Multiple Wick Product Chaos Processes1

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Abstract

Let u(x) x∈Rq be a symmetric nonnegative definite function which is bounded outside of all neighborhoods of zero but which may have u(0)=∞. Let px, δ(·) be the density of an Rq valued canonical normal random variable with mean x and variance δ and let {Gx, δ; (x, δ)∈Rq×[0,1]} be the mean zero Gaussian process with covariance

A finite positive measure μ on Rq is said to be in ℊr with respect to u, if

When μ ∈ ℊ|𝓂|, a multiple Wick product chaos 𝒞JOURNAL/jtpro/04.02/00044050-199912020-00011/OV0323/v/2017-10-11T050021Z/r/image-png, 1, 0, μ}(JOURNAL/jtpro/04.02/00044050-199912020-00011/OV0335/v/2017-10-11T050021Z/r/image-png) is defined to be the limit in L2, as δ→0, of

where JOURNAL/jtpro/04.02/00044050-199912020-00011/OV0323/v/2017-10-11T050021Z/r/image-png = (m1,…, mk) ∈ Zk+, and with |JOURNAL/jtpro/04.02/00044050-199912020-00011/OV0323/v/2017-10-11T050021Z/r/image-png| = def∑kj = 1mj,

Πp = 1mjGy + xj, p, δ, N: denotes the Wick product of the mj normal random variables Gy + xj, pδ}mjp = 1

Consider also the associated decoupled chaos processes 𝒢decr, 1, 0, μ, (x1,…, xr), r≤ |JOURNAL/jtpro/04.02/00044050-199912020-00011/OV0323/v/2017-10-11T050021Z/r/image-png| defined as the limit in L2, as δ→0, of

where {G(j)x, δ are independent copies of Gx,δ.

Define

Note that a neighborhood of the diagonals of JOURNAL/jtpro/04.02/00044050-199912020-00011/OV0335/v/2017-10-11T050021Z/r/image-png in (Rq)|JOURNAL/jtpro/04.02/00044050-199912020-00011/OV0323/v/2017-10-11T050021Z/r/image-png| 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 𝒞decr, 1, 0, μ, (x1,…,xr) is continuous on (Rq)r for all r≤ |JOURNAL/jtpro/04.02/00044050-199912020-00011/OV0323/v/2017-10-11T050021Z/r/image-png| then 𝒞}JOURNAL/jtpro/04.02/00044050-199912020-00011/OV0323/v/2017-10-11T050021Z/r/image-png, 1, 0, μ(JOURNAL/jtpro/04.02/00044050-199912020-00011/OV0335/v/2017-10-11T050021Z/r/image-png) is continuous on SJOURNAL/jtpro/04.02/00044050-199912020-00011/OV0323/v/2017-10-11T050021Z/r/image-png, ≠

When u satisfies some regularity conditions simple sufficient conditions are obtained for the continuity of 𝒞decr, 1, 0, μ(x1,…, xr) on (Rq)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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