We study the connection between PDEs and Lévy processes running with clocks given by time-changed Poisson processes with stochastic drifts. The random times we deal with are therefore given by time-changed Poissonian jumps related to some Frobenius-Perron operators $K$ associated to random translations. Moreover, we also consider their hitting times as a random clock. Thus, we study processes driven by equations involving time-fractional operators (modelling memory) and fractional powers of the difference operator $I-K$ (modelling jumps). For this large class of processes we also provide, in some cases, the explicit representation of the transition probability laws. To this aim, we show that a special role is played by the translation operator associated to the representation of the Poisson semigroup.
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机译：我们研究PDE和Lévy进程之间的联系，该进程与带有随机漂移的时变Poisson进程给出的时钟一起运行。因此，我们处理的随机时间是由与某些Frobenius-Perron算子$ K $有关的时变泊松跳跃给出的，后者与随机翻译有关。此外，我们还将其命中时间视为随机时钟。因此，我们研究了由方程驱动的过程，这些方程涉及时间分数算子（建模记忆）和差分算子$ I-K $的分数幂（建模跳跃）。对于此类大型过程，在某些情况下，我们还提供了过渡概率定律的明确表示。为此，我们表明与Poisson半群的表示形式相关的翻译运算符扮演着特殊的角色。
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