We consider a general class of integro-differential evolution equations which includes the governing equation of the generalized grey Brownian motion and the time- and space-fractional heat equation. We present a general relation between the parameters of the equation and the distribution of the underlying stochastic processes, as well as discuss different classes of processes providing stochastic solutions of these equations. For a subclass of evolution equations, containing Marichev-Saigo-Maeda time-fractional operators, we determine the parameters of the corresponding processes explicitly. Moreover, we explain how self-similar stochastic solutions with stationary increments can be obtained via linear fractional Lévy motion for suitable pseudo-differential operators in space.

我们考虑一类一般的积分微分演化方程，包括广义灰色布朗运动的控制方程和时间和空间分数热方程。我们提出了方程参数与潜在随机过程分布之间的一般关系，并讨论了提供这些方程随机解的不同类别的过程。对于包含 Marichev-Saigo-Maeda 时间分数算子的演化方程子类，我们明确地确定了相应过程的参数。此外，我们解释了如何通过线性分数 Lévy 运动为空间中合适的伪微分算子获得具有固定增量的自相似随机解。