Abstract Fractional relaxation equations, as well as relaxation functions time-changed by independent stochastic processes have been widely studied (see, for example, [21], [33] and [11]). We start here by proving that the upper-incomplete Gamma function satisfies the tempered-relaxation equation (of index ρ ∈ (0, 1)); thanks to this explicit form of the solution, we can then derive its spectral distribution, which extends the stable law. Accordingly, we define a new class of selfsimilar processes (by means of the n-times Laplace transform of its density) which is indexed by the parameter ρ: in the special case where ρ = 1, it reduces to the stable subordinator. Therefore the parameter ρ can be seen as a measure of the local deviation from the temporal dependence structure displayed in the standard stable case.

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回火松弛方程及相关广义稳定过程

摘要 分数松弛方程以及由独立随机过程随时间变化的松弛函数已得到广泛研究（例如，参见 [21]、[33] 和 [11]）。我们从这里开始，证明上不完全 Gamma 函数满足调和松弛方程（指数 ρ ∈ (0, 1)）；由于解的这种显式形式，我们可以推导出它的光谱分布，从而扩展了稳定定律。因此，我们定义了一类新的自相似过程（通过其密度的 n 次拉普拉斯变换），它由参数 ρ 索引：在 ρ = 1 的特殊情况下，它简化为稳定的从属过程。因此，参数 ρ 可以看作是对标准稳定情况下显示的时间相关结构的局部偏差的度量。