In this paper we focus on strong solutions of some heat-like problems with a non-local derivative in time induced by a Bernstein function and an elliptic operator given by the generator or the Fokker–Planck operator of a Pearson diffusion, covering a large class of important stochastic processes. Such kind of time-non-local equations naturally arise in the treatment of particle motion in heterogeneous media. In particular, we use spectral decomposition results for the usual Pearson diffusions to exploit explicit solutions of the aforementioned equations. Moreover, we provide stochastic representation of such solutions in terms of time-changed Pearson diffusions. Finally, we exploit some further properties of these processes, such as limit distributions and long/short-range dependence.

在本文中，我们专注于一些类热问题的强解，其非局部时间导数由伯恩斯坦函数和由发生器给出的椭圆算子或皮尔逊扩散的福克 - 普朗克算子引起，涵盖了一个大类重要的随机过程。这种非局部时间方程在处理非均质介质中的粒子运动时自然会出现。特别是，我们使用通常 Pearson 扩散的谱分解结果来利用上述方程的显式解。此外，我们根据时变 Pearson 扩散提供了此类解决方案的随机表示。最后，我们利用这些过程的一些进一步特性，例如极限分布和长/短程依赖性。