Page 99 - Special Topic Session (STS) - Volume 3

STS517 Jan Rosinski´

Infinitely divisible processes and their random
translations
Jan Rosinski´
University Of Tennessee, Usa

Abstract
The celebrated Cameron-Martin formula relates a Gaussian process to its
translation by a deterministic function and shows an isomorphism between
moments of nonlinear functionals of the original process and the translated
one. We show that a similar phenomenon occurs for more general infinitely
divisible processes when we allow random translations. Precise understanding
of Lévy measures of processes on path spaces is the key to produce rich classes
of admissible random translations. We illustrate this approach on examples of
squared Bessel processes, Feller diffusions, permanental processes, as well as
Lévy processes.

Keyword
path Lévy measure; admissible shifts; permanental processes; Lévy processes.

1. Introduction
be a centered Gaussian process over a set T. The Cameron-
Let = ( )
Martin Formula says that for every random variable in the -closure of the
2
subspace spanned by G and for any measurable functional : ℝ ↦ ℝ

Where ∅() = ( ). This formula has many applications, including stochastic

differential equations and stochastic partial driven by Gaussian random fields.
It is well-known that (1) does not extend to the Poissonian case. Indeed, it
∈[0,1] is a Poisson process, then there is no
is easy to show that if = ( )
function : [0,1] → ℝ, ≢ 0, such that

for all measurable functionals : ℝ [0,1] → ℝ and some random variable ≥ 0
with = 1.

There is however a rich class of random translations associated with each
infinitely divisible process. By definition, a process = ( ) over a general
∈
set is said to be infinitely divisible if for every ≥ 2 there exist i.i.d. processes
,
( ) ∈ , = 1, … , such that

88 | I S I W S C 2 0 1 9