strictly decreasing on (0,∞).
Given G = [0, 1]× (0,∞), consider the space of configurations X (G) and the Poisson
measure P with intensity measure Π = λ× Λ, where λ is a Lebesgue measure on [0, 1].
In accordance with (1), a Lévy process ξ with the Lévy measure Λ can be defined on
the probability space (X (G),B(X (G)), P ) by
ξ(t,X) = at+ (L2) lim
ε→0
( ∑
(s,x)∈X,
s�t,ε�x�1
x− t
∫ 1
ε
xΛ(dx)
)
+
∑
(s,x)∈X,
s�t,1<x<∞
x. (22)
The sum is taken over all points (s, x) of configurations X ∈ X . The relation between
the configurations and the corresponding trajectories of the random process is very sim-
ple. Namely, if the point (s, x) belongs to the configuration X, then the corresponding
trajectory ξ(·, X) of the random process ξ at the moment s has a jump which is equal to
x. We denote, by Ξ, the mapping X → D[0, 1] defined by (22) so that
Ξ(X) = ξ(·, X).
Note that the different configurations generate different trajectories of a random pro-
cess. Namely, if X1 �= X2, then
Ξ(X1) �= Ξ(X2). (23)
Let Pξ denote a measure generated by ξ in the space D[0, 1], i.e.,
Pξ = PΞ−1. (24)
Given a function z ∈ D[0, 1], we denote, by Δz, the function
Δz(t) = z(t)− z(t− 0), t ∈ [0, 1].
Further for f ∈ L+
1 [0, 1], we denote a mapping Ψf : D[0, 1]→ D[0, 1], by
[Ψf (z)](t) = z(t) +
∑
s�t
(
Tf(s)(Δz(s))−Δz(s)
)
, (25)
where Tt, t � 0 is a semigroup of transformations defined by (6), i.e., Tt(x) = U−1(U(x)+
t). Here, U is defined by (21) and, unlike the general case considered in (6), it does not
depend on t ∈ [0, 1].
Further, we consider the semigroup of nonsingular transformations
Φf , f ∈ L+
1 ([0, 1], λ),
of the configuration space X (G), G = [0, 1] × (0,∞) defined by (10). It follows from
(10), (22), and (25) that
Ψf ◦ Ξ = Ξ ◦ Φf . (26)
Now it follows from (26) that the mapping Ψf is correctly defined by (25) Pξ−a.s..
Moreover, it follows from (7) and (26) that, for every f, g ∈ L+
1 ([0, 1], λ),
Ψf+g = Ψf ◦Ψg.
THE MEASURE PRESERVING AND NONSINGULAR . . . 151
Further, for f ∈ L+
1 ([0, 1], λ), we define a subset Df of the set D[0, 1] by
Df = {z ∈ D[0, 1] : ∀t ∈ (0, 1] 0 � Δz(t) < U−1(f(t))}.
Theorem 3. For every f ∈ L+
1 ([0, 1], λ),
PξΨ−1
f Pξ,
and, moreover,
PξΨ−1
f = e
� 1
0 f(t)dt · Pξ
∣∣
Df
.
Proof. Using Theorem 1 and (26), we get
PΦ−1
f P,
therefore,
(PΦ−1
f )Ξ−1 PΞ−1. (27)
The RHS of (27) equals Pξ. Now we calculate the LHS of (27):
(PΦ−1
f )Ξ−1 = P (Ξ ◦ Φf )−1 = P (Ψf ◦ Ξ)−1 = (PΞ−1)Ψ−1
f = PξΨ−1
f .
Consider a Lévy process with both positive and negative jumps. The Lévy measure Λ
of such a process satisfies the conditions Λ((0,∞)) > 0 and Λ((−∞, 0)) > 0. We suppose
in addition that the measure Λ satisfies the conditions
1) Λ((0,∞)) =∞, and Λ((−∞, 0)) =∞,
2) the function U : R→ R defined by the formula
U(x) =
{
Λ((x,∞)), if x > 0;
−Λ((−∞, x)), if x < 0
is continuous and strictly decreasing on the intervals (0,∞) and (−∞, 0).
On the space of configurations X ([0, 1]×R), we consider the Poisson measure P with
intensity measure Π(dt, dx) = dtΛ(dx).
In accordance with (1), the Lévy process ξ(t) with the Lévy measure Λ can be defined
on the probability space (X ([0, 1]× R), P ) by
ξ(t,X) = at+ (L2) lim
ε→0
( ∑
(s,x)∈X,
s�t,ε�|x|�1
x− t
∫
ε�|x|�1
xdΛ
)
+
∑
(s,x)∈X,
s�t,|x|>1
x. (28)
Further, consider the group Tt, t ∈ (−∞,∞) defined by (16) and the corresponding
group Φ̃f , f ∈ L[0, 1], of transformations of X ([0, 1] × R). As above, using a group Φ̃f
and a mapping Ξ [defined by (28)], we construct a group Ψ̃f of transformations of the
space D[0, 1] so that
Ψ̃f ◦ Ξ = Ξ ◦ Φ̃f .
It is easy to prove that the mapping Ψ̃f can be defined by the formula
[Ψ̃f(z)](t) = z(t) +
∑
s�t,
Tf(s)(Δz(s)) �=∞
(Tf(s)(Δz(s))−Δz(s)), (29)
where
Δz(t) = z(t)− z(t− 0), t ∈ [0, 1].
152 NATALYA V. SMORODINA
Theorem 4. For every Borel function f : [0, 1]→ R, the mapping Ψ̃f is correctly defined
Pξ−a.s. and preserves the measure Pξ, i.e.,
PξΨ̃−1
f = Pξ.
Proof follows from Theorem 3.
6. One multidimensional generalization.
In this section, we consider the configuration space X (G), where G has the form
S × (Rd \ {0}), S is a complete separable metric space, and Rd = Rd ∪ {∞}. By Sd−1,
we denote a unit sphere in Rd and, by σ(dω), ω ∈ Sd−1, we denote the surface measure
on Sd−1.
Let Q denote the mapping from Rd \ {0} into Sd−1 × (0,∞) defined by
Q(x) = (
x
‖x‖ , ‖x‖). (30)
It follows from (30) that, for ω ∈ Sd−1, r ∈ (0,∞),
Q−1(ω, r) = rω ∈ R
d \ {0}.
Recall that the representation of a measure μ in polar coordinates on Rd\{0} is a measure
μQ−1 on Sd−1 × (0,∞) (that is, the image of the measure μ under the action of Q).
On the configuration space X (G) = X (S× (Rd \ {0})), we consider a Poisson measure
P with intensity measure Π of the form
Π(dθ, dx) = π(dθ)Γθ(dx), (31)
where, as above, π is a finite measure on S, and Γθ is a σ−finite measure on Rd \ {0}.
We suppose that Γθ(∞) = 0 for π−a.s. θ ∈ S, and the representation of the measure Γθ
in polar coordinates has the form
ΓθQ
−1(dω, dr) = σ(dω)γθ
ω(dr),
where, in turn, the measure γθ
ω satisfies, for σ−a.s. ω ∈ Sd−1, the following conditions:
1) γθ
ω((0,∞)) =∞,
2) for every r > 0 γθ
ω((r,∞)) <∞, and the function
Uθ
ω(r) = γθ
ω((r,∞))
is continuous and strictly decreasing.
For fixed θ ∈ S, we construct a d−parametric group of Γθ−preserving transformations
of Rd \ {0}. To this end, we define a mapping fθ
ω from (0,∞] into [0,∞) for ω ∈ Sd−1 by
the formula
fθ
ω(r) =
(
d · Uθ
ω(r)
)1/d (33)
for r ∈ (0,∞). Set fθ
ω(∞) = 0. It is easy to see that, for r ∈ (0,∞),
(fθ
ω)−1(r) =
(
Uθ
ω
)−1(rd
d
)
. (34)
Further, by μ, we denote a measure on [0,∞) of the form μ(dr) = rd−1dr.
THE MEASURE PRESERVING AND NONSINGULAR . . . 153
Lemma 1. For θ ∈ S, ω ∈ Sd−1, the mapping fθ
ω transforms the measure γθ
ω into the
measure μ and, conversely, the mapping (fθ
ω)−1 transforms the measure μ into γθ
ω, i.e.,
γθ
ω
(
fθ
ω
)−1 = μ
and
μfθ
ω = γθ
ω.
Proof. We denote the Lebesgue measure on [0,∞) by λ and the mapping from [0,∞)
into [0,∞) by g defined by the formula g(r) = rd
d . Note that fθ
ω = g−1 ◦ Uθ
ω.
It is easy to check that μg−1 = λ and γθ
ω(Uθ
ω)−1 = λ. So, we have
γθ
ω(fθ
ω)−1 = γθ
ω(g−1 ◦ Uθ
ω)−1 = μ.
By Fθ, we denote the mapping from Sd−1 × (0,∞] into Sd−1 × [0,∞) defined by the
formula
Fθ(ω, r) = (ω, fθ
ω(r)) (35)
It follows easily from Lemma 1 that the mapping Fθ transforms the measure ΓθQ
−1 into
the measure σ × μ, i.e.,
(ΓθQ
−1)F−1
θ = σ × μ. (36)
Further, by Eθ, Eθ : Rd \ {0} → Rd, we denote the mapping
Eθ = Q−1 ◦ Fθ ◦Q. (37)
It follows from (33) and (37) that
Eθ(x) =
x
‖x‖
(
dUθ
x
‖x‖
(‖x‖))1/d (38)
and
E−1
θ (x) =
x
‖x‖
(
Uθ
x
‖x‖
)−1(‖x‖d
d
)
. (39)
Now, for every t ∈ Rd, we define a mapping T θ
t , T
θ
t : Rd \ {0} → Rd \ {0} by
T θ
t (x) = E−1
θ (Eθ(x) + t). (40)
It is clear that, for every s, t ∈ Rd,
T θ
t+s = T θ
t ◦ T θ
s . (41)
We now show that the group T θ
t , t ∈ Rd is a d−parametric group of Γθ−preserving
transformations of Rd \ {0}, i.e., for every t ∈ R
d,
Γθ(T θ
t )−1 = Γθ. (42)
Notice that T θ
t is a superposition of three transformations, namely, Eθ, shift transfor-
mation, and E−1
θ . It follows from (36) and (37) that the transformationEθ transforms the
measure Γθ into the Lebesgue measure, the shift transformation preserves the Lebesgue
measure, and U−1
θ transforms the Lebesgue measure into Γθ.
By L(S,Rd), we denote the set of all Borel functions on S taking value in Rd. Given
f ∈ L(S,Rd), we define a mapping
τf : S × (Rd \ {0})→ S × (Rd \ {0}),
by
τf (θ, x) = (θ, T θ
f(θ)). (43)
It follows from (41) that, for every f, g ∈ L(S,Rd),
τf+g = τf ◦ τg, (44)
154 NATALYA V. SMORODINA
and (42) implies that, for every f ∈ L(S,Rd),
Πτ−1
f = Π.
Using the group of transformations τf , f ∈ L(S), we construct a group Φ̃f , f ∈
L(S,Rd) of transformations of X (S × (Rd \ {0})) by analogy with (10). In accordance
with (3), for f ∈ L(S,Rd), we define a mapping Φ̃f : X (S×(Rd\{0}))→ X (S×(Rd\{0}))
as a mapping generated by τf and such that
Φ̃f (X) = Y = {(θ, T θ
f(θ)(x))}.
The main result of this section is the following
Theorem 5.
1. For every f, g ∈ L(S,Rd), we have
Φ̃f+g = Φ̃f ◦ Φ̃g.
2. For every f ∈ L(S,Rd),
P Φ̃−1
f = P,
i.e., the group Φ̃f , f ∈ L(S,Rd), is a group of P− preserving transformations.
Proof. The statements of this theorem can be proved by the same arguments as Theorem
2.
Bibliography
1. I.M.Gel’fand, M.I.Graev, A.M.Vershik., Representations of the group of diffeomorphisms., Rus-
sian Math. Surveys 30 (1975), 3-50.
2. J.F.C.Kingman, Poisson Processes., Clarendon Press, Oxford, 1993.
3. J.Kerstan, K.Mattes, and J.Mecke, Infinite Divisible Point Processes, Akademie-Verlag, Berlin,
1978.
4. A.V.Skorokhod, On the differentiability of measures which correspond to stochastic processes
I, Theory Probab. Appl. II (1957) 629-649. 2 (1957), 407-423.
5. A.V.Skorokhod, Stochastic Processes with Independent Increments, Nauka, Moskow, 1986.
6. N.V.Smorodina, The invariant and quasi-invariant transformations of the stable Lévy pro-
cesses, Acta Appl. Math. 97 (2007), no. 1-3, 221-238.
7. Y.Takahashi, Absolute continuity of Poisson random fields, Publ. Res. Inst. Math. Sci. 26
(1990), 629-649.
8. A.Vershik, N.Tsilevich., Quasi-invariance of the gamma process and multiplicative properties
of the Poisson-Dirichlet measures, C.R.Acad.Sci.Paris Ser.I Math. (1999), no. 329, 163-168.
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