In the context of forward uncertainty quantification, we investigate the convergence of the Galerkin projections for random fractional differential equations. The governing system is formed by a finite set of independent input random parameters (a germ) and by a fractional derivative in the Caputo sense. Input uncertainty arises from biased measurements, and a fractional derivative, defined by a convolution, takes past history into account. While numerical experiments on the gPC-based Galerkin method are already available in the literature for random ordinary, partial and fractional differential equations, a theoretical analysis of mean-square convergence is still lacking for the fractional case. The aim of this contribution is to fill this gap, by establishing new inequalities and results and by raising new open problems.

在正向不确定性量化的背景下，我们研究了随机分数阶微分方程的伽辽金投影的收敛性。控制系统由一组有限的独立输入随机参数（胚芽）和卡普托意义上的分数阶导数构成。输入不确定性源于有偏差的测量，而由卷积定义的分数导数则考虑了过去的历史。虽然文献中已经针对随机常微分方程、偏微分方程和分数阶微分方程进行了基于 gPC 的伽辽金方法的数值实验，但仍然缺乏分数阶情况的均方收敛性的理论分析。本贡献的目的是通过建立新的不平等和结果以及提出新的开放问题来填补这一空白。