### Abstract

Let X =(X
_{t}
)
_{t≥0}
be a stochastic process which has a (not necessarily stationary) independent increment on a probability space (Ω, P). In this paper, we study the following Cauchy problem related to the stochastic process X: (Formula presented) We provide a sufficient condition on X (see Assumptions 2.1 and 2.2) to guarantee the unique solvability of equation (*) in L
_{p}
([0,T]; H
_{p}
^{φ}
), where H
_{p}
^{φ}
is a φ-potential space on R
^{d}
(see Definition 2.9). Furthermore we show that for this solution, (Formula presented), where N is independent of u and f.

Original language | English |
---|---|

Pages (from-to) | 3417-3450 |

Number of pages | 34 |

Journal | Transactions of the American Mathematical Society |

Volume | 371 |

Issue number | 5 |

DOIs | |

Publication status | Published - 2019 May 1 |

### Keywords

- Diffusion equation for jump process
- L -theory
- Non-stationary increment
- Pseudo-differential operator

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics