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

Research output: Contribution to journalArticle

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 languageEnglish
Pages (from-to)3417-3450
Number of pages34
JournalTransactions of the American Mathematical Society
Volume371
Issue number5
DOIs
Publication statusPublished - 2019 May 1

Fingerprint

Independent Increments
Random processes
Diffusion equation
Stochastic Processes
Unique Solvability
Probability Space
Cauchy Problem
Sufficient Conditions

Keywords

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

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

@article{8aa49766d7514a848d93584e9e566c72,
title = "An L p -theory for diffusion equations related to stochastic processes with non-stationary independent increment",
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.",
keywords = "Diffusion equation for jump process, L -theory, Non-stationary increment, Pseudo-differential operator",
author = "Ildoo Kim and Kim, {Kyeong Hun} and Panki Kim",
year = "2019",
month = "5",
day = "1",
doi = "10.1090/tran/7410",
language = "English",
volume = "371",
pages = "3417--3450",
journal = "Transactions of the American Mathematical Society",
issn = "0002-9947",
publisher = "American Mathematical Society",
number = "5",

}

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 -