## 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 |
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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 |

## Keywords

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

## ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics