Multiple Wick Product Chaos Processes

Abstract

Let u(x) x∈R ^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, δ}(·) be the density of an R ^q valued canonical normal random variable with mean x and variance δ and let {G _{x, δ}; (x, δ)∈R ^q×[0,1 ]} be the mean zero Gaussian process with covariance

$$EG_{x,\delta } G_{y,\delta } = \iint {u(s - t)p_{x,\delta } (s)}{ }p_{y,\delta } (t)ds{ }dt$$

A finite positive measure μ on R ^q is said to be in $G^r$ with respect to u, if

$$\iint {\left( {u\left( {x,y} \right)} \right)^r }d\mu \left( x \right){\text{ }}d\mu \left( y \right) < \infty $$

When $\mu \in G^{\left| {\bar m} \right|}$, a multiple Wick product chaos $\mathfrak{C}_{\bar m,1,0,\mu } (\bar X)$ is defined to be the limit in L ², as δ→0, of

$$\mathfrak{C}_{\bar m,1,0,\mu ,\delta } (\bar x)\mathop = \limits^{{def}} \int {\prod\limits_{j = 1}^k {:\prod\limits_{p = 1}^{mj} {G_{y + x_{j,p} ,\delta } :} } } du(y)$$

where

$$\bar m = (m_1 , \ldots ,m_k ) \in Z_ + ^k ,{and with }\left| {\bar m} \right| = ^{{def}} \sum\nolimits_{j = 1}^k {m_j } $$

,

$$\bar x = (x_{1,1} , \ldots ,x_{1,m_1 } , \ldots x_{k,1} , \ldots ,x_{k,m_k } ) \in (R^q )^{\left| {\bar m} \right|} $$

$:\prod\nolimits_{p = 1}^{m_j } {G_{y + x_{j,p} ,\delta ,N} :}$ denotes the Wick product of the m _j normal random variables ${\{ G_{y + x_{j,p} ,\delta } \} _{p = 1}^{m_j } }$.

Consider also the associated decoupled chaos processes $\mathfrak{C}_{r,1,0,\mu }^{dec} (x_1 , \ldots ,x_r )$, $r \leqslant \left| {\bar m} \right|$ defined as the limit in L ², as δ→0, of

$$\mathfrak{C}_{r,1,0,\mu ,\delta }^{dec} (x_1 , \ldots ,x_r )\mathop = \limits^{{def}} \int {\prod\limits_{j = 1}^r {G_{y + x_j ,\delta }^{(j)} d\mu (y)} } $$

where $\left\{ {G_{x,\delta }^{(j)} } \right\}$ are independent copies of G _x,δ.

Define

$$S_{\bar m,\varepsilon } = \left\{ {\bar x:\left| {x_{j,p} - x_{j',q} } \right| > \varepsilon ,\forall 1 \leqslant p \leqslant m_j ,1 \leqslant q \leqslant m_{j'} ,1 \leqslant j,j' \leqslant k,j \ne j'} \right\}$$

Note that a neighborhood of the diagonals of ${\bar x}$ in $(R^q )^{\left| {\bar m} \right|}$ is excluded, except those points on the diagonal which originate in the same Wick product in (i). Set

$$S_{\bar m, \ne } = \bigcup\limits_{\varepsilon > 0} {S_{\bar m,\varepsilon } } $$

One of the main results of this paper is:

Theorem A. If $\mathfrak{C}_{r,1,0,\mu ,\delta }^{dec} (x_1 , \ldots ,x_r )$ is continuous on (R ^q)^r for all $r \leqslant \left| {\bar m} \right|$ then $\mathfrak{C}_{\bar m,1,0,\mu } (\bar x)$ is continuous on $S_{\bar m, \ne }$.

When u satisfies some regularity conditions simple sufficient conditions are obtained for the continuity of $\mathfrak{C}_{r,1,0,\mu }^{dec} (x_1 , \ldots ,x_r )$ 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.