### Abstract

In this article, we present an Lp-theory (p ≥ 2) for the semi-linear stochastic partial differential equations (SPDEs) of type ∂ _{t} ^{α} u = L(ω,t,x)u +f (u)+∂ _{t} ^{β} ∞ ∑ k=1 ∫ _{0} ^{t} (∧^{k} (ω,t,x)u +g^{k}(u))dw _{t} ^{k}, where α ∈ (0, 2), β < α + 1/2 and ∂ _{t} ^{α} and ∂ _{t} ^{β} denote the Caputo derivatives of order α and β, respectively. The processes w _{t} ^{k}, k ∈ N = 1, 2, . . ., are independent one-dimensional Wiener processes, L is either divergence or nondivergence-type second-order operator, and ∧^{k} are linear operators of order up to two. This class of SPDEs can be used to describe random effects on transport of particles in medium with thermal memory or particles subject to sticking and trapping. We prove uniqueness and existence results of strong solutions in appropriate Sobolev spaces, and obtain maximal L_{p}-regularity of the solutions. By converting SPDEs driven by d-dimensional space-time white noise into the equations of above type, we also obtain an L_{p}-theory for SPDEs driven by space-time white noise if the space dimensiond < 4-2(2β ^{-1})α^{-1}. In particular, if β < 1/2 + α/4 then we can handle space-time white noise driven SPDEs with space dimension d = 1,2,3.

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

Pages (from-to) | 2087-2139 |

Number of pages | 53 |

Journal | Annals of Probability |

Volume | 47 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2019 Jan 1 |

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### Keywords

- Maximal Lp-regularity
- Multidimensional space-time white noise
- Stochastic partial differential equations
- Time fractional derivatives

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

**A Sobolev space theory for stochastic partial differential equations with time-fractional derivatives.** / Kim, Ildoo; Kim, Kyeong Hun; Lim, Sungbin.

Research output: Contribution to journal › Article

*Annals of Probability*, vol. 47, no. 4, pp. 2087-2139. https://doi.org/10.1214/18-AOP1303

}

TY - JOUR

T1 - A Sobolev space theory for stochastic partial differential equations with time-fractional derivatives

AU - Kim, Ildoo

AU - Kim, Kyeong Hun

AU - Lim, Sungbin

PY - 2019/1/1

Y1 - 2019/1/1

N2 - In this article, we present an Lp-theory (p ≥ 2) for the semi-linear stochastic partial differential equations (SPDEs) of type ∂ t α u = L(ω,t,x)u +f (u)+∂ t β ∞ ∑ k=1 ∫ 0 t (∧k (ω,t,x)u +gk(u))dw t k, where α ∈ (0, 2), β < α + 1/2 and ∂ t α and ∂ t β denote the Caputo derivatives of order α and β, respectively. The processes w t k, k ∈ N = 1, 2, . . ., are independent one-dimensional Wiener processes, L is either divergence or nondivergence-type second-order operator, and ∧k are linear operators of order up to two. This class of SPDEs can be used to describe random effects on transport of particles in medium with thermal memory or particles subject to sticking and trapping. We prove uniqueness and existence results of strong solutions in appropriate Sobolev spaces, and obtain maximal Lp-regularity of the solutions. By converting SPDEs driven by d-dimensional space-time white noise into the equations of above type, we also obtain an Lp-theory for SPDEs driven by space-time white noise if the space dimensiond < 4-2(2β -1)α-1. In particular, if β < 1/2 + α/4 then we can handle space-time white noise driven SPDEs with space dimension d = 1,2,3.

AB - In this article, we present an Lp-theory (p ≥ 2) for the semi-linear stochastic partial differential equations (SPDEs) of type ∂ t α u = L(ω,t,x)u +f (u)+∂ t β ∞ ∑ k=1 ∫ 0 t (∧k (ω,t,x)u +gk(u))dw t k, where α ∈ (0, 2), β < α + 1/2 and ∂ t α and ∂ t β denote the Caputo derivatives of order α and β, respectively. The processes w t k, k ∈ N = 1, 2, . . ., are independent one-dimensional Wiener processes, L is either divergence or nondivergence-type second-order operator, and ∧k are linear operators of order up to two. This class of SPDEs can be used to describe random effects on transport of particles in medium with thermal memory or particles subject to sticking and trapping. We prove uniqueness and existence results of strong solutions in appropriate Sobolev spaces, and obtain maximal Lp-regularity of the solutions. By converting SPDEs driven by d-dimensional space-time white noise into the equations of above type, we also obtain an Lp-theory for SPDEs driven by space-time white noise if the space dimensiond < 4-2(2β -1)α-1. In particular, if β < 1/2 + α/4 then we can handle space-time white noise driven SPDEs with space dimension d = 1,2,3.

KW - Maximal Lp-regularity

KW - Multidimensional space-time white noise

KW - Stochastic partial differential equations

KW - Time fractional derivatives

UR - http://www.scopus.com/inward/record.url?scp=85075854930&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85075854930&partnerID=8YFLogxK

U2 - 10.1214/18-AOP1303

DO - 10.1214/18-AOP1303

M3 - Article

AN - SCOPUS:85075854930

VL - 47

SP - 2087

EP - 2139

JO - Annals of Probability

JF - Annals of Probability

SN - 0091-1798

IS - 4

ER -