Multiple Wick Product Chaos Processes |
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Authors: | Michael B Marcus Jay Rosen |
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Institution: | (1) Department of Mathematics, The City College of CUNY, New York, New York, 10031;(2) Department of Mathematics, College of Staten Island, CUNY, Staten Island, New York, 10301 |
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Abstract: | Let u(x) xR
q
be a symmetric nonnegative definite function which is bounded outside of all neighborhoods of zero but which may have u(0)=. Let p
x, (·) be the density of an R
q
valued canonical normal random variable with mean x and variance and let {G
x, ; (x, )R
q
×0,1 ]} be the mean zero Gaussian process with covariance A finite positive measure on R
q
is said to be in
with respect to u, if
When
, a multiple Wick product chaos
is defined to be the limit in L
2, as 0, of where ,
denotes the Wick product of the m
j
normal random variables
.Consider also the associated decoupled chaos processes
,
defined as the limit in L
2, as 0, of where
are independent copies of G
x,.Define Note that a neighborhood of the diagonals of
in
is excluded, except those points on the diagonal which originate in the same Wick product in (i). Set
One of the main results of this paper is:
Theorem A. If
is continuous on (R
q
)
r
for all
then
is continuous on
.When u satisfies some regularity conditions simple sufficient conditions are obtained for the continuity of
on (R
q
)
r
. Also several variants of (i) are considered and related to different types of decoupled processes. These results have applications in the study of intersections of Lévy process and continuous additive functionals of several Lévy processes. |
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Keywords: | Gaussian chaos processes Levy processes Banach space |
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