In this talk, we introduce a fractional analogue of the Wiener chaos expansion. It is important to highlight that the fractional order relates to the order of chaos decomposition elements, and not to the process itself, which continues to be the standard Wiener process. The central instrument in our fractional analogue of the Wiener chaos expansion is the function denoted as $\mathcal{H}_\alpha(x,y)$, which is referred to herein as a power-normalised parabolic cylinder function ( and is very similar to the Hermite function).

Through careful analysis of several fundamental deterministic and stochastic properties, we affirm that this function essentially
serves as a fractional extension of the Hermite polynomial.

The power-normalised parabolic cylinder function $\mathcal{H}_\alpha(W_t,t)$ demonstrates martingale properties and can be interpreted as a fractional Itô integral with 1 as the integrand, thereby drawing parallels with its non-fractional counterpart.