I'm trying to understand the Wick renormalization in the framework of the Ito integral.

I saw the Wick theorem as presented on Wikipedia in a QFT course and I would like to understand how that is equivalent to the one that follows.

Firstly let me define the multiple Ito integral. Given an Hilbert space $H$, consider $H^{\hat\otimes n}$ that is the symmetrized tensor product of $H$ with itself $n$-times. Given an element $f_n\in H^{\hat\otimes n}$: \begin{equation} f_n=\sum_{finite}f_{i_1\cdots i_n}e_{i_1}^{k_1}\hat\otimes\cdots\hat\otimes e_{i_m}^{k_m} \end{equation} where $\{e_1,\ldots,e_n\}$ is an orthonormal basis of $H$ and $i_1,\ldots, i_n$ are all different, the multiple Ito integral $I_n(f_n)$ is: \begin{equation} I_n(f_n):=\sum_{finite}f_{i_1\cdots i_n} H_{k_1}(e_{i_1})\hat\otimes\cdots\hat\otimes H_{k_m}(e_{i_m}) \end{equation} where the $H_i$ are the Hermite polynomials.

Then given $f_n\in H^{\hat\otimes n} $, $g_m\in H^{\hat\otimes m}$ we define the Wick product $\diamond$: \begin{equation} I_n(f_n)\diamond I_m(g_m)=I_{n+m}(f_n\hat\otimes g_m) \end{equation} Therefore we can define powers of $f$ and polynomials using this Wick product that should be equivalent to the one mentioned earlier.

Can anyone give me some clues in understanding how these things are related?

Fore more details a possible reference is: http://arxiv.org/abs/0901.4911

  • 5
    $\begingroup$ ...and the question is? $\endgroup$ – ACuriousMind Jun 13 '16 at 21:54
  • $\begingroup$ Sorry, I don't know how but the last part was missing. Now should make more sense $\endgroup$ – popoolmica Jun 13 '16 at 22:18

I found a paper that helped me a bit in understanding how things work. So here is what I understood.

Given a random variable we define a formal power series and a formal derivation such that: \begin{equation} \frac{\partial}{\partial U}\left(\sum_{n=0}^\infty a_nU^n\right)=\sum_{n=0}^\infty (n+1)a_{n+1}U^n \end{equation} The "usual" Wick product $:\ :$ is recursively defined so that it satisfies: \begin{equation*} \begin{split} :U^0:&=1\\ \frac{\partial}{\partial U}:U^n:&=n:U^{n-1}:\\ \mathbb{E}(:U^n:)&=0 \end{split} \end{equation*} These relations imply: \begin{equation*} \begin{split} :U:&=U-\mathbb{E}(U)\\ :U^2:&=U^2-2U\mathbb{E}(U)-\mathbb{E}(U^2)+2\mathbb{E}(U)^2\\ \end{split} \end{equation*} and so on.

For a mean zero Gaussian random variable $U$ with variance $\sigma^2$ these relations imply: \begin{equation} :U^n:=\sigma^nH_n(\frac{U}{\sigma}) \end{equation} This shows that in the case of Gaussian random variables the "usual" Wick product reproduce the result obtained in the framework of It$\hat{o}$ calculus.

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