Multiple Wick Product Chaos Processes

Let u(x) x ∈ Rq be a symmetric non-negative definite function which is bounded away from 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 EGx,δGy,δ′ = ∫ ∫ u(s− t)px,δ(s)py,δ′(t) ds dt. A finite positive measure μ on Rq is said to be in Gr, with respect to u, if ∫ ∫ (u(x, y)) dμ(x) dμ(y) < ∞. When μ ∈ G|m̄|, a multiple Wick product chaos Gm̄,1,0,μ(x̄) is defined to be the limit in L2, as δ → 0, of Gm̄,1,0,μ,δ(x̄) def = ∫ k ∏