identified with the Hilbert space tensor product $\mathbb{K}^{N_1}\otimes\cdots\otimes\mathbb{K}^{N_q}$ by the unique linear isomorphism $\iota:\mathbb{K}^{N_1}\otimes\cdots\otimes\mathbb{K}^{N_q}\to\mathbb{K}^{N_1\times\cdots\times N_q}$ such that $\iota(x_1\otimes\cdots\otimes x_q)=(x_1^{i_1}\cdots x_q^{i_q})_{(i_1,\dots,i_q)\in\prod_{k=1}^q[N_k]}$ for $x_k\in\mathbb{K}^{N_k}$, $k=1,\dots,q$ (cf. Example E.10 of [@Janson1997]).

For two $\mathbb{K}$-valued arrays $S,T\in\mathbb{K}^{N_1\times\cdots\times N_q}$, we define their Hadamard-type product (i.e. entry-wise product) by $$S\circ T:=(S^{i_1,\dots,i_q}T^{i_1,\dots,i_q})_{(i_1,\dots,i_q)\in\prod_{k=1}^q[N_k]}\in\mathbb{K}^{N_1\times\cdots\times N_q}.$$ Also, we set $$\|T\|_{\ell_p}:=
\left\{\begin{array}{ll}
\displaystyle\left\{\sum_{(i_1,\dots,i_q)\in\prod_{k=1}^q[N_k]}|T^{i_1,\dots,i_q}|^p\right\}^{1/p} & \text{if }p\in(0,\infty),\\
\displaystyle\max_{(i_1,\dots,i_q)\in\prod_{k=1}^q[N_k]}|T^{i_1,\dots,i_q}| & \text{if }p=\infty.
\end{array}\right.$$

Now suppose that $V$ is a real Hilbert space. For $T\in V^{N_1\times\cdots\times N_q}$ and $x\in V$, we define Let $m$ be a positive integer. For each $j=1,\dots,m$, let $V_j$ be a real Hilbert space, $p_j\in\mathbb{N}$, $N_1^{(j)},\dots,N_{p_j}^{(j)}\in\mathbb{N}$ and $T_j\in V_j^{N^{(j)}_1\times\cdots\times N^{(j)}_{p_j}}$. Then we define In particular, we write $$T^{\otimes m}:=\underbrace{T\otimes\cdots\otimes T}_{m}.$$

Malliavin calculus {#sec:malliavin}
------------------

This subsection introduces some notation and concepts from Malliavin calculus used throughout the paper. We refer to @Nualart2006, Chapter 2 of @NP2012 and Chapter 15 of @Janson1997 for further details on this subject.

Given a probability space $(\Omega,\mathcal{F},P)$, let $\mathbb{W}=(\mathbb{W}(h))_{h\in H}$ be an isonormal Gaussian process over a real separable Hilbert space $H$.

Let $V$ be another real separable Hilbert space. For any real number $p\geq1$ and any integer $k\geq1$, $\mathbb{D}_{k,p}(V)$ denotes the stochastic Sobolev space of $V$-valued random variables which are $k$ times differentiable in 