Page 100 - Special Topic Session (STS) - Volume 3

STS517 Jan Rosinski´
where denotes the equality in finite dimensional distributions. Every infinitely

=
∈ has a Lévy measure ν defined on the path space
divisible process = ( )
ℝ , which characterizes the nonGaussian part of X. Suppose that a stochastic

∈ is independent of X and the distribution L(Z) of Z is
process = ( )
absolutely continuous with respect to ν. Then there exists a measurable
function g : ℝ ↦ ℝ such that for any measurable functional F : ℝ ↦ ℝ

+
+

There are two basic directions of applying identity (2). The first one is to
∈ of interest, associate with it (possibly easier to
start with a process = ( )
handle) infinitely divisible process = ( ) whose Lévy measure
∈
dominates the law of Z, and transfer path properties of X to Z via identity (2).
Using Dynkin’s Isomorphism Theorem, Marcus, M.B., &Rosen, J. (1992),
&(2006) derived many results for local times of Markov processes, including
Lévy processes. Another direction of applications of (2) is much harder, to
derive information about X by utilizing Z. One way to approach it is to consider
the “converse” version of (2) which expresses X as the process X +Z with
changed measure.
∈ is said to be Poissonian
Recall that a stochastic process = ( )
infinitely divisible if its all finite dimensional marginal distributions are infinitely
divisible without Gaussian part.
Throughout this short paper, an identity as (2) reads: if one side exists then the
other does and they are equal.

2. Methodology
Successful implementation of identities like (2) requires precise
understanding of Lévy measures of processes, which are defined on path
spaces with the usual cylindrical σ-algebras (as opposed to σ-rings in Lee, P.M.
(1967) and Maruyama, G. (1970)). We view Lévy measures as “laws of
processes” defined on possibly infinite measure spaces and call such
“processes” representations of Lévy measures. Properties of Lévy measures are
defined by properties of their representations. Transfer of regularity property
puts the Lévy measure on the same Borel function space where paths of the
corresponding infinitely divisible processes belong. This allows to relate path
properties of processes and representations of their Lévy measures.
Let ℝ be the space of all functions ∶ ↦ ℝ, and let B denote its

T
cylindrical (product) −algebra. Let 0T denotes the origin of ℝ , considerd as

a point or the one-point set, depending on the context. We give the following
definition of path Lévy measure.

T
Definition 1. A measure ν on (ℝ , B )is said to be a Lévy measure if the
T
following two conditions hold

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