In this paper, we introduce a fractional analogue of the Wiener polynomial 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 remains the standard Wiener process. The central instrument in our fractional analogue of the Wiener chaos expansion is the function denoted as \({\mathcal {H}}_\alpha (x,y)\), referred to herein as a power-normalised parabolic cylinder 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. In particular, the power-normalised parabolic cylinder function with the Wiener process and time as its arguments, \({\mathcal {H}}_\alpha (W_t,t)\), 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. To build a fractional analogue of polynomial Wiener chaos on the real line, we introduce a new function, which we call the extended Hermite function, by smoothly joining two power-normalized parabolic cylinder functions at zero. We form an orthogonal set of extended Hermite functions as a one-parameter family and use tensor products of the extended Hermite functions as building blocks in the fractional Wiener chaos expansion, in the same way that tensor products of Hermite polynomials are used as building blocks in the Wiener chaos polynomial expansion.

在本文中，我们介绍了 Wiener 多项式混沌展开的分数类比。需要强调的是，分数顺序与混沌分解元素的顺序有关，而不是与过程本身有关，该过程仍然是标准的 Wiener 过程。维纳混沌展开的分数阶模拟中的核心工具是表示为 \（{\mathcal {H}}_\alpha （x，y）\） 的函数，在本文中称为幂归一化抛物线圆柱函数。通过仔细分析几个基本的确定性和随机性，我们确认该函数本质上是 Hermite 多项式的分数扩展。特别是，以维纳过程和时间为参数的幂归一化抛物线圆柱函数 \（{\mathcal {H}}_\alpha （W_t，t）\） 展示了马丁格尔性质，可以解释为以 1 为被积函数的分数 Itô 积分，从而与其非分数对应物平行。为了在实线上构建多项式维纳混沌的分数阶模拟，我们引入了一个新函数，我们称之为扩展 Hermite 函数，通过在零处平滑连接两个幂归一化抛物线圆柱函数。我们形成一组正交的扩展 Hermite 函数作为单参数族，并使用扩展 Hermite 函数的张量积作为分数阶 Wiener 混沌展开的构建块，就像 Hermite 多项式的张量积用作 Wiener 混沌多项式扩展中的构建块一样。