### 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 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

**
An L
_{p}
-theory for diffusion equations related to stochastic processes with non-stationary independent increment
.** / Kim, Ildoo; Kim, Kyeong Hun; Kim, Panki.

Research output: Contribution to journal › Article

_{p}-theory for diffusion equations related to stochastic processes with non-stationary independent increment ',

*Transactions of the American Mathematical Society*, vol. 371, no. 5, pp. 3417-3450. https://doi.org/10.1090/tran/7410

}

TY - JOUR

T1 - An L p -theory for diffusion equations related to stochastic processes with non-stationary independent increment

AU - Kim, Ildoo

AU - Kim, Kyeong Hun

AU - Kim, Panki

PY - 2019/5/1

Y1 - 2019/5/1

N2 - 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.

AB - 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.

KW - Diffusion equation for jump process

KW - L -theory

KW - Non-stationary increment

KW - Pseudo-differential operator

UR - http://www.scopus.com/inward/record.url?scp=85062006426&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85062006426&partnerID=8YFLogxK

U2 - 10.1090/tran/7410

DO - 10.1090/tran/7410

M3 - Article

VL - 371

SP - 3417

EP - 3450

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 5

ER -